Generic graphs (common to directed/undirected)#
This module implements the base class for graphs and digraphs, and methods that can be applied on both. Here is what it can do:
Basic Graph operations:
Return a new 

Return an 

Create a dictionary encoding the graph. 

Return a copy of the graph. 

Export the graph to a file. 

Return the adjacency matrix of the (di)graph. 

Return an incidence matrix of the (di)graph 

Return the distance matrix of the (strongly) connected (di)graph 

Return the weighted adjacency matrix of the graph 

Return the Kirchhoff matrix (a.k.a. the Laplacian) of the graph. 

Return whether there are loops in the (di)graph 

Return whether loops are permitted in the (di)graph 

Change whether loops are permitted in the (di)graph 

Return a list of all loops in the (di)graph 

Return a list of all loops in the (di)graph 

Return the number of edges that are loops 

Return a list of vertices with loops 

Remove loops on vertices in 

Return whether there are multiple edges in the (di)graph. 

Return whether multiple edges are permitted in the (di)graph. 

Change whether multiple edges are permitted in the (di)graph. 

Return any multiple edges in the (di)graph. 

Return or set the graph’s name. 

Return whether the graph is immutable. 

Whether the (di)graph is to be considered as a weighted (di)graph. 

Test whether the graph is antisymmetric 

Return the density 

Return the number of vertices. 

Return the number of edges. 

Create an isolated vertex. 

Add vertices to the (di)graph from an iterable container of vertices 

Delete vertex, removing all incident edges. 

Delete vertices from the (di)graph taken from an iterable container of vertices. 

Check if 

Return a random vertex of 

Return an iterator over random vertices of 

Return a random edge of 

Return an iterator over random edges of 

Return a list of all vertices in the external boundary of 

Associate arbitrary objects with each vertex 

Associate an arbitrary object with a vertex. 

Retrieve the object associated with a given vertex. 

Return a dictionary of the objects associated to each vertex. 

Return an iterator over the given vertices. 

Return an iterator over neighbors of 

Return a list of the vertices. 

Return a list of neighbors (in and out if directed) of 

Merge vertices. 

Add an edge from 

Add edges from an iterable container. 

Subdivide an edge \(k\) times. 

Subdivide \(k\) times edges from an iterable container. 

Delete the edge from 

Delete edges from an iterable container. 

Contract an edge from 

Contract edges from an iterable container. 

Delete all edges from 

Set the edge label of a given edge. 

Check whether 

Return a 

Return a list of edges 

Return an iterator over edges. 

Return incident edges to some vertices. 

Return the label of an edge. 

Return a list of the labels of all edges in 

Remove all multiple edges, retaining one edge for each. 

Empty the graph of vertices and edges and removes name, associated objects, and position information. 

Return the degree (in + out for digraphs) of a vertex or of vertices. 

Return the average degree of the graph. 

Return a list, whose ith entry is the frequency of degree i. 

Return an iterator over the degrees of the (di)graph. 

Return the degree sequence of this (di)graph. 

Return a random subgraph containing each vertex with probability 

Add a clique to the graph with the given vertices. 

Add a cycle to the graph with the given vertices. 

Add a path to the graph with the given vertices. 

Return the complement of the (di)graph. 

Return the line graph of the (di)graph. 

Return a simple version of itself (i.e., undirected and loops and multiple edges are removed). 

Return the disjoint union of self and other. 

Return the union of self and other. 

Relabel the vertices of 

Return the number of edges from vertex to an edge in cell. 

Return the subgraph containing the given vertices and edges. 

Check whether 
Graph products:
Return the Cartesian product of self and other. 

Return the tensor product, also called the categorical product, of self and other. 

Return the lexicographic product of self and other. 

Return the strong product of self and other. 

Return the disjunctive product of self and other. 
Paths and cycles:
Return a DiGraph which is an Eulerian orientation of the current graph. 

Return a list of edges forming an Eulerian circuit if one exists. 

Return a minimum weight cycle basis of the graph. 

Return a list of cycles which form a basis of the cycle space of 

Return a list of all paths (also lists) between a pair of vertices in the (di)graph. 

Return the number of triangles in the (di)graph. 

Return an iterator over the simple paths between a pair of vertices. 
Linear algebra:
Return a list of the eigenvalues of the adjacency matrix. 

Return the right eigenvectors of the adjacency matrix of the graph. 

Return the right eigenspaces of the adjacency matrix of the graph. 
Some metrics:
Return the number of triangles for the set nbunch of vertices as a dictionary keyed by vertex. 

Return the average clustering coefficient. 

Return the clustering coefficient for each vertex in nbunch 

Return the transitivity (fraction of transitive triangles) of the graph. 

Return the Szeged index of the graph. 

Return the katz centrality of the vertex u of the graph. 

Return the katz matrix of the graph. 

Return the PageRank of the vertices of 
Automorphism group:
Return the coarsest partition which is finer than the input partition, and equitable with respect to self. 

Return the largest subgroup of the automorphism group of the (di)graph whose orbit partition is finer than the partition given. 

Return whether the automorphism group of self is transitive within the partition provided 

Test for isomorphism between self and other. 

Return the canonical graph. 

Check whether the graph is a Cayley graph. 
Graph properties:
Return 

Check whether the graph is planar. 

Check whether the graph is circular planar (outerplanar) 

Return 

Check whether the given graph is chordal. 

Test whether the given graph is bipartite. 

Check whether the graph is a circulant graph. 

Check whether the graph is an interval graph. 

Return whether the current graph is a Gallai tree. 

Check whether a set of vertices is a clique 

Check whether 

Check whether 

Test whether the digraph is transitively reduced. 

Check whether the given partition is equitable with respect to self. 

Check whether the graph is selfcomplementary. 
Traversals:
Return an iterator over the vertices in a breadthfirst ordering. 

Return an iterator over the vertices in a depthfirst ordering. 

Perform a lexicographic breadth first search (LexBFS) on the graph. 

Perform a lexicographic UP search (LexUP) on the graph. 

Perform a lexicographic depth first search (LexDFS) on the graph. 

Perform a lexicographic DOWN search (LexDOWN) on the graph. 
Distances:
Return the betweenness centrality 

Returns the closeness centrality (1/average distance to all vertices) 

Return the (directed) distance from u to v in the (di)graph 

Return the distances between all pairs of vertices. 

Return the distances distribution of the (di)graph in a dictionary. 

Return the girth of the graph. 

Return the odd girth of the graph. 

Return a list of vertices representing some shortest path from \(u\) to \(v\) 

Return the minimal length of paths from u to v 

Return a dictionary associating to each vertex v a shortest path from u to v, if it exists. 

Return a dictionary of shortest path lengths keyed by targets that are connected by a path from u. 

Compute a shortest path between each pair of vertices. 

Return the Wiener index of the graph. 

Return the average distance between vertices of the graph. 
Flows, connectivity, trees:
Test whether the (di)graph is connected. 

Return the list of connected components 

Return the number of connected components. 

Return a list of connected components as graph objects. 

Return a list of the vertices connected to vertex. 

Return the sizes of the connected components as a list. 

Compute the blocks and cut vertices of the graph. 

Compute the blocksandcuts tree of the graph. 

Return True if the input edge is a cutedge or a bridge. 

Return True if the input vertex is a cutvertex. 

Return a minimum edge cut between vertices \(s\) and \(t\) 

Return a minimum vertex cut between nonadjacent vertices \(s\) and \(t\) 

Return a maximum flow in the graph from 

Return a \(k\)nowhere zero flow of the (di)graph. 

Return a list of edgedisjoint paths between two vertices 

Return a list of vertexdisjoint paths between two vertices 

Return the edge connectivity of the graph. 

Return the vertex connectivity of the graph. 

Compute the transitive closure of a graph and returns it. 

Return a transitive reduction of a graph. 

Return the edges of a minimum spanning tree. 

Return the number of spanning trees in a graph. 

Returns a dominator tree of the graph. 

Iterator over the induced connected subgraphs of order at most \(k\) 
Plot/embeddingrelated methods:
Set a combinatorial embedding dictionary to 

Return the attribute _embedding if it exists. 

Return the faces of an embedded graph. 

Return the number of faces of an embedded graph. 

Return the planar dual of an embedded graph. 

Return the position dictionary 

Set the position dictionary. 

Compute a planar layout of the graph using Schnyder’s algorithm. 

Check whether the position dictionary gives a planar embedding. 

Return an instance of 

Set multiple options for rendering a graph with LaTeX. 

Return a layout for the vertices of this graph. 

Return a spring layout for this graph 

Return a ranked layout for this graph 

Extend randomly a partial layout 

Return a circular layout for this graph 

Return an ordered tree layout for this graph 

Return an ordered forest layout for this graph 

Call 


Set some vertices on a circle in the embedding of this graph. 

Set some vertices on a line in the embedding of this graph. 
Return a 

Return a 

Show the (di)graph. 

Plot the graph in three dimensions. 

Plot the graph using 

Return a representation in the 

Write a representation in the 
Algorithmically hard stuff:
Return a tree of minimum weight connecting the given set of vertices. 

Return the desired number of edgedisjoint spanning trees/arborescences. 

Compute the minimum feedback vertex set of a (di)graph. 

Return a minimum edge multiway cut 

Return a maximum edge cut of the graph. 

Return a longest path of 

Solve the traveling salesman problem (TSP) 

Test whether the current graph is Hamiltonian. 

Return a Hamiltonian cycle/circuit of the current graph/digraph 

Return a Hamiltonian path of the current graph/digraph 

Solve a multicommodity flow problem. 

Return a set of disjoint routed paths. 

Return a minimum dominating set of the graph 

Return a greedy distance\(k\) dominating set of the graph. 

Return a copy of 

Return the number of labelled occurrences of 

Return an iterator over the labelled copies of 

Return the characteristic polynomial of the adjacency matrix of the (di)graph. 

Return the minimal genus of the graph. 

Return the crossing number of the graph. 
Miscellaneous
Return the edge polytope of 

Return the symmetric edge polytope of 
Methods#
 class sage.graphs.generic_graph.GenericGraph#
Bases:
GenericGraph_pyx
Base class for graphs and digraphs.
 __eq__(other)#
Compare self and other for equality.
Do not call this method directly. That is, for
G.__eq__(H)
writeG == H
. Two graphs are considered equal if the following hold:
they are either both directed, or both undirected;
they have the same settings for loops, multiedges, and weightedness;
they have the same set of vertices;
they have the same (multi)set of arrows/edges, where labels of arrows/edges are taken into account if and only if the graphs are considered weighted. See
weighted()
.
Note that this is not an isomorphism test.
EXAMPLES:
sage: G = graphs.EmptyGraph() sage: H = Graph() sage: G == H True sage: G.to_directed() == H.to_directed() True sage: G = graphs.RandomGNP(8, .9999) sage: H = graphs.CompleteGraph(8) sage: G == H # random  most often true True sage: G = Graph({0: [1, 2, 3, 4, 5, 6, 7]} ) sage: H = Graph({1: [0], 2: [0], 3: [0], 4: [0], 5: [0], 6: [0], 7: [0]} ) sage: G == H True sage: G.allow_loops(True) sage: G == H False sage: G = graphs.RandomGNP(9, .3).to_directed() sage: H = graphs.RandomGNP(9, .3).to_directed() sage: G == H # most often false False sage: G = Graph(multiedges=True, sparse=True) sage: G.add_edge(0, 1) sage: H = copy(G) sage: H.add_edge(0, 1) sage: G == H False
Note that graphs must be considered weighted, or Sage will not pay attention to edge label data in equality testing:
sage: foo = Graph(sparse=True) sage: foo.add_edges([(0, 1, 1), (0, 2, 2)]) sage: bar = Graph(sparse=True) sage: bar.add_edges([(0, 1, 2), (0, 2, 1)]) sage: foo == bar True sage: foo.weighted(True) sage: foo == bar False sage: bar.weighted(True) sage: foo == bar False
 add_clique(vertices, loops=False)#
Add a clique to the graph with the given vertices.
If the vertices are already present, only the edges are added.
INPUT:
vertices
– an iterable container of vertices for the clique to be added, e.g. a list, set, graph, etc.loops
– boolean (default:False
); whether to add edges from every given vertex to itself. This is allowed only if the (di)graph allows loops.
EXAMPLES:
sage: G = Graph() sage: G.add_clique(range(4)) sage: G.is_isomorphic(graphs.CompleteGraph(4)) True sage: D = DiGraph() sage: D.add_clique(range(4)) sage: D.is_isomorphic(digraphs.Complete(4)) True sage: D = DiGraph(loops=True) sage: D.add_clique(range(4), loops=True) sage: D.is_isomorphic(digraphs.Complete(4, loops=True)) True sage: D = DiGraph(loops=False) sage: D.add_clique(range(4), loops=True) Traceback (most recent call last): ... ValueError: cannot add edge from 0 to 0 in graph without loops
If the list of vertices contains repeated elements, a loop will be added at that vertex, even if
loops=False
:sage: G = Graph(loops=True) sage: G.add_clique([1, 1]) sage: G.edges(sort=True) [(1, 1, None)]
This is equivalent to:
sage: G = Graph(loops=True) sage: G.add_clique([1], loops=True) sage: G.edges(sort=True) [(1, 1, None)]
 add_cycle(vertices)#
Add a cycle to the graph with the given vertices.
If the vertices are already present, only the edges are added.
For digraphs, adds the directed cycle, whose orientation is determined by the list. Adds edges
(vertices[u], vertices[u+1])
and(vertices[1], vertices[0])
.INPUT:
vertices
– an ordered list of the vertices of the cycle to be added
EXAMPLES:
sage: G = Graph() sage: G.add_vertices(range(10)); G Graph on 10 vertices sage: show(G) # needs sage.plot sage: G.add_cycle(list(range(10, 20))) sage: show(G) # needs sage.plot sage: G.add_cycle(list(range(10))) sage: show(G) # needs sage.plot
sage: D = DiGraph() sage: D.add_cycle(list(range(4))) sage: D.edges(sort=True) [(0, 1, None), (1, 2, None), (2, 3, None), (3, 0, None)]
 add_edge(u, v=None, label=None)#
Add an edge from
u
tov
.INPUT: The following forms are all accepted:
G.add_edge( 1, 2 )
G.add_edge( (1, 2) )
G.add_edges( [ (1, 2) ])
G.add_edge( 1, 2, ‘label’ )
G.add_edge( (1, 2, ‘label’) )
G.add_edges( [ (1, 2, ‘label’) ] )
WARNING: The following intuitive input results in nonintuitive output:
sage: G = Graph() sage: G.add_edge((1, 2), 'label') sage: G.edges(sort=False) [('label', (1, 2), None)]
You must either use the
label
keyword:sage: G = Graph() sage: G.add_edge((1, 2), label="label") sage: G.edges(sort=False) [(1, 2, 'label')]
Or use one of these:
sage: G = Graph() sage: G.add_edge(1, 2, 'label') sage: G.edges(sort=False) [(1, 2, 'label')] sage: G = Graph() sage: G.add_edge((1, 2, 'label')) sage: G.edges(sort=False) [(1, 2, 'label')]
Vertex name cannot be
None
, so:sage: G = Graph() sage: G.add_edge(None, 4) sage: G.vertices(sort=True) [0, 4]
 add_edges(edges, loops=True)#
Add edges from an iterable container.
INPUT:
edges
– an iterable of edges, given either as(u, v)
or(u, v, label)
.loops
– boolean (default:True
); ifFalse
, remove all loops(v, v)
from the input iterator. IfNone
, remove loops unless the graph allows loops.
EXAMPLES:
sage: G = graphs.DodecahedralGraph() sage: H = Graph() sage: H.add_edges(G.edge_iterator()); H Graph on 20 vertices sage: G = graphs.DodecahedralGraph().to_directed() sage: H = DiGraph() sage: H.add_edges(G.edge_iterator()); H Digraph on 20 vertices sage: H.add_edges(iter([])) sage: H = Graph() sage: H.add_edges([(0, 1), (0, 2, "label")]) sage: H.edges(sort=True) [(0, 1, None), (0, 2, 'label')]
We demonstrate the
loops
argument:sage: H = Graph() sage: H.add_edges([(0, 0)], loops=False); H.edges(sort=True) [] sage: H.add_edges([(0, 0)], loops=None); H.edges(sort=True) [] sage: H.add_edges([(0, 0)]); H.edges(sort=True) Traceback (most recent call last): ... ValueError: cannot add edge from 0 to 0 in graph without loops sage: H = Graph(loops=True) sage: H.add_edges([(0, 0)], loops=False); H.edges(sort=True) [] sage: H.add_edges([(0, 0)], loops=None); H.edges(sort=True) [(0, 0, None)] sage: H.add_edges([(0, 0)]); H.edges(sort=True) [(0, 0, None)]
 add_path(vertices)#
Add a path to the graph with the given vertices.
If the vertices are already present, only the edges are added.
For digraphs, adds the directed path
vertices[0], ..., vertices[1]
.INPUT:
vertices
– an ordered list of the vertices of the path to be added
EXAMPLES:
sage: G = Graph() sage: G.add_vertices(range(10)); G Graph on 10 vertices sage: show(G) # needs sage.plot sage: G.add_path(list(range(10, 20))) sage: show(G) # needs sage.plot sage: G.add_path(list(range(10))) sage: show(G) # needs sage.plot
sage: D = DiGraph() sage: D.add_path(list(range(4))) sage: D.edges(sort=True) [(0, 1, None), (1, 2, None), (2, 3, None)]
 add_vertex(name=None)#
Create an isolated vertex.
If the vertex already exists, then nothing is done.
INPUT:
name
– an immutable object (default:None
); when no name is specified (default), then the new vertex will be represented by the least integer not already representing a vertex.name
must be an immutable object (e.g., an integer, a tuple, etc.).
As it is implemented now, if a graph \(G\) has a large number of vertices with numeric labels, then
G.add_vertex()
could potentially be slow, ifname=None
.OUTPUT:
If
name=None
, the new vertex name is returned.None
otherwise.EXAMPLES:
sage: G = Graph(); G.add_vertex(); G 0 Graph on 1 vertex
sage: D = DiGraph(); D.add_vertex(); D 0 Digraph on 1 vertex
 add_vertices(vertices)#
Add vertices to the (di)graph from an iterable container of vertices.
Vertices that already exist in the graph will not be added again.
INPUT:
vertices
– iterator container of vertex labels. A new label is created, used and returned in the output list for allNone
values invertices
.
OUTPUT:
Generated names of new vertices if there is at least one
None
value present invertices
.None
otherwise.EXAMPLES:
sage: d = {0: [1,4,5], 1: [2,6], 2: [3,7], 3: [4,8], 4: [9], 5: [7,8], 6: [8,9], 7: [9]} sage: G = Graph(d) sage: G.add_vertices([10,11,12]) sage: G.vertices(sort=True) [0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12] sage: G.add_vertices(graphs.CycleGraph(25).vertex_iterator()) sage: G.vertices(sort=True) [0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24]
sage: G = Graph() sage: G.add_vertices([1, 2, 3]) sage: G.add_vertices([4, None, None, 5]) [0, 6]
 adjacency_matrix(sparse, vertices=None, base_ring=None, **kwds)#
Return the adjacency matrix of the (di)graph.
By default, the matrix returned is over the integers.
INPUT:
sparse
– boolean (default:None
); whether to represent with a sparse matrixvertices
– list (default:None
); the ordering of the vertices defining how they should appear in the matrix. By default, the ordering given byGenericGraph.vertices()
withsort=True
is used. If the vertices are not comparable, the keywordvertices
must be used to specify an ordering, or a TypeError exception will be raised.base_ring
– a ring (default:ZZ
); the base ring of the matrix space to use.**kwds
– other keywords to pass tomatrix()
.
EXAMPLES:
sage: G = graphs.CubeGraph(4) sage: G.adjacency_matrix() # needs sage.modules [0 1 1 0 1 0 0 0 1 0 0 0 0 0 0 0] [1 0 0 1 0 1 0 0 0 1 0 0 0 0 0 0] [1 0 0 1 0 0 1 0 0 0 1 0 0 0 0 0] [0 1 1 0 0 0 0 1 0 0 0 1 0 0 0 0] [1 0 0 0 0 1 1 0 0 0 0 0 1 0 0 0] [0 1 0 0 1 0 0 1 0 0 0 0 0 1 0 0] [0 0 1 0 1 0 0 1 0 0 0 0 0 0 1 0] [0 0 0 1 0 1 1 0 0 0 0 0 0 0 0 1] [1 0 0 0 0 0 0 0 0 1 1 0 1 0 0 0] [0 1 0 0 0 0 0 0 1 0 0 1 0 1 0 0] [0 0 1 0 0 0 0 0 1 0 0 1 0 0 1 0] [0 0 0 1 0 0 0 0 0 1 1 0 0 0 0 1] [0 0 0 0 1 0 0 0 1 0 0 0 0 1 1 0] [0 0 0 0 0 1 0 0 0 1 0 0 1 0 0 1] [0 0 0 0 0 0 1 0 0 0 1 0 1 0 0 1] [0 0 0 0 0 0 0 1 0 0 0 1 0 1 1 0]
sage: matrix(GF(2), G) # matrix over GF(2) # needs sage.modules sage.rings.finite_rings [0 1 1 0 1 0 0 0 1 0 0 0 0 0 0 0] [1 0 0 1 0 1 0 0 0 1 0 0 0 0 0 0] [1 0 0 1 0 0 1 0 0 0 1 0 0 0 0 0] [0 1 1 0 0 0 0 1 0 0 0 1 0 0 0 0] [1 0 0 0 0 1 1 0 0 0 0 0 1 0 0 0] [0 1 0 0 1 0 0 1 0 0 0 0 0 1 0 0] [0 0 1 0 1 0 0 1 0 0 0 0 0 0 1 0] [0 0 0 1 0 1 1 0 0 0 0 0 0 0 0 1] [1 0 0 0 0 0 0 0 0 1 1 0 1 0 0 0] [0 1 0 0 0 0 0 0 1 0 0 1 0 1 0 0] [0 0 1 0 0 0 0 0 1 0 0 1 0 0 1 0] [0 0 0 1 0 0 0 0 0 1 1 0 0 0 0 1] [0 0 0 0 1 0 0 0 1 0 0 0 0 1 1 0] [0 0 0 0 0 1 0 0 0 1 0 0 1 0 0 1] [0 0 0 0 0 0 1 0 0 0 1 0 1 0 0 1] [0 0 0 0 0 0 0 1 0 0 0 1 0 1 1 0]
sage: D = DiGraph({0: [1, 2, 3], 1: [0, 2], 2: [3], ....: 3: [4], 4: [0, 5], 5: [1]}) sage: D.adjacency_matrix() # needs sage.modules [0 1 1 1 0 0] [1 0 1 0 0 0] [0 0 0 1 0 0] [0 0 0 0 1 0] [1 0 0 0 0 1] [0 1 0 0 0 0]
A different ordering of the vertices:
sage: graphs.PathGraph(5).adjacency_matrix(vertices=[2, 4, 1, 3, 0]) # needs sage.modules [0 0 1 1 0] [0 0 0 1 0] [1 0 0 0 1] [1 1 0 0 0] [0 0 1 0 0]
A different base ring:
sage: graphs.PathGraph(5).adjacency_matrix(base_ring=RDF) # needs sage.modules [0.0 1.0 0.0 0.0 0.0] [1.0 0.0 1.0 0.0 0.0] [0.0 1.0 0.0 1.0 0.0] [0.0 0.0 1.0 0.0 1.0] [0.0 0.0 0.0 1.0 0.0] sage: type(_) # needs sage.modules <class 'sage.matrix.matrix_real_double_dense.Matrix_real_double_dense'>
A different matrix implementation:
sage: graphs.PathGraph(5).adjacency_matrix(sparse=False, # needs sage.modules ....: implementation='numpy') [0 1 0 0 0] [1 0 1 0 0] [0 1 0 1 0] [0 0 1 0 1] [0 0 0 1 0] sage: type(_) <class 'sage.matrix.matrix_numpy_integer_dense.Matrix_numpy_integer_dense'>
As an immutable matrix:
sage: M = graphs.PathGraph(5).adjacency_matrix(sparse=False, # needs sage.modules ....: immutable=True); M [0 1 0 0 0] [1 0 1 0 0] [0 1 0 1 0] [0 0 1 0 1] [0 0 0 1 0] sage: M[2, 2] = 1 # needs sage.modules Traceback (most recent call last): ... ValueError: matrix is immutable; please change a copy instead (i.e., use copy(M) to change a copy of M).
 all_paths(G, start, end, use_multiedges=False, report_edges=False, labels=False)#
Return the list of all paths between a pair of vertices.
If
start
is the same vertex asend
, then[[start]]
is returned – a list containing the 1vertex, 0edge path “start
”.If
G
has multiple edges, a path will be returned as many times as the product of the multiplicity of the edges along that path depending on the value of the flaguse_multiedges
.INPUT:
start
– a vertex of a graph, where to startend
– a vertex of a graph, where to enduse_multiedges
– boolean (default:False
); this parameter is used only if the graph has multiple edges.If
False
, the graph is considered as simple and an edge label is arbitrarily selected for each edge as insage.graphs.generic_graph.GenericGraph.to_simple()
ifreport_edges
isTrue
If
True
, a path will be reported as many times as the edges multiplicities along that path (whenreport_edges = False
orlabels = False
), or with all possible combinations of edge labels (whenreport_edges = True
andlabels = True
)
report_edges
– boolean (default:False
); whether to report paths as list of vertices (default) or list of edges, ifFalse
thenlabels
parameter is ignoredlabels
– boolean (default:False
); ifFalse
, each edge is simply a pair(u, v)
of vertices. Otherwise a list of edges along with its edge labels are used to represent the path.
EXAMPLES:
sage: eg1 = Graph({0:[1, 2], 1:[4], 2:[3, 4], 4:[5], 5:[6]}) sage: eg1.all_paths(0, 6) [[0, 1, 4, 5, 6], [0, 2, 4, 5, 6]] sage: eg2 = graphs.PetersenGraph() sage: sorted(eg2.all_paths(1, 4)) [[1, 0, 4], [1, 0, 5, 7, 2, 3, 4], [1, 0, 5, 7, 2, 3, 8, 6, 9, 4], [1, 0, 5, 7, 9, 4], [1, 0, 5, 7, 9, 6, 8, 3, 4], [1, 0, 5, 8, 3, 2, 7, 9, 4], [1, 0, 5, 8, 3, 4], [1, 0, 5, 8, 6, 9, 4], [1, 0, 5, 8, 6, 9, 7, 2, 3, 4], [1, 2, 3, 4], [1, 2, 3, 8, 5, 0, 4], [1, 2, 3, 8, 5, 7, 9, 4], [1, 2, 3, 8, 6, 9, 4], [1, 2, 3, 8, 6, 9, 7, 5, 0, 4], [1, 2, 7, 5, 0, 4], [1, 2, 7, 5, 8, 3, 4], [1, 2, 7, 5, 8, 6, 9, 4], [1, 2, 7, 9, 4], [1, 2, 7, 9, 6, 8, 3, 4], [1, 2, 7, 9, 6, 8, 5, 0, 4], [1, 6, 8, 3, 2, 7, 5, 0, 4], [1, 6, 8, 3, 2, 7, 9, 4], [1, 6, 8, 3, 4], [1, 6, 8, 5, 0, 4], [1, 6, 8, 5, 7, 2, 3, 4], [1, 6, 8, 5, 7, 9, 4], [1, 6, 9, 4], [1, 6, 9, 7, 2, 3, 4], [1, 6, 9, 7, 2, 3, 8, 5, 0, 4], [1, 6, 9, 7, 5, 0, 4], [1, 6, 9, 7, 5, 8, 3, 4]] sage: dg = DiGraph({0:[1, 3], 1:[3], 2:[0, 3]}) sage: sorted(dg.all_paths(0, 3)) [[0, 1, 3], [0, 3]] sage: ug = dg.to_undirected() sage: sorted(ug.all_paths(0, 3)) [[0, 1, 3], [0, 2, 3], [0, 3]] sage: g = Graph([(0, 1), (0, 1), (1, 2), (1, 2)], multiedges=True) sage: g.all_paths(0, 2, use_multiedges=True) [[0, 1, 2], [0, 1, 2], [0, 1, 2], [0, 1, 2]] sage: dg = DiGraph({0:[1, 2, 1], 3:[0, 0]}, multiedges=True) sage: dg.all_paths(3, 1, use_multiedges=True) [[3, 0, 1], [3, 0, 1], [3, 0, 1], [3, 0, 1]] sage: g = Graph([(0, 1, 'a'), (0, 1, 'b'), (1, 2, 'c'), (1, 2, 'd')], multiedges=True) sage: g.all_paths(0, 2, use_multiedges=False) [[0, 1, 2]] sage: g.all_paths(0, 2, use_multiedges=True) [[0, 1, 2], [0, 1, 2], [0, 1, 2], [0, 1, 2]] sage: g.all_paths(0, 2, use_multiedges=True, report_edges=True) [[(0, 1), (1, 2)], [(0, 1), (1, 2)], [(0, 1), (1, 2)], [(0, 1), (1, 2)]] sage: g.all_paths(0, 2, use_multiedges=True, report_edges=True, labels=True) [((0, 1, 'b'), (1, 2, 'd')), ((0, 1, 'b'), (1, 2, 'c')), ((0, 1, 'a'), (1, 2, 'd')), ((0, 1, 'a'), (1, 2, 'c'))] sage: g.all_paths(0, 2, use_multiedges=False, report_edges=True, labels=True) [((0, 1, 'b'), (1, 2, 'd'))] sage: g.all_paths(0, 2, use_multiedges=False, report_edges=False, labels=True) [[0, 1, 2]] sage: g.all_paths(0, 2, use_multiedges=True, report_edges=False, labels=True) [[0, 1, 2], [0, 1, 2], [0, 1, 2], [0, 1, 2]]
 allow_loops(new, check=True)#
Change whether loops are permitted in the (di)graph
INPUT:
new
– booleancheck
– boolean (default:True
); whether to remove existing loops from the (di)graph when the new status isFalse
EXAMPLES:
sage: G = Graph(loops=True); G Looped graph on 0 vertices sage: G.has_loops() False sage: G.allows_loops() True sage: G.add_edge((0, 0)) sage: G.has_loops() True sage: G.loops() [(0, 0, None)] sage: G.allow_loops(False); G Graph on 1 vertex sage: G.has_loops() False sage: G.edges(sort=True) [] sage: D = DiGraph(loops=True); D Looped digraph on 0 vertices sage: D.has_loops() False sage: D.allows_loops() True sage: D.add_edge((0, 0)) sage: D.has_loops() True sage: D.loops() [(0, 0, None)] sage: D.allow_loops(False); D Digraph on 1 vertex sage: D.has_loops() False sage: D.edges(sort=True) []
 allow_multiple_edges(new, check=True, keep_label='any')#
Change whether multiple edges are permitted in the (di)graph.
INPUT:
new
– boolean; ifTrue
, the new graph will allow multiple edgescheck
– boolean (default:True
); ifTrue
andnew
isFalse
, we remove all multiple edges from the graphkeep_label
– string (default:'any'
); used only ifnew
isFalse
andcheck
isTrue
. If there are multiple edges with different labels, this variable defines which label should be kept:'any'
– any label'min'
– the smallest label'max'
– the largest label
Warning
'min'
and'max'
only works if the labels can be compared. ATypeError
might be raised when working with noncomparable objects in Python 3.EXAMPLES:
The standard behavior with undirected graphs:
sage: G = Graph(multiedges=True, sparse=True); G Multigraph on 0 vertices sage: G.has_multiple_edges() False sage: G.allows_multiple_edges() True sage: G.add_edges([(0, 1, 1), (0, 1, 2), (0, 1, 3)]) sage: G.has_multiple_edges() True sage: G.multiple_edges(sort=True) [(0, 1, 1), (0, 1, 2), (0, 1, 3)] sage: G.allow_multiple_edges(False); G Graph on 2 vertices sage: G.has_multiple_edges() False sage: G.edges(sort=True) [(0, 1, 3)]
If we ask for the minimum label:
sage: G = Graph([(0, 1, 1), (0, 1, 2), (0, 1, 3)], multiedges=True, sparse=True) sage: G.allow_multiple_edges(False, keep_label='min') sage: G.edges(sort=True) [(0, 1, 1)]
If we ask for the maximum label:
sage: G = Graph([(0, 1, 1), (0, 1, 2), (0, 1, 3)], multiedges=True, sparse=True) sage: G.allow_multiple_edges(False, keep_label='max') sage: G.edges(sort=True) [(0, 1, 3)]
The standard behavior with digraphs:
sage: D = DiGraph(multiedges=True, sparse=True); D Multidigraph on 0 vertices sage: D.has_multiple_edges() False sage: D.allows_multiple_edges() True sage: D.add_edges([(0, 1)] * 3) sage: D.has_multiple_edges() True sage: D.multiple_edges() [(0, 1, None), (0, 1, None), (0, 1, None)] sage: D.allow_multiple_edges(False); D Digraph on 2 vertices sage: D.has_multiple_edges() False sage: D.edges(sort=True) [(0, 1, None)]
 allows_loops()#
Return whether loops are permitted in the (di)graph
EXAMPLES:
sage: G = Graph(loops=True); G Looped graph on 0 vertices sage: G.has_loops() False sage: G.allows_loops() True sage: G.add_edge((0, 0)) sage: G.has_loops() True sage: G.loops() [(0, 0, None)] sage: G.allow_loops(False); G Graph on 1 vertex sage: G.has_loops() False sage: G.edges(sort=True) [] sage: D = DiGraph(loops=True); D Looped digraph on 0 vertices sage: D.has_loops() False sage: D.allows_loops() True sage: D.add_edge((0, 0)) sage: D.has_loops() True sage: D.loops() [(0, 0, None)] sage: D.allow_loops(False); D Digraph on 1 vertex sage: D.has_loops() False sage: D.edges(sort=True) []
 allows_multiple_edges()#
Return whether multiple edges are permitted in the (di)graph.
EXAMPLES:
sage: G = Graph(multiedges=True, sparse=True); G Multigraph on 0 vertices sage: G.has_multiple_edges() False sage: G.allows_multiple_edges() True sage: G.add_edges([(0, 1)] * 3) sage: G.has_multiple_edges() True sage: G.multiple_edges() [(0, 1, None), (0, 1, None), (0, 1, None)] sage: G.allow_multiple_edges(False); G Graph on 2 vertices sage: G.has_multiple_edges() False sage: G.edges(sort=True) [(0, 1, None)] sage: D = DiGraph(multiedges=True, sparse=True); D Multidigraph on 0 vertices sage: D.has_multiple_edges() False sage: D.allows_multiple_edges() True sage: D.add_edges([(0, 1)] * 3) sage: D.has_multiple_edges() True sage: D.multiple_edges() [(0, 1, None), (0, 1, None), (0, 1, None)] sage: D.allow_multiple_edges(False); D Digraph on 2 vertices sage: D.has_multiple_edges() False sage: D.edges(sort=True) [(0, 1, None)]
 am(sparse, vertices=None, base_ring=None, **kwds)#
Return the adjacency matrix of the (di)graph.
By default, the matrix returned is over the integers.
INPUT:
sparse
– boolean (default:None
); whether to represent with a sparse matrixvertices
– list (default:None
); the ordering of the vertices defining how they should appear in the matrix. By default, the ordering given byGenericGraph.vertices()
withsort=True
is used. If the vertices are not comparable, the keywordvertices
must be used to specify an ordering, or a TypeError exception will be raised.base_ring
– a ring (default:ZZ
); the base ring of the matrix space to use.**kwds
– other keywords to pass tomatrix()
.
EXAMPLES:
sage: G = graphs.CubeGraph(4) sage: G.adjacency_matrix() # needs sage.modules [0 1 1 0 1 0 0 0 1 0 0 0 0 0 0 0] [1 0 0 1 0 1 0 0 0 1 0 0 0 0 0 0] [1 0 0 1 0 0 1 0 0 0 1 0 0 0 0 0] [0 1 1 0 0 0 0 1 0 0 0 1 0 0 0 0] [1 0 0 0 0 1 1 0 0 0 0 0 1 0 0 0] [0 1 0 0 1 0 0 1 0 0 0 0 0 1 0 0] [0 0 1 0 1 0 0 1 0 0 0 0 0 0 1 0] [0 0 0 1 0 1 1 0 0 0 0 0 0 0 0 1] [1 0 0 0 0 0 0 0 0 1 1 0 1 0 0 0] [0 1 0 0 0 0 0 0 1 0 0 1 0 1 0 0] [0 0 1 0 0 0 0 0 1 0 0 1 0 0 1 0] [0 0 0 1 0 0 0 0 0 1 1 0 0 0 0 1] [0 0 0 0 1 0 0 0 1 0 0 0 0 1 1 0] [0 0 0 0 0 1 0 0 0 1 0 0 1 0 0 1] [0 0 0 0 0 0 1 0 0 0 1 0 1 0 0 1] [0 0 0 0 0 0 0 1 0 0 0 1 0 1 1 0]
sage: matrix(GF(2), G) # matrix over GF(2) # needs sage.modules sage.rings.finite_rings [0 1 1 0 1 0 0 0 1 0 0 0 0 0 0 0] [1 0 0 1 0 1 0 0 0 1 0 0 0 0 0 0] [1 0 0 1 0 0 1 0 0 0 1 0 0 0 0 0] [0 1 1 0 0 0 0 1 0 0 0 1 0 0 0 0] [1 0 0 0 0 1 1 0 0 0 0 0 1 0 0 0] [0 1 0 0 1 0 0 1 0 0 0 0 0 1 0 0] [0 0 1 0 1 0 0 1 0 0 0 0 0 0 1 0] [0 0 0 1 0 1 1 0 0 0 0 0 0 0 0 1] [1 0 0 0 0 0 0 0 0 1 1 0 1 0 0 0] [0 1 0 0 0 0 0 0 1 0 0 1 0 1 0 0] [0 0 1 0 0 0 0 0 1 0 0 1 0 0 1 0] [0 0 0 1 0 0 0 0 0 1 1 0 0 0 0 1] [0 0 0 0 1 0 0 0 1 0 0 0 0 1 1 0] [0 0 0 0 0 1 0 0 0 1 0 0 1 0 0 1] [0 0 0 0 0 0 1 0 0 0 1 0 1 0 0 1] [0 0 0 0 0 0 0 1 0 0 0 1 0 1 1 0]
sage: D = DiGraph({0: [1, 2, 3], 1: [0, 2], 2: [3], ....: 3: [4], 4: [0, 5], 5: [1]}) sage: D.adjacency_matrix() # needs sage.modules [0 1 1 1 0 0] [1 0 1 0 0 0] [0 0 0 1 0 0] [0 0 0 0 1 0] [1 0 0 0 0 1] [0 1 0 0 0 0]
A different ordering of the vertices:
sage: graphs.PathGraph(5).adjacency_matrix(vertices=[2, 4, 1, 3, 0]) # needs sage.modules [0 0 1 1 0] [0 0 0 1 0] [1 0 0 0 1] [1 1 0 0 0] [0 0 1 0 0]
A different base ring:
sage: graphs.PathGraph(5).adjacency_matrix(base_ring=RDF) # needs sage.modules [0.0 1.0 0.0 0.0 0.0] [1.0 0.0 1.0 0.0 0.0] [0.0 1.0 0.0 1.0 0.0] [0.0 0.0 1.0 0.0 1.0] [0.0 0.0 0.0 1.0 0.0] sage: type(_) # needs sage.modules <class 'sage.matrix.matrix_real_double_dense.Matrix_real_double_dense'>
A different matrix implementation:
sage: graphs.PathGraph(5).adjacency_matrix(sparse=False, # needs sage.modules ....: implementation='numpy') [0 1 0 0 0] [1 0 1 0 0] [0 1 0 1 0] [0 0 1 0 1] [0 0 0 1 0] sage: type(_) <class 'sage.matrix.matrix_numpy_integer_dense.Matrix_numpy_integer_dense'>
As an immutable matrix:
sage: M = graphs.PathGraph(5).adjacency_matrix(sparse=False, # needs sage.modules ....: immutable=True); M [0 1 0 0 0] [1 0 1 0 0] [0 1 0 1 0] [0 0 1 0 1] [0 0 0 1 0] sage: M[2, 2] = 1 # needs sage.modules Traceback (most recent call last): ... ValueError: matrix is immutable; please change a copy instead (i.e., use copy(M) to change a copy of M).
 antisymmetric()#
Check whether the graph is antisymmetric.
A graph represents an antisymmetric relation if the existence of a path from a vertex \(x\) to a vertex \(y\) implies that there is not a path from \(y\) to \(x\) unless \(x = y\).
EXAMPLES:
A directed acyclic graph is antisymmetric:
sage: G = digraphs.RandomDirectedGNR(20, 0.5) # needs networkx sage: G.antisymmetric() # needs networkx True
Loops are allowed:
sage: G.allow_loops(True) # needs networkx sage: G.add_edge(0, 0) # needs networkx sage: G.antisymmetric() # needs networkx True
An undirected graph is never antisymmetric unless it is just a union of isolated vertices (with possible loops):
sage: graphs.RandomGNP(20, 0.5).antisymmetric() # needs networkx False sage: Graph(3).antisymmetric() True sage: Graph([(i, i) for i in range(3)], loops=True).antisymmetric() True sage: DiGraph([(i, i) for i in range(3)], loops=True).antisymmetric() True
 automorphism_group(partition=None, verbosity=0, edge_labels=False, order=False, return_group=True, orbits=False, algorithm=None)#
Return the automorphism group of the graph.
With
partition
this can also return the largest subgroup of the automorphism group of the (di)graph whose orbit partition is finer than the partition given.INPUT:
partition
– default is the unit partition, otherwise computes the subgroup of the full automorphism group respecting the partition.edge_labels
– defaultFalse
, otherwise allows only permutations respecting edge labels.order
– (defaultFalse
) ifTrue
, compute the order of the automorphism groupreturn_group
– defaultTrue
orbits
– returns the orbits of the group acting on the vertices of the graphalgorithm
– Ifalgorithm = "bliss"
, the automorphism group is computed using the optional package bliss (http://www.tcs.tkk.fi/Software/bliss/index.html). Setting it to"sage"
uses Sage’s implementation. If set toNone
(default), bliss is used when available.
OUTPUT: The order of the output is group, order, orbits. However, there are options to turn each of these on or off.
EXAMPLES:
Graphs:
sage: # needs sage.groups sage: graphs_query = GraphQuery(display_cols=['graph6'],num_vertices=4) sage: L = graphs_query.get_graphs_list() sage: graphs_list.show_graphs(L) # needs sage.plot sage: for g in L: ....: G = g.automorphism_group() ....: G.order(), G.gens() (24, ((2,3), (1,2), (0,1))) (4, ((2,3), (0,1))) (2, ((1,2),)) (6, ((1,2), (0,1))) (6, ((2,3), (1,2))) (8, ((1,2), (0,1)(2,3))) (2, ((0,1)(2,3),)) (2, ((1,2),)) (8, ((2,3), (0,1), (0,2)(1,3))) (4, ((2,3), (0,1))) (24, ((2,3), (1,2), (0,1))) sage: C = graphs.CubeGraph(4) sage: G = C.automorphism_group() sage: M = G.character_table() # random order of rows, thus abs() below sage: QQ(M.determinant()).abs() 712483534798848 sage: G.order() 384
sage: # needs sage.groups sage: D = graphs.DodecahedralGraph() sage: G = D.automorphism_group() sage: A5 = AlternatingGroup(5) sage: Z2 = CyclicPermutationGroup(2) sage: H = A5.direct_product(Z2)[0] #see documentation for direct_product to explain the [0] sage: G.is_isomorphic(H) True
Multigraphs:
sage: G = Graph(multiedges=True,sparse=True) sage: G.add_edge(('a', 'b')) sage: G.add_edge(('a', 'b')) sage: G.add_edge(('a', 'b')) sage: G.automorphism_group() # needs sage.groups Permutation Group with generators [('a','b')]
Digraphs:
sage: D = DiGraph( { 0:[1], 1:[2], 2:[3], 3:[4], 4:[0] } ) sage: D.automorphism_group() # needs sage.groups Permutation Group with generators [(0,1,2,3,4)]
Edge labeled graphs:
sage: G = Graph(sparse=True) sage: G.add_edges( [(0,1,'a'),(1,2,'b'),(2,3,'c'),(3,4,'b'),(4,0,'a')] ) sage: G.automorphism_group(edge_labels=True) # needs sage.groups Permutation Group with generators [(1,4)(2,3)] sage: G.automorphism_group(edge_labels=True, algorithm="bliss") # optional  bliss Permutation Group with generators [(1,4)(2,3)] sage: G.automorphism_group(edge_labels=True, algorithm="sage") # needs sage.groups Permutation Group with generators [(1,4)(2,3)]
sage: G = Graph({0 : {1 : 7}}) sage: G.automorphism_group(edge_labels=True) # needs sage.groups Permutation Group with generators [(0,1)] sage: # needs sage.groups sage: foo = Graph(sparse=True) sage: bar = Graph(sparse=True) sage: foo.add_edges([(0,1,1),(1,2,2), (2,3,3)]) sage: bar.add_edges([(0,1,1),(1,2,2), (2,3,3)]) sage: foo.automorphism_group(edge_labels=True) Permutation Group with generators [()] sage: foo.automorphism_group() Permutation Group with generators [(0,3)(1,2)] sage: bar.automorphism_group(edge_labels=True) Permutation Group with generators [()]
You can also ask for just the order of the group:
sage: G = graphs.PetersenGraph() sage: G.automorphism_group(return_group=False, order=True) # needs sage.groups 120
Or, just the orbits (note that each graph here is vertex transitive)
sage: # needs sage.groups sage: G = graphs.PetersenGraph() sage: G.automorphism_group(return_group=False, orbits=True, algorithm='sage') [[0, 1, 2, 3, 4, 5, 6, 7, 8, 9]] sage: orb = G.automorphism_group(partition=[[0],list(range(1,10))], ....: return_group=False, orbits=True, algorithm='sage') sage: sorted([sorted(o) for o in orb], key=len) [[0], [1, 4, 5], [2, 3, 6, 7, 8, 9]] sage: C = graphs.CubeGraph(3) sage: orb = C.automorphism_group(orbits=True, return_group=False, algorithm='sage') sage: [sorted(o) for o in orb] [['000', '001', '010', '011', '100', '101', '110', '111']]
One can also use the faster algorithm for computing the automorphism group of the graph  bliss:
sage: # optional  bliss sage: G = graphs.HallJankoGraph() sage: A1 = G.automorphism_group() # needs sage.groups sage: A2 = G.automorphism_group(algorithm='bliss') sage: A1.is_isomorphic(A2) # needs sage.groups True
 average_degree()#
Return the average degree of the graph.
The average degree of a graph \(G=(V,E)\) is equal to \(\frac{2E}{V}\).
EXAMPLES:
The average degree of a regular graph is equal to the degree of any vertex:
sage: g = graphs.CompleteGraph(5) sage: g.average_degree() == 4 True
The average degree of a tree is always strictly less than \(2\):
sage: tree = graphs.RandomTree(20) sage: tree.average_degree() < 2 True
For any graph, it is equal to \(\frac{2E}{V}\):
sage: g = graphs.RandomGNP(20, .4) sage: g.average_degree() == 2 * g.size() / g.order() True
 average_distance(by_weight=False, algorithm=None, weight_function=None, check_weight=True)#
Return the average distance between vertices of the graph.
Formally, for a graph \(G\) this value is equal to \(\frac 1 {n(n1)} \sum_{u,v\in G} d(u,v)\) where \(d(u,v)\) denotes the distance between vertices \(u\) and \(v\) and \(n\) is the number of vertices in \(G\).
For more information on the input variables and more examples, we refer to
wiener_index()
andshortest_path_all_pairs()
, which have very similar input variables.INPUT:
by_weight
– boolean (default:False
); ifTrue
, the edges in the graph are weighted, otherwise all edges have weight 1algorithm
– string (default:None
); one of the algorithms available for methodwiener_index()
weight_function
– function (default:None
); a function that takes as input an edge(u, v, l)
and outputs its weight. If notNone
,by_weight
is automatically set toTrue
. IfNone
andby_weight
isTrue
, we use the edge labell
, ifl
is notNone
, else1
as a weight.check_weight
– boolean (default:True
); ifTrue
, we check that the weight_function outputs a number for each edge
EXAMPLES:
From [GYLL1993]:
sage: g=graphs.PathGraph(10) sage: w=lambda x: (x*(x*x 1)/6)/(x*(x1)/2) sage: g.average_distance()==w(10) True
Average distance of a circuit:
sage: g = digraphs.Circuit(6) sage: g.average_distance() 3
 blocks_and_cut_vertices(G, algorithm='Tarjan_Boost', sort=False, key=None)#
Return the blocks and cut vertices of the graph.
In the case of a digraph, this computation is done on the underlying graph.
A cut vertex is one whose deletion increases the number of connected components. A block is a maximal induced subgraph which itself has no cut vertices. Two distinct blocks cannot overlap in more than a single cut vertex.
INPUT:
algorithm
– string (default:"Tarjan_Boost"
); the algorithm to use among:"Tarjan_Boost"
(default) – Tarjan’s algorithm (Boost implementation)"Tarjan_Sage"
– Tarjan’s algorithm (Sage implementation)
sort
– boolean (default:False
); whether to sort vertices inside the components and the list of cut vertices currently only available for ``”Tarjan_Sage”``key
– a function (default:None
); a function that takes a vertex as its one argument and returns a value that can be used for comparisons in the sorting algorithm (we must havesort=True
)
OUTPUT:
(B, C)
, whereB
is a list of blocks  each is a list of vertices and the blocks are the corresponding induced subgraphs  andC
is a list of cut vertices.ALGORITHM:
We implement the algorithm proposed by Tarjan in [Tarjan72]. The original version is recursive. We emulate the recursion using a stack.
See also
EXAMPLES:
We construct a trivial example of a graph with one cut vertex:
sage: from sage.graphs.connectivity import blocks_and_cut_vertices sage: rings = graphs.CycleGraph(10) sage: rings.merge_vertices([0, 5]) sage: blocks_and_cut_vertices(rings) ([[0, 1, 4, 2, 3], [0, 6, 9, 7, 8]], [0]) sage: rings.blocks_and_cut_vertices() ([[0, 1, 4, 2, 3], [0, 6, 9, 7, 8]], [0]) sage: B, C = blocks_and_cut_vertices(rings, algorithm="Tarjan_Sage", sort=True) sage: B, C ([[0, 1, 2, 3, 4], [0, 6, 7, 8, 9]], [0]) sage: B2, C2 = blocks_and_cut_vertices(rings, algorithm="Tarjan_Sage", sort=False) sage: Set(map(Set, B)) == Set(map(Set, B2)) and set(C) == set(C2) True
The Petersen graph is biconnected, hence has no cut vertices:
sage: blocks_and_cut_vertices(graphs.PetersenGraph()) ([[0, 1, 4, 5, 2, 6, 3, 7, 8, 9]], [])
Decomposing paths to pairs:
sage: g = graphs.PathGraph(4) + graphs.PathGraph(5) sage: blocks_and_cut_vertices(g) ([[2, 3], [1, 2], [0, 1], [7, 8], [6, 7], [5, 6], [4, 5]], [1, 2, 5, 6, 7])
A disconnected graph:
sage: g = Graph({1: {2: 28, 3: 10}, 2: {1: 10, 3: 16}, 4: {}, 5: {6: 3, 7: 10, 8: 4}}) sage: blocks_and_cut_vertices(g) ([[1, 2, 3], [5, 6], [5, 7], [5, 8], [4]], [5])
A directed graph with Boost’s algorithm (github issue #25994):
sage: rings = graphs.CycleGraph(10) sage: rings.merge_vertices([0, 5]) sage: rings = rings.to_directed() sage: blocks_and_cut_vertices(rings, algorithm="Tarjan_Boost") ([[0, 1, 4, 2, 3], [0, 6, 9, 7, 8]], [0])
 blocks_and_cuts_tree(G)#
Return the blocksandcuts tree of
self
.This new graph has two different kinds of vertices, some representing the blocks (type B) and some other the cut vertices of the graph (type C).
There is an edge between a vertex \(u\) of type B and a vertex \(v\) of type C if the cutvertex corresponding to \(v\) is in the block corresponding to \(u\).
The resulting graph is a tree, with the additional characteristic property that the distance between two leaves is even. When
self
is not connected, the resulting graph is a forest.When
self
is biconnected, the tree is reduced to a single node of type \(B\).We referred to [HarPri] and [Gallai] for blocks and cuts tree.
EXAMPLES:
sage: from sage.graphs.connectivity import blocks_and_cuts_tree sage: T = blocks_and_cuts_tree(graphs.KrackhardtKiteGraph()); T Graph on 5 vertices sage: T.is_isomorphic(graphs.PathGraph(5)) True sage: from sage.graphs.connectivity import blocks_and_cuts_tree sage: T = graphs.KrackhardtKiteGraph().blocks_and_cuts_tree(); T Graph on 5 vertices
The distance between two leaves is even:
sage: T = blocks_and_cuts_tree(graphs.RandomTree(40)) sage: T.is_tree() True sage: leaves = [v for v in T if T.degree(v) == 1] sage: all(T.distance(u,v) % 2 == 0 for u in leaves for v in leaves) True
The tree of a biconnected graph has a single vertex, of type \(B\):
sage: T = blocks_and_cuts_tree(graphs.PetersenGraph()) sage: T.vertices(sort=True) [('B', (0, 1, 4, 5, 2, 6, 3, 7, 8, 9))]
 breadth_first_search(start, ignore_direction=False, distance=None, neighbors=None, report_distance=False, edges=False)#
Return an iterator over the vertices in a breadthfirst ordering.
INPUT:
start
– vertex or list of vertices from which to start the traversalignore_direction
– boolean (default:False
); only applies to directed graphs. IfTrue
, searches across edges in either direction.distance
– integer (default:None
); the maximum distance from thestart
nodes to traverse. Thestart
nodes are at distance zero from themselves.neighbors
– function (default:None
); a function that inputs a vertex and return a list of vertices. For an undirected graph,neighbors
is by default theneighbors()
function. For a digraph, theneighbors
function defaults to theneighbor_out_iterator()
function of the graph.report_distance
– boolean (default:False
); ifTrue
, reports pairs(vertex, distance)
wheredistance
is the distance from thestart
nodes. IfFalse
only the vertices are reported.edges
– boolean (default:False
); whether to return the edges of the BFS tree in the order of visit or the vertices (default). Edges are directed in root to leaf orientation of the tree.Note that parameters
edges
andreport_distance
cannot beTrue
simultaneously.
See also
breadth_first_search
– breadthfirst search for fast compiled graphs.depth_first_search
– depthfirst search for fast compiled graphs.depth_first_search()
– depthfirst search for generic graphs.
EXAMPLES:
sage: G = Graph({0: [1], 1: [2], 2: [3], 3: [4], 4: [0]}) sage: list(G.breadth_first_search(0)) [0, 1, 4, 2, 3]
By default, the edge direction of a digraph is respected, but this can be overridden by the
ignore_direction
parameter:sage: D = DiGraph({0: [1, 2, 3], 1: [4, 5], 2: [5], 3: [6], 5: [7], 6: [7], 7: [0]}) sage: list(D.breadth_first_search(0)) [0, 1, 2, 3, 4, 5, 6, 7] sage: list(D.breadth_first_search(0, ignore_direction=True)) [0, 1, 2, 3, 7, 4, 5, 6]
You can specify a maximum distance in which to search. A distance of zero returns the
start
vertices:sage: D = DiGraph({0: [1, 2, 3], 1: [4, 5], 2: [5], 3: [6], 5: [7], 6: [7], 7: [0]}) sage: list(D.breadth_first_search(0, distance=0)) [0] sage: list(D.breadth_first_search(0, distance=1)) [0, 1, 2, 3]
Multiple starting vertices can be specified in a list:
sage: D = DiGraph({0: [1, 2, 3], 1: [4, 5], 2: [5], 3: [6], 5: [7], 6: [7], 7: [0]}) sage: list(D.breadth_first_search([0])) [0, 1, 2, 3, 4, 5, 6, 7] sage: list(D.breadth_first_search([0, 6])) [0, 6, 1, 2, 3, 7, 4, 5] sage: list(D.breadth_first_search([0, 6], distance=0)) [0, 6] sage: list(D.breadth_first_search([0, 6], distance=1)) [0, 6, 1, 2, 3, 7] sage: list(D.breadth_first_search(6, ignore_direction=True, distance=2)) [6, 3, 7, 0, 5]
More generally, you can specify a
neighbors
function. For example, you can traverse the graph backwards by settingneighbors
to be theneighbors_in()
function of the graph:sage: D = DiGraph({0: [1, 2, 3], 1: [4, 5], 2: [5], 3: [6], 5: [7], 6: [7], 7: [0]}) sage: list(D.breadth_first_search(5, neighbors=D.neighbors_in, distance=2)) [5, 1, 2, 0] sage: list(D.breadth_first_search(5, neighbors=D.neighbors_out, distance=2)) [5, 7, 0] sage: list(D.breadth_first_search(5 ,neighbors=D.neighbors, distance=2)) [5, 1, 2, 7, 0, 4, 6]
It is possible (github issue #16470) using the keyword
report_distance
to get pairs(vertex, distance)
encoding the distance from the starting vertices:sage: G = graphs.PetersenGraph() sage: list(G.breadth_first_search(0, report_distance=True)) [(0, 0), (1, 1), (4, 1), (5, 1), (2, 2), (6, 2), (3, 2), (9, 2), (7, 2), (8, 2)] sage: list(G.breadth_first_search(0, report_distance=False)) [0, 1, 4, 5, 2, 6, 3, 9, 7, 8] sage: D = DiGraph({0: [1, 3], 1: [0, 2], 2: [0, 3], 3: [4]}) sage: D.show() # needs sage.plot sage: list(D.breadth_first_search(4, neighbors=D.neighbor_in_iterator, ....: report_distance=True)) [(4, 0), (3, 1), (0, 2), (2, 2), (1, 3)] sage: C = graphs.CycleGraph(4) sage: list(C.breadth_first_search([0, 1], report_distance=True)) [(0, 0), (1, 0), (3, 1), (2, 1)]
You can get edges of the BFS tree instead of the vertices using the
edges
parameter:sage: D = DiGraph({1:[2,3],2:[4],3:[4],4:[1],5:[2,6]}) sage: list(D.breadth_first_search(1, edges=True)) [(1, 2), (1, 3), (2, 4)]
 canonical_label(partition=None, certificate=False, edge_labels=False, algorithm=None, return_graph=True)#
Return the canonical graph.
A canonical graph is the representative graph of an isomorphism class by some canonization function \(c\). If \(G\) and \(H\) are graphs, then \(G \cong c(G)\), and \(c(G) == c(H)\) if and only if \(G \cong H\).
See the Wikipedia article Graph_canonization for more information.
INPUT:
partition
– if given, the canonical label with respect to this set partition will be computed. The default is the unit set partition.certificate
– boolean (default:False
). When set toTrue
, a dictionary mapping from the vertices of the (di)graph to its canonical label will also be returned.edge_labels
– boolean (default:False
). When set toTrue
, allows only permutations respecting edge labels.algorithm
– a string (default:None
). The algorithm to use; currently available:'bliss'
: use the optional package bliss (http://www.tcs.tkk.fi/Software/bliss/index.html);'sage'
: always use Sage’s implementation.None
(default): use bliss when available and possibleNote
Make sure you always compare canonical forms obtained by the same algorithm.
return_graph
– boolean (default:True
). When set toFalse
, returns the list of edges of the canonical graph instead of the canonical graph; only available when'bliss'
is explicitly set as algorithm.
EXAMPLES:
Canonization changes isomorphism to equality:
sage: g1 = graphs.GridGraph([2,3]) sage: g2 = Graph({1: [2, 4], 3: [2, 6], 5: [4, 2, 6]}) sage: g1 == g2 False sage: g1.is_isomorphic(g2) True sage: g1.canonical_label() == g2.canonical_label() True
We can get the relabeling used for canonization:
sage: g, c = g1.canonical_label(algorithm='sage', certificate=True) sage: g Grid Graph for [2, 3]: Graph on 6 vertices sage: c {(0, 0): 3, (0, 1): 4, (0, 2): 2, (1, 0): 0, (1, 1): 5, (1, 2): 1}
Multigraphs and directed graphs work too:
sage: G = Graph(multiedges=True,sparse=True) sage: G.add_edge((0,1)) sage: G.add_edge((0,1)) sage: G.add_edge((0,1)) sage: G.canonical_label() Multigraph on 2 vertices sage: Graph('A?').canonical_label() Graph on 2 vertices sage: P = graphs.PetersenGraph() sage: DP = P.to_directed() sage: DP.canonical_label(algorithm='sage').adjacency_matrix() # needs sage.modules [0 0 0 0 0 0 0 1 1 1] [0 0 0 0 1 0 1 0 0 1] [0 0 0 1 0 0 1 0 1 0] [0 0 1 0 0 1 0 0 0 1] [0 1 0 0 0 1 0 0 1 0] [0 0 0 1 1 0 0 1 0 0] [0 1 1 0 0 0 0 1 0 0] [1 0 0 0 0 1 1 0 0 0] [1 0 1 0 1 0 0 0 0 0] [1 1 0 1 0 0 0 0 0 0]
Edge labeled graphs:
sage: G = Graph(sparse=True) sage: G.add_edges( [(0,1,'a'),(1,2,'b'),(2,3,'c'),(3,4,'b'),(4,0,'a')] ) sage: G.canonical_label(edge_labels=True) Graph on 5 vertices sage: G.canonical_label(edge_labels=True, algorithm="bliss", # optional  bliss ....: certificate=True) (Graph on 5 vertices, {0: 4, 1: 3, 2: 1, 3: 0, 4: 2}) sage: G.canonical_label(edge_labels=True, algorithm="sage", ....: certificate=True) (Graph on 5 vertices, {0: 4, 1: 3, 2: 0, 3: 1, 4: 2})
Another example where different canonization algorithms give different graphs:
sage: g = Graph({'a': ['b'], 'c': ['d']}) sage: g_sage = g.canonical_label(algorithm='sage') sage: g_bliss = g.canonical_label(algorithm='bliss') # optional  bliss sage: g_sage.edges(sort=True, labels=False) [(0, 3), (1, 2)] sage: g_bliss.edges(sort=True, labels=False) # optional  bliss [(0, 1), (2, 3)]
 cartesian_product(other)#
Return the Cartesian product of
self
andother
.The Cartesian product of \(G\) and \(H\) is the graph \(L\) with vertex set \(V(L)\) equal to the Cartesian product of the vertices \(V(G)\) and \(V(H)\), and \(((u,v), (w,x))\) is an edge iff either  \((u, w)\) is an edge of self and \(v = x\), or  \((v, x)\) is an edge of other and \(u = w\).
See also
is_cartesian_product()
– factorization of graphs according to the Cartesian productgraph_products
– a module on graph products
 categorical_product(other)#
Return the tensor product of
self
andother
.The tensor product of \(G\) and \(H\) is the graph \(L\) with vertex set \(V(L)\) equal to the Cartesian product of the vertices \(V(G)\) and \(V(H)\), and \(((u,v), (w,x))\) is an edge iff  \((u, w)\) is an edge of self, and  \((v, x)\) is an edge of other.
The tensor product is also known as the categorical product and the Kronecker product (referring to the Kronecker matrix product). See the Wikipedia article Kronecker_product.
EXAMPLES:
sage: Z = graphs.CompleteGraph(2) sage: C = graphs.CycleGraph(5) sage: T = C.tensor_product(Z); T Graph on 10 vertices sage: T.size() 10 sage: T.plot() # long time Graphics object consisting of 21 graphics primitives
sage: D = graphs.DodecahedralGraph() sage: P = graphs.PetersenGraph() sage: T = D.tensor_product(P); T Graph on 200 vertices sage: T.size() 900 sage: T.plot() # long time Graphics object consisting of 1101 graphics primitives
 centrality_betweenness(k=None, normalized=True, weight=None, endpoints=False, seed=None, exact=False, algorithm=None)#
Return the betweenness centrality.
The betweenness centrality of a vertex is the fraction of number of shortest paths that go through each vertex. The betweenness is normalized by default to be in range (0,1).
Measures of the centrality of a vertex within a graph determine the relative importance of that vertex to its graph. Vertices that occur on more shortest paths between other vertices have higher betweenness than vertices that occur on less.
INPUT:
normalized
– boolean (default:True
); if set toFalse
, result is not normalized.k
– integer (default:None
); if set to an integer, usek
node samples to estimate betweenness. Higher values give better approximations. Not available whenalgorithm="Sage"
.weight
– string (default:None
); if set to a string, use that attribute of the nodes as weight.weight = True
is equivalent toweight = "weight"
. Not available whenalgorithm="Sage"
.endpoints
– boolean (default:False
); if set toTrue
it includes the endpoints in the shortest paths count. Not available whenalgorithm="Sage"
.exact
– boolean (default:False
); whether to compute over rationals or ondouble
C variables. Not available whenalgorithm="NetworkX"
.algorithm
– string (default:None
); can be either"Sage"
(seecentrality
),"NetworkX"
or"None"
. In the latter case, Sage’s algorithm will be used whenever possible.
EXAMPLES:
sage: g = graphs.ChvatalGraph() sage: g.centrality_betweenness() # abs tol 1e10 {0: 0.06969696969696969, 1: 0.06969696969696969, 2: 0.0606060606060606, 3: 0.0606060606060606, 4: 0.06969696969696969, 5: 0.06969696969696969, 6: 0.0606060606060606, 7: 0.0606060606060606, 8: 0.0606060606060606, 9: 0.0606060606060606, 10: 0.0606060606060606, 11: 0.0606060606060606} sage: g.centrality_betweenness(normalized=False) # abs tol 1e10 {0: 3.833333333333333, 1: 3.833333333333333, 2: 3.333333333333333, 3: 3.333333333333333, 4: 3.833333333333333, 5: 3.833333333333333, 6: 3.333333333333333, 7: 3.333333333333333, 8: 3.333333333333333, 9: 3.333333333333333, 10: 3.333333333333333, 11: 3.333333333333333} sage: D = DiGraph({0:[1,2,3], 1:[2], 3:[0,1]}) sage: D.show(figsize=[2,2]) # needs sage.plot sage: D = D.to_undirected() sage: D.show(figsize=[2,2]) # needs sage.plot sage: D.centrality_betweenness() # abs tol abs 1e10 {0: 0.16666666666666666, 1: 0.16666666666666666, 2: 0.0, 3: 0.0}
 centrality_closeness(vert=None, by_weight=False, algorithm=None, weight_function=None, check_weight=True)#
Return the closeness centrality of all vertices in
vert
.In a (strongly) connected graph, the closeness centrality of a vertex \(v\) is equal to the inverse of the average distance between \(v\) and other vertices. If the graph is disconnected, the closeness centrality of \(v\) is multiplied by the fraction of reachable vertices in the graph: this way, central vertices should also reach several other vertices in the graph [OLJ2014]. In formulas,
\[c(v)=\frac{r(v)1}{\sum_{w \in R(v)} d(v,w)}\frac{r(v)1}{n1}\]where \(R(v)\) is the set of vertices reachable from \(v\), and \(r(v)\) is the cardinality of \(R(v)\).
‘Closeness centrality may be defined as the total graphtheoretic distance of a given vertex from all other vertices… Closeness is an inverse measure of centrality in that a larger value indicates a less central actor while a smaller value indicates a more central actor,’ [Bor1995].
For more information, see the Wikipedia article Centrality.
INPUT:
vert
– the vertex or the list of vertices we want to analyze. IfNone
(default), all vertices are considered.by_weight
– boolean (default:False
); ifTrue
, the edges in the graph are weighted, and otherwise all edges have weight 1algorithm
– string (default:None
); one of the following algorithms:'BFS'
: performs a BFS from each vertex that has to be analyzed. Does not work with edge weights.'NetworkX'
: the NetworkX algorithm (works only with positive weights).'Dijkstra_Boost'
: the Dijkstra algorithm, implemented in Boost (works only with positive weights).'FloydWarshallCython'
: the Cython implementation of the FloydWarshall algorithm. Works only ifby_weight==False
and all centralities are needed.'FloydWarshallPython'
: the Python implementation of the FloydWarshall algorithm. Works only if all centralities are needed, but it can deal with weighted graphs, even with negative weights (but no negative cycle is allowed).'Johnson_Boost'
: the Johnson algorithm, implemented in Boost (works also with negative weights, if there is no negative cycle).None
(default): Sage chooses the best algorithm:'BFS'
ifby_weight
isFalse
,'Dijkstra_Boost'
if all weights are positive,'Johnson_Boost'
otherwise.
weight_function
– function (default:None
); a function that takes as input an edge(u, v, l)
and outputs its weight. If notNone
,by_weight
is automatically set toTrue
. IfNone
andby_weight
isTrue
, we use the edge labell
as a weight, ifl
is notNone
, else1
as a weight.check_weight
– boolean (default:True
); ifTrue
, we check that theweight_function
outputs a number for each edge.
OUTPUT:
If
vert
is a vertex, the closeness centrality of that vertex. Otherwise, a dictionary associating to each vertex invert
its closeness centrality. If a vertex has (out)degree 0, its closeness centrality is not defined, and the vertex is not included in the output.EXAMPLES:
Standard examples:
sage: (graphs.ChvatalGraph()).centrality_closeness() {0: 0.61111111111111..., 1: 0.61111111111111..., 2: 0.61111111111111..., 3: 0.61111111111111..., 4: 0.61111111111111..., 5: 0.61111111111111..., 6: 0.61111111111111..., 7: 0.61111111111111..., 8: 0.61111111111111..., 9: 0.61111111111111..., 10: 0.61111111111111..., 11: 0.61111111111111...} sage: D = DiGraph({0:[1,2,3], 1:[2], 3:[0,1]}) sage: D.show(figsize=[2,2]) # needs sage.plot sage: D.centrality_closeness(vert=[0,1]) {0: 1.0, 1: 0.3333333333333333} sage: D = D.to_undirected() sage: D.show(figsize=[2,2]) # needs sage.plot sage: D.centrality_closeness() {0: 1.0, 1: 1.0, 2: 0.75, 3: 0.75}
In a (strongly) connected (di)graph, the closeness centrality of \(v\) is inverse of the average distance between \(v\) and all other vertices:
sage: g = graphs.PathGraph(5) sage: g.centrality_closeness(0) 0.4 sage: dist = g.shortest_path_lengths(0).values() sage: float(len(dist)1) / sum(dist) 0.4 sage: d = g.to_directed() sage: d.centrality_closeness(0) 0.4 sage: dist = d.shortest_path_lengths(0).values() sage: float(len(dist)1) / sum(dist) 0.4
If a vertex has (out)degree 0, its closeness centrality is not defined:
sage: g = Graph(5) sage: g.centrality_closeness() {} sage: print(g.centrality_closeness(0)) None
Weighted graphs:
sage: D = graphs.GridGraph([2,2]) sage: weight_function = lambda e:10 sage: D.centrality_closeness([(0,0),(0,1)]) # tol abs 1e12 {(0, 0): 0.75, (0, 1): 0.75} sage: D.centrality_closeness((0,0), weight_function=weight_function) # tol abs 1e12 0.075
 characteristic_polynomial(var='x', laplacian=False)#
Return the characteristic polynomial of the adjacency matrix of the (di)graph.
Let \(G\) be a (simple) graph with adjacency matrix \(A\). Let \(I\) be the identity matrix of dimensions the same as \(A\). The characteristic polynomial of \(G\) is defined as the determinant \(\det(xI  A)\).
Note
characteristic_polynomial
andcharpoly
are aliases and thus provide exactly the same method.INPUT:
x
– (default:'x'
); the variable of the characteristic polynomiallaplacian
– boolean (default:False
); ifTrue
, use the Laplacian matrix
See also
EXAMPLES:
sage: P = graphs.PetersenGraph() sage: P.characteristic_polynomial() # needs sage.modules x^10  15*x^8 + 75*x^6  24*x^5  165*x^4 + 120*x^3 + 120*x^2  160*x + 48 sage: P.charpoly() # needs sage.modules x^10  15*x^8 + 75*x^6  24*x^5  165*x^4 + 120*x^3 + 120*x^2  160*x + 48 sage: P.characteristic_polynomial(laplacian=True) # needs sage.modules x^10  30*x^9 + 390*x^8  2880*x^7 + 13305*x^6  39882*x^5 + 77640*x^4  94800*x^3 + 66000*x^2  20000*x
 charpoly(var='x', laplacian=False)#
Return the characteristic polynomial of the adjacency matrix of the (di)graph.
Let \(G\) be a (simple) graph with adjacency matrix \(A\). Let \(I\) be the identity matrix of dimensions the same as \(A\). The characteristic polynomial of \(G\) is defined as the determinant \(\det(xI  A)\).
Note
characteristic_polynomial
andcharpoly
are aliases and thus provide exactly the same method.INPUT:
x
– (default:'x'
); the variable of the characteristic polynomiallaplacian
– boolean (default:False
); ifTrue
, use the Laplacian matrix
See also
EXAMPLES:
sage: P = graphs.PetersenGraph() sage: P.characteristic_polynomial() # needs sage.modules x^10  15*x^8 + 75*x^6  24*x^5  165*x^4 + 120*x^3 + 120*x^2  160*x + 48 sage: P.charpoly() # needs sage.modules x^10  15*x^8 + 75*x^6  24*x^5  165*x^4 + 120*x^3 + 120*x^2  160*x + 48 sage: P.characteristic_polynomial(laplacian=True) # needs sage.modules x^10  30*x^9 + 390*x^8  2880*x^7 + 13305*x^6  39882*x^5 + 77640*x^4  94800*x^3 + 66000*x^2  20000*x
 clear()#
Empties the graph of vertices and edges and removes name, associated objects, and position information.
EXAMPLES:
sage: G = graphs.CycleGraph(4) sage: G.set_vertices({0:'vertex0'}) sage: print(G.order(), G.size()) 4 4 sage: G.name() 'Cycle graph' sage: G.get_vertex(0) 'vertex0' sage: H = G.copy(sparse=True) sage: H.clear() sage: print(H.order(), H.size()) 0 0 sage: H.name() '' sage: H.get_vertex(0) sage: H = G.copy(sparse=False) sage: H.clear() sage: print(H.order(), H.size()) 0 0 sage: H.name() '' sage: H.get_vertex(0)
 cluster_transitivity()#
Return the transitivity (fraction of transitive triangles) of the graph.
Transitivity is the fraction of all existing triangles over all connected triples (triads), \(T = 3\times\frac{\text{triangles}}{\text{triads}}\).
See also section “Clustering” in chapter “Algorithms” of [HSS].
EXAMPLES:
sage: graphs.FruchtGraph().cluster_transitivity() # needs networkx 0.25
 cluster_triangles(nbunch=None, implementation=None)#
Return the number of triangles for the set \(nbunch\) of vertices as a dictionary keyed by vertex.
See also section “Clustering” in chapter “Algorithms” of [HSS].
INPUT:
nbunch
– a list of vertices (default:None); the vertices to inspect. If ``nbunch=None
, returns data for all vertices in the graph.implementation
– string (default:None
); one of'sparse_copy'
,'dense_copy'
,'networkx'
orNone
(default). In the latter case, the best algorithm available is used. Note that'networkx'
does not support directed graphs.
EXAMPLES:
sage: F = graphs.FruchtGraph() sage: list(F.cluster_triangles().values()) [1, 1, 0, 1, 1, 1, 1, 1, 0, 1, 1, 0] sage: F.cluster_triangles() {0: 1, 1: 1, 2: 0, 3: 1, 4: 1, 5: 1, 6: 1, 7: 1, 8: 0, 9: 1, 10: 1, 11: 0} sage: F.cluster_triangles(nbunch=[0, 1, 2]) {0: 1, 1: 1, 2: 0}
sage: G = graphs.RandomGNP(20, .3) sage: d1 = G.cluster_triangles(implementation="networkx") # needs networkx sage: d2 = G.cluster_triangles(implementation="dense_copy") sage: d3 = G.cluster_triangles(implementation="sparse_copy") sage: d1 == d2 and d1 == d3 # needs networkx True
 clustering_average(implementation=None)#
Return the average clustering coefficient.
The clustering coefficient of a node \(i\) is the fraction of existing triangles containing node \(i\) over all possible triangles containing \(i\): \(c_i = T(i) / \binom {k_i} 2\) where \(T(i)\) is the number of existing triangles through \(i\), and \(k_i\) is the degree of vertex \(i\).
A coefficient for the whole graph is the average of the \(c_i\).
See also section “Clustering” in chapter “Algorithms” of [HSS].
INPUT:
implementation
– string (default:None
); one of'boost'
,'sparse_copy'
,'dense_copy'
,'networkx'
orNone
(default). In the latter case, the best algorithm available is used. Note that only'networkx'
supports directed graphs.
EXAMPLES:
sage: (graphs.FruchtGraph()).clustering_average() 1/4 sage: (graphs.FruchtGraph()).clustering_average(implementation='networkx') # needs networkx 0.25
 clustering_coeff(nodes=None, weight=False, implementation=None)#
Return the clustering coefficient for each vertex in
nodes
as a dictionary keyed by vertex.For an unweighted graph, the clustering coefficient of a node \(i\) is the fraction of existing triangles containing node \(i\) over all possible triangles containing \(i\): \(c_i = T(i) / \binom {k_i} 2\) where \(T(i)\) is the number of existing triangles through \(i\), and \(k_i\) is the degree of vertex \(i\).
For weighted graphs the clustering is defined as the geometric average of the subgraph edge weights, normalized by the maximum weight in the network.
The value of \(c_i\) is assigned \(0\) if \(k_i < 2\).
See also section “Clustering” in chapter “Algorithms” of [HSS].
INPUT:
nodes
– an iterable container of vertices (default:None
); the vertices to inspect. By default, returns data on all vertices in graphweight
– string or boolean (default:False
); if it is a string it uses the indicated edge property as weight.weight = True
is equivalent toweight = 'weight'
implementation
– string (default:None
); one of'boost'
,'sparse_copy'
,'dense_copy'
,'networkx'
orNone
(default). In the latter case, the best algorithm available is used. Note that only'networkx'
supports directed or weighted graphs, and that'sparse_copy'
and'dense_copy'
do not supportnode
different fromNone
EXAMPLES:
sage: graphs.FruchtGraph().clustering_coeff() {0: 1/3, 1: 1/3, 2: 0, 3: 1/3, 4: 1/3, 5: 1/3, 6: 1/3, 7: 1/3, 8: 0, 9: 1/3, 10: 1/3, 11: 0} sage: (graphs.FruchtGraph()).clustering_coeff(weight=True) # needs networkx {0: 0.3333333333333333, 1: 0.3333333333333333, 2: 0, 3: 0.3333333333333333, 4: 0.3333333333333333, 5: 0.3333333333333333, 6: 0.3333333333333333, 7: 0.3333333333333333, 8: 0, 9: 0.3333333333333333, 10: 0.3333333333333333, 11: 0} sage: (graphs.FruchtGraph()).clustering_coeff(nodes=[0,1,2]) {0: 0.3333333333333333, 1: 0.3333333333333333, 2: 0.0} sage: (graphs.FruchtGraph()).clustering_coeff(nodes=[0,1,2], # needs networkx ....: weight=True) {0: 0.3333333333333333, 1: 0.3333333333333333, 2: 0} sage: (graphs.GridGraph([5,5])).clustering_coeff(nodes=[(0,0),(0,1),(2,2)]) {(0, 0): 0.0, (0, 1): 0.0, (2, 2): 0.0}
 coarsest_equitable_refinement(partition, sparse=True)#
Return the coarsest partition which is finer than the input partition, and equitable with respect to self.
A partition is equitable with respect to a graph if for every pair of cells \(C_1\), \(C_2\) of the partition, the number of edges from a vertex of \(C_1\) to \(C_2\) is the same, over all vertices in \(C_1\).
A partition \(P_1\) is finer than \(P_2\) (\(P_2\) is coarser than \(P_1\)) if every cell of \(P_1\) is a subset of a cell of \(P_2\).
INPUT:
partition
– a list of listssparse
– boolean (default:False
); whether to use sparse ordense representation  for small graphs, use dense for speed
EXAMPLES:
sage: G = graphs.PetersenGraph() sage: G.coarsest_equitable_refinement([[0],list(range(1,10))]) [[0], [2, 3, 6, 7, 8, 9], [1, 4, 5]] sage: G = graphs.CubeGraph(3) sage: verts = G.vertices(sort=True) sage: Pi = [verts[:1], verts[1:]] sage: Pi [['000'], ['001', '010', '011', '100', '101', '110', '111']] sage: [sorted(cell) for cell in G.coarsest_equitable_refinement(Pi)] [['000'], ['011', '101', '110'], ['111'], ['001', '010', '100']]
Note that given an equitable partition, this function returns that partition:
sage: P = graphs.PetersenGraph() sage: prt = [[0], [1, 4, 5], [2, 3, 6, 7, 8, 9]] sage: P.coarsest_equitable_refinement(prt) [[0], [1, 4, 5], [2, 3, 6, 7, 8, 9]]
sage: ss = (graphs.WheelGraph(6)).line_graph(labels=False) sage: prt = [[(0, 1)], [(0, 2), (0, 3), (0, 4), (1, 2), (1, 4)], [(2, 3), (3, 4)]] sage: ss.coarsest_equitable_refinement(prt) Traceback (most recent call last): ... TypeError: partition ([[(0, 1)], [(0, 2), (0, 3), (0, 4), (1, 2), (1, 4)], [(2, 3), (3, 4)]]) is not valid for this graph: vertices are incorrect
sage: ss = (graphs.WheelGraph(5)).line_graph(labels=False) sage: ss.coarsest_equitable_refinement(prt) [[(0, 1)], [(1, 2), (1, 4)], [(0, 3)], [(0, 4), (0, 2)], [(2, 3), (3, 4)]]
ALGORITHM: Brendan D. McKay’s Master’s Thesis, University of Melbourne, 1976.
 complement()#
Return the complement of the (di)graph.
The complement of a graph has the same vertices, but exactly those edges that are not in the original graph. This is not well defined for graphs with multiple edges.
EXAMPLES:
sage: P = graphs.PetersenGraph() sage: P.plot() # long time # needs sage.plot Graphics object consisting of 26 graphics primitives sage: PC = P.complement() sage: PC.plot() # long time # needs sage.plot Graphics object consisting of 41 graphics primitives
sage: graphs.TetrahedralGraph().complement().size() 0 sage: graphs.CycleGraph(4).complement().edges(sort=True) [(0, 2, None), (1, 3, None)] sage: graphs.CycleGraph(4).complement() complement(Cycle graph): Graph on 4 vertices sage: G = Graph(multiedges=True, sparse=True) sage: G.add_edges([(0, 1)] * 3) sage: G.complement() Traceback (most recent call last): ... ValueError: This method is not known to work on graphs with multiedges. Perhaps this method can be updated to handle them, but in the meantime if you want to use it please disallow multiedges using allow_multiple_edges().
 connected_component_containing_vertex(G, vertex, sort=None, key=None)#
Return a list of the vertices connected to vertex.
INPUT:
G
– the input graphv
– the vertex to search forsort
– boolean (default:None
); ifTrue
, vertices inside the component are sorted according to the default orderingAs of github issue #35889, this argument must be explicitly specified (unless a
key
is given); otherwise a warning is printed andsort=True
is used. The default will eventually be changed toFalse
.key
– a function (default:None
); a function that takes a vertex as its one argument and returns a value that can be used for comparisons in the sorting algorithm (we must havesort=True
)
EXAMPLES:
sage: from sage.graphs.connectivity import connected_component_containing_vertex sage: G = Graph({0: [1, 3], 1: [2], 2: [3], 4: [5, 6], 5: [6]}) sage: connected_component_containing_vertex(G, 0, sort=True) [0, 1, 2, 3] sage: G.connected_component_containing_vertex(0, sort=True) [0, 1, 2, 3] sage: D = DiGraph({0: [1, 3], 1: [2], 2: [3], 4: [5, 6], 5: [6]}) sage: connected_component_containing_vertex(D, 0, sort=True) [0, 1, 2, 3] sage: connected_component_containing_vertex(D, 0, sort=True, key=lambda x: x) [3, 2, 1, 0]
 connected_components(G, sort=None, key=None)#
Return the list of connected components.
This returns a list of lists of vertices, each list representing a connected component. The list is ordered from largest to smallest component.
INPUT:
G
– the input graphsort
– boolean (default:None
); ifTrue
, vertices inside each component are sorted according to the default orderingAs of github issue #35889, this argument must be explicitly specified (unless a
key
is given); otherwise a warning is printed andsort=True
is used. The default will eventually be changed toFalse
.key
– a function (default:None
); a function that takes a vertex as its one argument and returns a value that can be used for comparisons in the sorting algorithm (we must havesort=True
)
EXAMPLES:
sage: from sage.graphs.connectivity import connected_components sage: G = Graph({0: [1, 3], 1: [2], 2: [3], 4: [5, 6], 5: [6]}) sage: connected_components(G, sort=True) [[0, 1, 2, 3], [4, 5, 6]] sage: G.connected_components(sort=True) [[0, 1, 2, 3], [4, 5, 6]] sage: D = DiGraph({0: [1, 3], 1: [2], 2: [3], 4: [5, 6], 5: [6]}) sage: connected_components(D, sort=True) [[0, 1, 2, 3], [4, 5, 6]] sage: connected_components(D, sort=True, key=lambda x: x) [[3, 2, 1, 0], [6, 5, 4]]
 connected_components_number(G)#
Return the number of connected components.
INPUT:
G
– the input graph
EXAMPLES:
sage: from sage.graphs.connectivity import connected_components_number sage: G = Graph({0: [1, 3], 1: [2], 2: [3], 4: [5, 6], 5: [6]}) sage: connected_components_number(G) 2 sage: G.connected_components_number() 2 sage: D = DiGraph({0: [1, 3], 1: [2], 2: [3], 4: [5, 6], 5: [6]}) sage: connected_components_number(D) 2
 connected_components_sizes(G)#
Return the sizes of the connected components as a list.
The list is sorted from largest to lower values.
EXAMPLES:
sage: from sage.graphs.connectivity import connected_components_sizes sage: for x in graphs(3): ....: print(connected_components_sizes(x)) [1, 1, 1] [2, 1] [3] [3] sage: for x in graphs(3): ....: print(x.connected_components_sizes()) [1, 1, 1] [2, 1] [3] [3]
 connected_components_subgraphs(G)#
Return a list of connected components as graph objects.
EXAMPLES:
sage: from sage.graphs.connectivity import connected_components_subgraphs sage: G = Graph({0: [1, 3], 1: [2], 2: [3], 4: [5, 6], 5: [6]}) sage: L = connected_components_subgraphs(G) sage: graphs_list.show_graphs(L) # needs sage.plot sage: D = DiGraph({0: [1, 3], 1: [2], 2: [3], 4: [5, 6], 5: [6]}) sage: L = connected_components_subgraphs(D) sage: graphs_list.show_graphs(L) # needs sage.plot sage: L = D.connected_components_subgraphs() sage: graphs_list.show_graphs(L) # needs sage.plot
 connected_subgraph_iterator(G, k=None, vertices_only=False, edges_only=False, labels=False, induced=True, exactly_k=False)#
Return an terator over the induced connected subgraphs of order at most \(k\).
This method implements a iterator over the induced connected subgraphs of the input (di)graph. An induced subgraph of a graph is another graph, formed from a subset of the vertices of the graph and all of the edges connecting pairs of vertices in that subset (Wikipedia article Induced_subgraph).
As for method
sage.graphs.generic_graph.connected_components()
, edge orientation is ignored. Hence, the directed graph with a single arc \(0 \to 1\) is considered connected.INPUT:
G
– aGraph
or aDiGraph
; loops and multiple edges are allowedk
– (optional) integer; maximum order of the connected subgraphs to report; by default, the method iterates over all connected subgraphs (equivalent tok == n
)vertices_only
– boolean (default:False
); whether to return (Di)Graph or list of vertices. This parameter is ignored wheninduced
isTrue
.edges_only
– boolean (default:False
); whether to return (Di)Graph or list of edges. Whenvertices_only
isTrue
, this parameter is ignored.labels
– boolean (default:False
); whether to return labelled edges or not. This parameter is used only whenvertices_only
isFalse
andedges_only
isTrue
.induced
– boolean (default:True
); whether to return induced connected sub(di)graph only or also noninduced sub(di)graphs. This parameter can be set toFalse
for simple (di)graphs only.exactly_k
– boolean (default:False
);True
if we only return graphs of order \(k\),False
if we return graphs of order at most \(k\).
EXAMPLES:
sage: G = DiGraph([(1, 2), (2, 3), (3, 4), (4, 2)]) sage: list(G.connected_subgraph_iterator()) [Subgraph of (): Digraph on 1 vertex, Subgraph of (): Digraph on 2 vertices, Subgraph of (): Digraph on 3 vertices, Subgraph of (): Digraph on 4 vertices, Subgraph of (): Digraph on 3 vertices, Subgraph of (): Digraph on 1 vertex, Subgraph of (): Digraph on 2 vertices, Subgraph of (): Digraph on 3 vertices, Subgraph of (): Digraph on 2 vertices, Subgraph of (): Digraph on 1 vertex, Subgraph of (): Digraph on 2 vertices, Subgraph of (): Digraph on 1 vertex] sage: list(G.connected_subgraph_iterator(vertices_only=True)) [[1], [1, 2], [1, 2, 3], [1, 2, 3, 4], [1, 2, 4], [2], [2, 3], [2, 3, 4], [2, 4], [3], [3, 4], [4]] sage: list(G.connected_subgraph_iterator(k=2)) [Subgraph of (): Digraph on 1 vertex, Subgraph of (): Digraph on 2 vertices, Subgraph of (): Digraph on 1 vertex, Subgraph of (): Digraph on 2 vertices, Subgraph of (): Digraph on 2 vertices, Subgraph of (): Digraph on 1 vertex, Subgraph of (): Digraph on 2 vertices, Subgraph of (): Digraph on 1 vertex] sage: list(G.connected_subgraph_iterator(k=3, vertices_only=True, exactly_k=True)) [[1, 2, 3], [1, 2, 4], [2, 3, 4]] sage: list(G.connected_subgraph_iterator(k=2, vertices_only=True)) [[1], [1, 2], [2], [2, 3], [2, 4], [3], [3, 4], [4]] sage: G = DiGraph([(1, 2), (2, 1)]) sage: list(G.connected_subgraph_iterator()) [Subgraph of (): Digraph on 1 vertex, Subgraph of (): Digraph on 2 vertices, Subgraph of (): Digraph on 1 vertex] sage: list(G.connected_subgraph_iterator(vertices_only=True)) [[1], [1, 2], [2]] sage: G = graphs.CompleteGraph(3) sage: len(list(G.connected_subgraph_iterator())) 7 sage: len(list(G.connected_subgraph_iterator(vertices_only=True))) 7 sage: len(list(G.connected_subgraph_iterator(edges_only=True))) 7 sage: len(list(G.connected_subgraph_iterator(induced=False))) 10 sage: G = DiGraph([(0, 1), (1, 0), (1, 2), (2, 1)]) sage: len(list(G.connected_subgraph_iterator())) 6 sage: len(list(G.connected_subgraph_iterator(vertices_only=True))) 6 sage: len(list(G.connected_subgraph_iterator(edges_only=True))) 6 sage: len(list(G.connected_subgraph_iterator(induced=False))) 18
 contract_edge(u, v=None, label=None)#
Contract an edge from
u
tov
.This method returns silently if the edge does not exist.
INPUT: The following forms are all accepted:
G.contract_edge( 1, 2 )
G.contract_edge( (1, 2) )
G.contract_edge( [ (1, 2) ] )
G.contract_edge( 1, 2, ‘label’ )
G.contract_edge( (1, 2, ‘label’) )
G.contract_edge( [ (1, 2, ‘label’) ] )
EXAMPLES:
sage: G = graphs.CompleteGraph(4) sage: G.contract_edge((0, 1)); G.edges(sort=True) [(0, 2, None), (0, 3, None), (2, 3, None)] sage: G = graphs.CompleteGraph(4) sage: G.allow_loops(True); G.allow_multiple_edges(True) sage: G.contract_edge((0, 1)); G.edges(sort=True) [(0, 2, None), (0, 2, None), (0, 3, None), (0, 3, None), (2, 3, None)] sage: G.contract_edge((0, 2)); G.edges(sort=True) [(0, 0, None), (0, 3, None), (0, 3, None), (0, 3, None)]
sage: G = graphs.CompleteGraph(4).to_directed() sage: G.allow_loops(True) sage: G.contract_edge(0, 1); G.edges(sort=True) [(0, 0, None), (0, 2, None), (0, 3, None), (2, 0, None), (2, 3, None), (3, 0, None), (3, 2, None)]
 contract_edges(edges)#
Contract edges from an iterable container.
If \(e\) is an edge that is not contracted but the vertices of \(e\) are merged by contraction of other edges, then \(e\) will become a loop.
INPUT:
edges
– a list containing 2tuples or 3tuples that represent edges
EXAMPLES:
sage: G = graphs.CompleteGraph(4) sage: G.allow_loops(True); G.allow_multiple_edges(True) sage: G.contract_edges([(0, 1), (1, 2), (0, 2)]); G.edges(sort=True) [(0, 3, None), (0, 3, None), (0, 3, None)] sage: G.contract_edges([(1, 3), (2, 3)]); G.edges(sort=True) [(0, 3, None), (0, 3, None), (0, 3, None)] sage: G = graphs.CompleteGraph(4) sage: G.allow_loops(True); G.allow_multiple_edges(True) sage: G.contract_edges([(0, 1), (1, 2), (0, 2), (1, 3), (2, 3)]); G.edges(sort=True) [(0, 0, None)]
sage: D = digraphs.Complete(4) sage: D.allow_loops(True); D.allow_multiple_edges(True) sage: D.contract_edges([(0, 1), (1, 0), (0, 2)]); D.edges(sort=True) [(0, 0, None), (0, 0, None), (0, 0, None), (0, 3, None), (0, 3, None), (0, 3, None), (3, 0, None), (3, 0, None), (3, 0, None)]
 copy(weighted=None, data_structure=None, sparse=None, immutable=None, hash_labels=None)#
Change the graph implementation
INPUT:
weighted
– boolean (default:None
); weightedness for the copy. Might change the equality class if notNone
.sparse
– boolean (default:None
);sparse=True
is an alias fordata_structure="sparse"
, andsparse=False
is an alias fordata_structure="dense"
. Only used whendata_structure=None
.data_structure
– string (default:None
); one of"sparse"
,"static_sparse"
, or"dense"
. See the documentation ofGraph
orDiGraph
.immutable
– boolean (default:None
); whether to create a mutable/immutable copy. Only used whendata_structure=None
.immutable=None
(default) means that the graph and its copy will behave the same way.immutable=True
is a shortcut fordata_structure='static_sparse'
immutable=False
means that the created graph is mutable. When used to copy an immutable graph, the data structure used is"sparse"
unless anything else is specified.
hash_labels
– boolean (default:None
); whether to include edge labels during hashing of the copy. This parameter defaults toTrue
if the graph is weighted. This parameter is ignored when parameterimmutable
is notTrue
. Beware that trying to hash unhashable labels will raise an error.
Note
If the graph uses
StaticSparseBackend
and the_immutable
flag, thenself
is returned rather than a copy (unless one of the optional arguments is used).OUTPUT:
A Graph object.
Warning
Please use this method only if you need to copy but change the underlying data structure or weightedness. Otherwise simply do
copy(g)
instead ofg.copy()
.Warning
If
weighted
is passed and is not the weightedness of the original, then the copy will not equal the original.EXAMPLES:
sage: g = Graph({0: [0, 1, 1, 2]}, loops=True, multiedges=True, sparse=True) sage: g == copy(g) True sage: g = DiGraph({0: [0, 1, 1, 2], 1: [0, 1]}, loops=True, multiedges=True, sparse=True) sage: g == copy(g) True
Note that vertex associations are also kept:
sage: d = {0: graphs.DodecahedralGraph(), 1: graphs.FlowerSnark(), 2: graphs.MoebiusKantorGraph(), 3: graphs.PetersenGraph()} sage: T = graphs.TetrahedralGraph() sage: T.set_vertices(d) sage: T2 = copy(T) sage: T2.get_vertex(0) Dodecahedron: Graph on 20 vertices
Notice that the copy is at least as deep as the objects:
sage: T2.get_vertex(0) is T.get_vertex(0) False
Examples of the keywords in use:
sage: G = graphs.CompleteGraph(9) sage: H = G.copy() sage: H == G; H is G True False sage: G1 = G.copy(sparse=True) sage: G1 == G True sage: G1 is G False sage: G2 = copy(G) sage: G2 is G False
Argument
weighted
affects the equality class:sage: G = graphs.CompleteGraph(5) sage: H1 = G.copy(weighted=False) sage: H2 = G.copy(weighted=True) sage: [G.weighted(), H1.weighted(), H2.weighted()] [False, False, True] sage: [G == H1, G == H2, H1 == H2] [True, False, False] sage: G.weighted(True) sage: [G == H1, G == H2, H1 == H2] [False, True, False]
 crossing_number()#
Return the crossing number of the graph.
The crossing number of a graph is the minimum number of edge crossings needed to draw the graph on a plane. It can be seen as a measure of nonplanarity; a planar graph has crossing number zero.
See the Wikipedia article Crossing_number for more information.
EXAMPLES:
sage: P = graphs.PetersenGraph() sage: P.crossing_number() 2
ALGORITHM:
This is slow brute force implementation: for every \(k\) pairs of edges try adding a new vertex for a crossing point for them. If the result is not planar in any of those, try \(k+1\) pairs.
Computing the crossing number is NPhard problem.
 cycle_basis(output='vertex')#
Return a list of cycles which form a basis of the cycle space of
self
.A basis of cycles of a graph is a minimal collection of cycles (considered as sets of edges) such that the edge set of any cycle in the graph can be written as a \(Z/2Z\) sum of the cycles in the basis.
See the Wikipedia article Cycle_basis for more information.
INPUT:
output
– string (default:'vertex'
); whether every cycle is given as a list of vertices (output == 'vertex'
) or a list of edges (output == 'edge'
)
OUTPUT:
A list of lists, each of them representing the vertices (or the edges) of a cycle in a basis.
ALGORITHM:
Uses the NetworkX library for graphs without multiple edges.
Otherwise, by the standard algorithm using a spanning tree.
EXAMPLES:
A cycle basis in Petersen’s Graph
sage: g = graphs.PetersenGraph() sage: g.cycle_basis() # needs networkx [[1, 6, 8, 5, 0], [4, 9, 6, 8, 5, 0], [7, 9, 6, 8, 5], [4, 3, 8, 5, 0], [1, 2, 3, 8, 5, 0], [7, 2, 3, 8, 5]]
One can also get the result as a list of lists of edges:
sage: g.cycle_basis(output='edge') # needs networkx [[(1, 6, None), (6, 8, None), (8, 5, None), (5, 0, None), (0, 1, None)], [(4, 9, None), (9, 6, None), (6, 8, None), (8, 5, None), (5, 0, None), (0, 4, None)], [(7, 9, None), (9, 6, None), (6, 8, None), (8, 5, None), (5, 7, None)], [(4, 3, None), (3, 8, None), (8, 5, None), (5, 0, None), (0, 4, None)], [(1, 2, None), (2, 3, None), (3, 8, None), (8, 5, None), (5, 0, None), (0, 1, None)], [(7, 2, None), (2, 3, None), (3, 8, None), (8, 5, None), (5, 7, None)]]
Checking the given cycles are algebraically free:
sage: g = graphs.RandomGNP(30, .4) # needs networkx sage: basis = g.cycle_basis() # needs networkx
Building the space of (directed) edges over \(Z/2Z\). On the way, building a dictionary associating a unique vector to each undirected edge:
sage: m = g.size() sage: edge_space = VectorSpace(FiniteField(2), m) # needs sage.modules sage.rings.finite_rings sage: edge_vector = dict(zip(g.edges(labels=False, sort=False), # needs sage.modules sage.rings.finite_rings ....: edge_space.basis())) sage: for (u, v), vec in list(edge_vector.items()): # needs sage.modules sage.rings.finite_rings ....: edge_vector[(v, u)] = vec
Defining a lambda function associating a vector to the vertices of a cycle:
sage: vertices_to_edges = lambda x: zip(x, x[1:] + [x[0]]) sage: cycle_to_vector = lambda x: sum(edge_vector[e] for e in vertices_to_edges(x))
Finally checking the cycles are a free set:
sage: basis_as_vectors = [cycle_to_vector(_) for _ in basis] # needs networkx sage.modules sage.rings.finite_rings sage: edge_space.span(basis_as_vectors).rank() == len(basis) # needs networkx sage.modules sage.rings.finite_rings True
For undirected graphs with multiple edges:
sage: G = Graph([(0, 2, 'a'), (0, 2, 'b'), (0, 1, 'c'), (1, 2, 'd')], ....: multiedges=True) sage: G.cycle_basis() # needs networkx [[0, 2], [2, 1, 0]] sage: G.cycle_basis(output='edge') # needs networkx [[(0, 2, 'b'), (2, 0, 'a')], [(2, 1, 'd'), (1, 0, 'c'), (0, 2, 'a')]] sage: H = Graph([(1, 2), (2, 3), (2, 3), (3, 4), (1, 4), ....: (1, 4), (4, 5), (5, 6), (4, 6), (6, 7)], multiedges=True) sage: H.cycle_basis() # needs networkx [[1, 4], [2, 3], [4, 3, 2, 1], [6, 5, 4]]
Disconnected graph:
sage: G.add_cycle(["Hey", "Wuuhuu", "Really ?"]) sage: [sorted(c) for c in G.cycle_basis()] # needs networkx [['Hey', 'Really ?', 'Wuuhuu'], [0, 2], [0, 1, 2]] sage: [sorted(c) for c in G.cycle_basis(output='edge')] # needs networkx [[('Hey', 'Wuuhuu', None), ('Really ?', 'Hey', None), ('Wuuhuu', 'Really ?', None)], [(0, 2, 'a'), (2, 0, 'b')], [(0, 2, 'b'), (1, 0, 'c'), (2, 1, 'd')]]
Graph that allows multiple edges but does not contain any:
sage: G = graphs.CycleGraph(3) sage: G.allow_multiple_edges(True) sage: G.cycle_basis() # needs networkx [[2, 1, 0]]
Not yet implemented for directed graphs:
sage: G = DiGraph([(0, 2, 'a'), (0, 1, 'c'), (1, 2, 'd')]) sage: G.cycle_basis() # needs networkx Traceback (most recent call last): ... NotImplementedError: not implemented for directed graphs
 degree(vertices=None, labels=False)#
Return the degree (in + out for digraphs) of a vertex or of vertices.
INPUT:
vertices
– a vertex or an iterable container of vertices (default:None
); ifvertices
is a single vertex, returns the number of neighbors of that vertex. Ifvertices
is an iterable container of vertices, returns a list of degrees. Ifvertices
isNone
, same as listing all vertices.labels
– boolean (default:False
); whenTrue
, return a dictionary mapping each vertex invertices
to its degree. Otherwise, return the degree of a single vertex or a list of the degrees of each vertex invertices
OUTPUT:
When
vertices
is a single vertex andlabels
isFalse
, returns the degree of that vertex as an integerWhen
vertices
is an iterable container of vertices (orNone
) andlabels
isFalse
, returns a list of integers. The \(i\)th value is the degree of the \(i\)th vertex in the listvertices
. Whenvertices
isNone
, the \(i\)th value is the degree of \(i\)th vertex in the orderinglist(self)
, which might be different from the ordering of the vertices given byg.vertices(sort=True)
.When
labels
isTrue
, returns a dictionary mapping each vertex invertices
to its degree
EXAMPLES:
sage: P = graphs.PetersenGraph() sage: P.degree(5) 3
sage: K = graphs.CompleteGraph(9) sage: K.degree() [8, 8, 8, 8, 8, 8, 8, 8, 8]
sage: D = DiGraph({0: [1, 2, 3], 1: [0, 2], 2: [3], 3: [4], 4: [0,5], 5: [1]}) sage: D.degree(vertices=[0, 1, 2], labels=True) {0: 5, 1: 4, 2: 3} sage: D.degree() [5, 4, 3, 3, 3, 2]
When
vertices=None
andlabels=False
, the \(i\)th value of the returned list is the degree of the \(i\)th vertex in the listlist(self)
:sage: # needs sage.combinat sage: D = digraphs.DeBruijn(4, 2) sage: D.delete_vertex('20') sage: print(D.degree()) [7, 7, 6, 7, 8, 8, 7, 8, 8, 7, 8, 8, 8, 7, 8] sage: print(D.degree(vertices=list(D))) [7, 7, 6, 7, 8, 8, 7, 8, 8, 7, 8, 8, 8, 7, 8] sage: print(D.degree(vertices=D.vertices(sort=False))) [7, 7, 6, 7, 8, 8, 7, 8, 8, 7, 8, 8, 8, 7, 8]
 degree_histogram()#
Return a list, whose \(i\)th entry is the frequency of degree \(i\).
EXAMPLES:
sage: G = graphs.Grid2dGraph(9, 12) sage: G.degree_histogram() [0, 0, 4, 34, 70]
sage: G = graphs.Grid2dGraph(9, 12).to_directed() sage: G.degree_histogram() [0, 0, 0, 0, 4, 0, 34, 0, 70]
 degree_iterator(vertices=None, labels=False)#
Return an iterator over the degrees of the (di)graph.
In the case of a digraph, the degree is defined as the sum of the indegree and the outdegree, i.e. the total number of edges incident to a given vertex.
INPUT:
vertices
– a vertex or an iterable container of vertices (default:None
); ifvertices
is a single vertex, the iterator will yield the number of neighbors of that vertex. Ifvertices
is an iterable container of vertices, return an iterator over the degrees of these vertices. Ifvertices
isNone
, same as listing all vertices.labels
– boolean (default:False
); whether to return an iterator over degrees (labels=False
), or over tuples(vertex, degree)
Note
The returned iterator yields values in order specified by
list(vertices)
. Whenvertices
isNone
, it yields values in the same order aslist(self)
, which might be different from the ordering of the vertices given byg.vertices(sort=True)
.EXAMPLES:
sage: G = graphs.Grid2dGraph(3, 4) sage: for i in G.degree_iterator(): ....: print(i) 2 3 3 ... 3 2 sage: for i in G.degree_iterator(labels=True): ....: print(i) ((0, 0), 2) ((0, 1), 3) ((0, 2), 3) ... ((2, 2), 3) ((2, 3), 2)
sage: D = graphs.Grid2dGraph(2,4).to_directed() sage: for i in D.degree_iterator(): ....: print(i) 4 6 ... 6 4 sage: for i in D.degree_iterator(labels=True): ....: print(i) ((0, 0), 4) ((0, 1), 6) ... ((1, 2), 6) ((1, 3), 4)
When
vertices=None
yields values in the order oflist(D)
:sage: V = list(D) sage: D = digraphs.DeBruijn(4, 2) # needs sage.combinat sage: D.delete_vertex('20') # needs sage.combinat sage: print(list(D.degree_iterator())) # needs sage.combinat [7, 7, 6, 7, 8, 8, 7, 8, 8, 7, 8, 8, 8, 7, 8] sage: print([D.degree(v) for v in D]) # needs sage.combinat [7, 7, 6, 7, 8, 8, 7, 8, 8, 7, 8, 8, 8, 7, 8]
 degree_sequence()#
Return the degree sequence of this (di)graph.
EXAMPLES:
The degree sequence of an undirected graph:
sage: g = Graph({1: [2, 5], 2: [1, 5, 3, 4], 3: [2, 5], 4: [3], 5: [2, 3]}) sage: g.degree_sequence() [4, 3, 3, 2, 2]
The degree sequence of a digraph:
sage: g = DiGraph({1: [2, 5, 6], 2: [3, 6], 3: [4, 6], 4: [6], 5: [4, 6]}) sage: g.degree_sequence() [5, 3, 3, 3, 3, 3]
Degree sequences of some common graphs:
sage: graphs.PetersenGraph().degree_sequence() [3, 3, 3, 3, 3, 3, 3, 3, 3, 3] sage: graphs.HouseGraph().degree_sequence() [3, 3, 2, 2, 2] sage: graphs.FlowerSnark().degree_sequence() [3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3]
 degree_to_cell(vertex, cell)#
Returns the number of edges from vertex to an edge in cell. In the case of a digraph, returns a tuple (in_degree, out_degree).
EXAMPLES:
sage: G = graphs.CubeGraph(3) sage: cell = G.vertices(sort=True)[:3] sage: G.degree_to_cell('011', cell) 2 sage: G.degree_to_cell('111', cell) 0
sage: D = DiGraph({ 0:[1,2,3], 1:[3,4], 3:[4,5]}) sage: cell = [0,1,2] sage: D.degree_to_cell(5, cell) (0, 0) sage: D.degree_to_cell(3, cell) (2, 0) sage: D.degree_to_cell(0, cell) (0, 2)
 delete_edge(u, v=None, label=None)#
Delete the edge from
u
tov
.This method returns silently if vertices or edge does not exist.
INPUT: The following forms are all accepted:
G.delete_edge( 1, 2 )
G.delete_edge( (1, 2) )
G.delete_edges( [ (1, 2) ] )
G.delete_edge( 1, 2, ‘label’ )
G.delete_edge( (1, 2, ‘label’) )
G.delete_edges( [ (1, 2, ‘label’) ] )
EXAMPLES:
sage: G = graphs.CompleteGraph(9) sage: G.size() 36 sage: G.delete_edge( 1, 2 ) sage: G.delete_edge( (3, 4) ) sage: G.delete_edges( [ (5, 6), (7, 8) ] ) sage: G.size() 32
sage: G.delete_edge( 2, 3, 'label' ) sage: G.delete_edge( (4, 5, 'label') ) sage: G.delete_edges( [ (6, 7, 'label') ] ) sage: G.size() 32 sage: G.has_edge( (4, 5) ) # correct! True sage: G.has_edge( (4, 5, 'label') ) # correct! False
sage: C = digraphs.Complete(9) sage: C.size() 72 sage: C.delete_edge( 1, 2 ) sage: C.delete_edge( (3, 4) ) sage: C.delete_edges( [ (5, 6), (7, 8) ] ) sage: C.size() 68
sage: C.delete_edge( 2, 3, 'label' ) sage: C.delete_edge( (4, 5, 'label') ) sage: C.delete_edges( [ (6, 7, 'label') ] ) sage: C.size() # correct! 68 sage: C.has_edge( (4, 5) ) # correct! True sage: C.has_edge( (4, 5, 'label') ) # correct! False
 delete_edges(edges)#
Delete edges from an iterable container.
EXAMPLES:
sage: K12 = graphs.CompleteGraph(12) sage: K4 = graphs.CompleteGraph(4) sage: K12.size() 66 sage: K12.delete_edges(K4.edge_iterator()) sage: K12.size() 60
sage: K12 = digraphs.Complete(12) sage: K4 = digraphs.Complete(4) sage: K12.size() 132 sage: K12.delete_edges(K4.edge_iterator()) sage: K12.size() 120
 delete_multiedge(u, v)#
Delete all edges from
u
tov
.EXAMPLES:
sage: G = Graph(multiedges=True, sparse=True) sage: G.add_edges([(0, 1), (0, 1), (0, 1), (1, 2), (2, 3)]) sage: G.edges(sort=True) [(0, 1, None), (0, 1, None), (0, 1, None), (1, 2, None), (2, 3, None)] sage: G.delete_multiedge(0, 1) sage: G.edges(sort=True) [(1, 2, None), (2, 3, None)]
sage: D = DiGraph(multiedges=True, sparse=True) sage: D.add_edges([(0, 1, 1), (0, 1, 2), (0, 1, 3), (1, 0, None), (1, 2, None), (2, 3, None)]) sage: D.edges(sort=True) [(0, 1, 1), (0, 1, 2), (0, 1, 3), (1, 0, None), (1, 2, None), (2, 3, None)] sage: D.delete_multiedge(0, 1) sage: D.edges(sort=True) [(1, 0, None), (1, 2, None), (2, 3, None)]
 delete_vertex(vertex, in_order=False)#
Delete vertex, removing all incident edges.
Deleting a nonexistent vertex will raise an exception.
INPUT:
in_order
– boolean (default:False
); ifTrue
, this deletes the \(i\)th vertex in the sorted list of vertices, i.e.G.vertices(sort=True)[i]
EXAMPLES:
sage: G = Graph(graphs.WheelGraph(9)) sage: G.delete_vertex(0) sage: G.show() # needs sage.plot
sage: D = DiGraph({0: [1, 2, 3, 4, 5], 1: [2], 2: [3], 3: [4], 4: [5], 5: [1]}) sage: D.delete_vertex(0); D Digraph on 5 vertices sage: D.vertices(sort=True) [1, 2, 3, 4, 5] sage: D.delete_vertex(0) Traceback (most recent call last): ... ValueError: vertex (0) not in the graph
sage: G = graphs.CompleteGraph(4).line_graph(labels=False) sage: G.vertices(sort=True) [(0, 1), (0, 2), (0, 3), (1, 2), (1, 3), (2, 3)] sage: G.delete_vertex(0, in_order=True) sage: G.vertices(sort=True) [(0, 2), (0, 3), (1, 2), (1, 3), (2, 3)] sage: G = graphs.PathGraph(5) sage: G.set_vertices({0: 'no delete', 1: 'delete'}) sage: G.delete_vertex(1) sage: G.get_vertices() {0: 'no delete', 2: None, 3: None, 4: None} sage: G.get_pos() {0: (0, 0), 2: (2, 0), 3: (3, 0), 4: (4, 0)}
 delete_vertices(vertices)#
Delete vertices from the (di)graph taken from an iterable container of vertices.
Deleting a nonexistent vertex will raise an exception, in which case none of the vertices in
vertices
is deleted.EXAMPLES:
sage: D = DiGraph({0: [1, 2, 3, 4, 5], 1: [2], 2: [3], 3: [4], 4: [5], 5: [1]}) sage: D.delete_vertices([1, 2, 3, 4, 5]); D Digraph on 1 vertex sage: D.vertices(sort=False) [0] sage: D.delete_vertices([1]) Traceback (most recent call last): ... ValueError: vertex (1) not in the graph
 density()#
Return the density of the (di)graph.
The density of a (di)graph is defined as the number of edges divided by number of possible edges.
In the case of a multigraph, raises an error, since there is an infinite number of possible edges.
EXAMPLES:
sage: d = {0: [1,4,5], 1: [2,6], 2: [3,7], 3: [4,8], 4: [9], 5: [7, 8], 6: [8,9], 7: [9]} sage: G = Graph(d); G.density() 1/3 sage: G = Graph({0: [1, 2], 1: [0]}); G.density() 2/3 sage: G = DiGraph({0: [1, 2], 1: [0]}); G.density() 1/2
Note that there are more possible edges on a looped graph:
sage: G.allow_loops(True) sage: G.density() 1/3
 depth_first_search(start, ignore_direction=False, neighbors=None, edges=False)#
Return an iterator over the vertices in a depthfirst ordering.
INPUT:
start
– vertex or list of vertices from which to start the traversalignore_direction
– boolean (default:False
); only applies to directed graphs. IfTrue
, searches across edges in either direction.neighbors
– function (default:None
); a function that inputs a vertex and return a list of vertices. For an undirected graph,neighbors
is by default theneighbors()
function. For a digraph, theneighbors
function defaults to theneighbor_out_iterator()
function of the graph.edges
– boolean (default:False
); whether to return the edges of the DFS tree in the order of visit or the vertices (default). Edges are directed in root to leaf orientation of the tree.
See also
breadth_first_search
– breadthfirst search for fast compiled graphs.depth_first_search
– depthfirst search for fast compiled graphs.
EXAMPLES:
sage: G = Graph({0: [1], 1: [2], 2: [3], 3: [4], 4: [0]}) sage: list(G.depth_first_search(0)) [0, 4, 3, 2, 1]
By default, the edge direction of a digraph is respected, but this can be overridden by the
ignore_direction
parameter:sage: D = DiGraph({0: [1, 2, 3], 1: [4, 5], 2: [5], 3: [6], 5: [7], 6: [7], 7: [0]}) sage: list(D.depth_first_search(0)) [0, 3, 6, 7, 2, 5, 1, 4] sage: list(D.depth_first_search(0, ignore_direction=True)) [0, 7, 6, 3, 5, 2, 1, 4]
Multiple starting vertices can be specified in a list:
sage: D = DiGraph({0: [1, 2, 3], 1: [4, 5], 2: [5], 3: [6], 5: [7], 6: [7], 7: [0]}) sage: list(D.depth_first_search([0])) [0, 3, 6, 7, 2, 5, 1, 4] sage: list(D.depth_first_search([0, 6])) [0, 3, 6, 7, 2, 5, 1, 4]
More generally, you can specify a
neighbors
function. For example, you can traverse the graph backwards by settingneighbors
to be theneighbors_in()
function of the graph:sage: D = digraphs.Path(10) sage: D.add_path([22, 23, 24, 5]) sage: D.add_path([5, 33, 34, 35]) sage: list(D.depth_first_search(5, neighbors=D.neighbors_in)) [5, 4, 3, 2, 1, 0, 24, 23, 22] sage: list(D.breadth_first_search(5, neighbors=D.neighbors_in)) [5, 24, 4, 23, 3, 22, 2, 1, 0] sage: list(D.depth_first_search(5, neighbors=D.neighbors_out)) [5, 6, 7, 8, 9, 33, 34, 35] sage: list(D.breadth_first_search(5, neighbors=D.neighbors_out)) [5, 33, 6, 34, 7, 35, 8, 9]
You can get edges of the DFS tree instead of the vertices using the
edges
parameter:sage: D = digraphs.Path(5) sage: list(D.depth_first_search(2, edges=True)) [(2, 3), (3, 4)] sage: list(D.depth_first_search(2, edges=True, ignore_direction=True)) [(2, 3), (3, 4), (2, 1), (1, 0)]
 disjoint_routed_paths(pairs, solver, verbose=None, integrality_tolerance=0)#
Return a set of disjoint routed paths.
Given a set of pairs \((s_i,t_i)\), a set of disjoint routed paths is a set of \(s_it_i\) paths which can intersect at their endpoints and are vertexdisjoint otherwise.
INPUT:
pairs
– list of pairs of verticessolver
– string (default:None
); specify a Mixed Integer Linear Programming (MILP) solver to be used. If set toNone
, the default one is used. For more information on MILP solvers and which default solver is used, see the methodsolve
of the classMixedIntegerLinearProgram
.verbose
– integer (default:0
); sets the level of verbosity. Set to 0 by default, which means quiet.integrality_tolerance
– float; parameter for use with MILP solvers over an inexact base ring; seeMixedIntegerLinearProgram.get_values()
.
EXAMPLES:
Given a grid, finding two vertexdisjoint paths, the first one from the topleft corner to the bottomleft corner, and the second from the topright corner to the bottomright corner is easy:
sage: g = graphs.Grid2dGraph(5, 5) sage: p1,p2 = g.disjoint_routed_paths([((0, 0), (0, 4)), ((4, 4), (4, 0))]) # needs sage.numerical.mip
Though there is obviously no solution to the problem in which each corner is sending information to the opposite one:
sage: g = graphs.Grid2dGraph(5, 5) sage: p1,p2 = g.disjoint_routed_paths([((0, 0), (4, 4)), ((0, 4), (4, 0))]) # needs sage.numerical.mip Traceback (most recent call last): ... EmptySetError: the disjoint routed paths do not exist
 disjoint_union(other, labels='pairs', immutable=None)#
Return the disjoint union of
self
andother
.INPUT:
labels
– string (default:'pairs'
); if set to'pairs'
, each elementv
in the first graph will be named(0, v)
and each elementu
inother
will be named(1, u)
in the result. If set to'integers'
, the elements of the result will be relabeled with consecutive integers.immutable
– boolean (default:None
); whether to create a mutable/immutable disjoint union.immutable=None
(default) means that the graphs and their disjoint union will behave the same way.
EXAMPLES:
sage: G = graphs.CycleGraph(3) sage: H = graphs.CycleGraph(4) sage: J = G.disjoint_union(H); J Cycle graph disjoint_union Cycle graph: Graph on 7 vertices sage: J.vertices(sort=True) [(0, 0), (0, 1), (0, 2), (1, 0), (1, 1), (1, 2), (1, 3)] sage: J = G.disjoint_union(H, labels='integers'); J Cycle graph disjoint_union Cycle graph: Graph on 7 vertices sage: J.vertices(sort=True) [0, 1, 2, 3, 4, 5, 6] sage: (G + H).vertices(sort=True) # '+'operator is a shortcut [0, 1, 2, 3, 4, 5, 6]
sage: G = Graph({'a': ['b']}) sage: G.name("Custom path") sage: G.name() 'Custom path' sage: H = graphs.CycleGraph(3) sage: J = G.disjoint_union(H); J Custom path disjoint_union Cycle graph: Graph on 5 vertices sage: J.vertices(sort=True) [(0, 'a'), (0, 'b'), (1, 0), (1, 1), (1, 2)]
 disjunctive_product(other)#
Return the disjunctive product of
self
andother
.The disjunctive product of \(G\) and \(H\) is the graph \(L\) with vertex set \(V(L)=V(G)\times V(H)\), and \(((u,v), (w,x))\) is an edge iff either :
\((u, w)\) is an edge of \(G\), or
\((v, x)\) is an edge of \(H\).
EXAMPLES:
sage: Z = graphs.CompleteGraph(2) sage: D = Z.disjunctive_product(Z); D Graph on 4 vertices sage: D.plot() # long time Graphics object consisting of 11 graphics primitives
sage: C = graphs.CycleGraph(5) sage: D = C.disjunctive_product(Z); D Graph on 10 vertices sage: D.plot() # long time Graphics object consisting of 46 graphics primitives
 distance(u, v, by_weight=False, weight_function=None, check_weight=True)#
Return the (directed) distance from
u
tov
in the (di)graph.The distance is the length of the shortest path from
u
tov
.This method simply calls
shortest_path_length()
, with default arguments. For more information, and for more option, we refer to that method.INPUT:
by_weight
– boolean (default:False
); ifFalse
, the graph is considered unweighted, and the distance is the number of edges in a shortest path. IfTrue
, the distance is the sum of edge labels (which are assumed to be numbers).weight_function
– function (default:None
); a function that takes as input an edge(u, v, l)
and outputs its weight. If notNone
,by_weight
is automatically set toTrue
. IfNone
andby_weight
isTrue
, we use the edge labell
, ifl
is notNone
, else1
as a weight.check_weight
– boolean (default:True
); whether to check that theweight_function
outputs a number for each edge.
EXAMPLES:
sage: G = graphs.CycleGraph(9) sage: G.distance(0,1) 1 sage: G.distance(0,4) 4 sage: G.distance(0,5) 4 sage: G = Graph({0:[], 1:[]}) sage: G.distance(0,1) +Infinity sage: G = Graph({ 0: {1: 1}, 1: {2: 1}, 2: {3: 1}, 3: {4: 2}, 4: {0: 2}}, sparse = True) sage: G.plot(edge_labels=True).show() # long time sage: G.distance(0, 3) 2 sage: G.distance(0, 3, by_weight=True) 3
 distance_all_pairs(by_weight=False, algorithm=None, weight_function=None, check_weight=True)#
Return the distances between all pairs of vertices.
INPUT:
by_weight
boolean (default: \(False`\)); ifTrue
, the edges in the graph are weighted; ifFalse
, all edges have weight 1.algorithm
– string (default:None
); one of the following algorithms:'BFS'
: the computation is done through a BFS centered on each vertex successively. Works only ifby_weight==False
.'FloydWarshallCython'
: the Cython implementation of the FloydWarshall algorithm. Works only ifby_weight==False
.'FloydWarshallPython'
: the Python implementation of the FloydWarshall algorithm. Works also with weighted graphs, even with negative weights (but no negative cycle is allowed).'Dijkstra_NetworkX'
: the Dijkstra algorithm, implemented in NetworkX. It works with weighted graphs, but no negative weight is allowed.'Dijkstra_Boost'
: the Dijkstra algorithm, implemented in Boost (works only with positive weights).'Johnson_Boost'
: the Johnson algorithm, implemented in Boost (works also with negative weights, if there is no negative cycle).None
(default): Sage chooses the best algorithm:'BFS'
ifby_weight
isFalse
,'Dijkstra_Boost'
if all weights are positive,'FloydWarshallCython'
otherwise.
weight_function
– function (default:None
); a function that takes as input an edge(u, v, l)
and outputs its weight. If notNone
,by_weight
is automatically set toTrue
. IfNone
andby_weight
isTrue
, we use the edge labell
, ifl
is notNone
, else1
as a weight.check_weight
– boolean (default:True
); whether to check that theweight_function
outputs a number for each edge.
OUTPUT:
A doubly indexed dictionary
Note
There is a Cython version of this method that is usually much faster for large graphs, as most of the time is actually spent building the final double dictionary. Everything on the subject is to be found in the
distances_all_pairs
module.Note
This algorithm simply calls
GenericGraph.shortest_path_all_pairs()
, and we suggest to look at that method for more information and examples.EXAMPLES:
The Petersen Graph:
sage: g = graphs.PetersenGraph() sage: print(g.distance_all_pairs()) {0: {0: 0, 1: 1, 2: 2, 3: 2, 4: 1, 5: 1, 6: 2, 7: 2, 8: 2, 9: 2}, 1: {0: 1, 1: 0, 2: 1, 3: 2, 4: 2, 5: 2, 6: 1, 7: 2, 8: 2, 9: 2}, 2: {0: 2, 1: 1, 2: 0, 3: 1, 4: 2, 5: 2, 6: 2, 7: 1, 8: 2, 9: 2}, 3: {0: 2, 1: 2, 2: 1, 3: 0, 4: 1, 5: 2, 6: 2, 7: 2, 8: 1, 9: 2}, 4: {0: 1, 1: 2, 2: 2, 3: 1, 4: 0, 5: 2, 6: 2, 7: 2, 8: 2, 9: 1}, 5: {0: 1, 1: 2, 2: 2, 3: 2, 4: 2, 5: 0, 6: 2, 7: 1, 8: 1, 9: 2}, 6: {0: 2, 1: 1, 2: 2, 3: 2, 4: 2, 5: 2, 6: 0, 7: 2, 8: 1, 9: 1}, 7: {0: 2, 1: 2, 2: 1, 3: 2, 4: 2, 5: 1, 6: 2, 7: 0, 8: 2, 9: 1}, 8: {0: 2, 1: 2, 2: 2, 3: 1, 4: 2, 5: 1, 6: 1, 7: 2, 8: 0, 9: 2}, 9: {0: 2, 1: 2, 2: 2, 3: 2, 4: 1, 5: 2, 6: 1, 7: 1, 8: 2, 9: 0}}
Testing on Random Graphs:
sage: g = graphs.RandomGNP(20,.3) sage: distances = g.distance_all_pairs() sage: all((g.distance(0,v) == Infinity and v not in distances[0]) or ....: g.distance(0,v) == distances[0][v] for v in g) True
 distance_matrix(vertices, base_ring=None, **kwds)#
Return the distance matrix of (di)graph.
The (di)graph is expected to be (strongly) connected.
The distance matrix of a (strongly) connected (di)graph is a matrix whose rows and columns are by default (
vertices == None
) indexed with the positions of the vertices of the (di)graph in the orderingvertices()
. Whenvertices
is set, the position of the vertices in this ordering is used. The intersection of row \(i\) and column \(j\) contains the shortest path distance from the vertex at the \(i\)th position to the vertex at the \(j\)th position.Note that even when the vertices are consecutive integers starting from one, usually the vertex is not equal to its index.
INPUT:
vertices
– list (default:None
); the ordering of the vertices defining how they should appear in the matrix. By default, the ordering given byvertices()
is used. Becausevertices()
only works if the vertices can be sorted, usingvertices
is useful when working with possibly nonsortable objects in Python 3.base_ring
– a ring (default: determined from the weights); the base ring of the matrix space to use.**kwds
– other keywords to pass to the subfunctiondistance_all_pairs()
or tomatrix()
EXAMPLES:
sage: d = DiGraph({1: [2, 3], 2: [3], 3: [4], 4: [1]}) sage: d.distance_matrix() # needs sage.modules [0 1 1 2] [3 0 1 2] [2 3 0 1] [1 2 2 0] sage: d.distance_matrix(vertices=[4, 3, 2, 1]) # needs sage.modules [0 2 2 1] [1 0 3 2] [2 1 0 3] [2 1 1 0] sage: G = graphs.CubeGraph(3) sage: G.distance_matrix() # needs sage.modules [0 1 1 2 1 2 2 3] [1 0 2 1 2 1 3 2] [1 2 0 1 2 3 1 2] [2 1 1 0 3 2 2 1] [1 2 2 3 0 1 1 2] [2 1 3 2 1 0 2 1] [2 3 1 2 1 2 0 1] [3 2 2 1 2 1 1 0]
The well known result of Graham and Pollak states that the determinant of the distance matrix of any tree of order \(n\) is \((1)^{n1}(n1)2^{n2}\):
sage: all(T.distance_matrix().det() == (1)^9*(9)*2^8 # needs sage.modules ....: for T in graphs.trees(10)) True
See also
distance_all_pairs()
– computes the distance between any two vertices.
 distances_distribution(G)#
Return the distances distribution of the (di)graph in a dictionary.
This method ignores all edge labels, so that the distance considered is the topological distance.
OUTPUT:
A dictionary
d
such that the number of pairs of vertices at distancek
(if any) is equal to \(d[k] \cdot V(G) \cdot (V(G)1)\).Note
We consider that two vertices that do not belong to the same connected component are at infinite distance, and we do not take the trivial pairs of vertices \((v, v)\) at distance \(0\) into account. Empty (di)graphs and (di)graphs of order 1 have no paths and so we return the empty dictionary
{}
.EXAMPLES:
An empty Graph:
sage: g = Graph() sage: g.distances_distribution() {}
A Graph of order 1:
sage: g = Graph() sage: g.add_vertex(1) sage: g.distances_distribution() {}
A Graph of order 2 without edge:
sage: g = Graph() sage: g.add_vertices([1,2]) sage: g.distances_distribution() {+Infinity: 1}
The Petersen Graph:
sage: g = graphs.PetersenGraph() sage: g.distances_distribution() {1: 1/3, 2: 2/3}
A graph with multiple disconnected components:
sage: g = graphs.PetersenGraph() sage: g.add_edge('good','wine') sage: g.distances_distribution() {1: 8/33, 2: 5/11, +Infinity: 10/33}
The de Bruijn digraph dB(2,3):
sage: D = digraphs.DeBruijn(2,3) # needs sage.combinat sage: D.distances_distribution() # needs sage.combinat {1: 1/4, 2: 11/28, 3: 5/14}
 dominating_set(g, k, independent=1, total=False, value_only=False, solver=False, verbose=None, integrality_tolerance=0)#
Return a minimum distance\(k\) dominating set of the graph.
A minimum dominating set \(S\) of a graph \(G\) is a set of its vertices of minimal cardinality such that any vertex of \(G\) is in \(S\) or has one of its neighbors in \(S\). See the Wikipedia article Dominating_set.
A minimum distance\(k\) dominating set is a set \(S\) of vertices of \(G\) of minimal cardinality such that any vertex of \(G\) is in \(S\) or at distance at most \(k\) from a vertex in \(S\). A distance\(0\) dominating set is the set of vertices itself, and when \(k\) is the radius of the graph, any vertex dominates all the other vertices.
As an optimization problem, it can be expressed as follows, where \(N^k(u)\) denotes the set of vertices at distance at most \(k\) from \(u\) (the set of neighbors when \(k=1\)):
\[\begin{split}\mbox{Minimize : }&\sum_{v\in G} b_v\\ \mbox{Such that : }&\forall v \in G, b_v+\sum_{u \in N^k(v)} b_u\geq 1\\ &\forall x\in G, b_x\mbox{ is a binary variable}\end{split}\]INPUT:
k
– a nonnegative integer (default:1
); the domination distanceindependent
– boolean (default:False
); whenTrue
, computes a minimum independent dominating set, that is a minimum dominating set that is also an independent set (see alsoindependent_set()
)total
– boolean (default:False
); whenTrue
, computes a total dominating set (see the See the Wikipedia article Dominating_set)value_only
– boolean (default:False
); whether to only return the cardinality of the computed dominating set, or to return its list of vertices (default)solver
– string (default:None
); specify a Mixed Integer Linear Programming (MILP) solver to be used. If set toNone
, the default one is used. For more information on MILP solvers and which default solver is used, see the methodsolve
of the classMixedIntegerLinearProgram
.verbose
– integer (default:0
); sets the level of verbosity. Set to 0 by default, which means quiet.integrality_tolerance
– float; parameter for use with MILP solvers over an inexact base ring; seeMixedIntegerLinearProgram.get_values()
.
EXAMPLES:
A basic illustration on a
PappusGraph
:sage: g = graphs.PappusGraph() sage: g.dominating_set(value_only=True) # needs sage.numerical.mip 5
If we build a graph from two disjoint stars, then link their centers we will find a difference between the cardinality of an independent set and a stable independent set:
sage: g = 2 * graphs.StarGraph(5) sage: g.add_edge(0, 6) sage: len(g.dominating_set()) # needs sage.numerical.mip 2 sage: len(g.dominating_set(independent=True)) # needs sage.numerical.mip 6
The total dominating set of the Petersen graph has cardinality 4:
sage: G = graphs.PetersenGraph() sage: G.dominating_set(total=True, value_only=True) # needs sage.numerical.mip 4
The dominating set is calculated for both the directed and undirected graphs (modification introduced in github issue #17905):
sage: g = digraphs.Path(3) sage: g.dominating_set(value_only=True) # needs sage.numerical.mip 2 sage: g = graphs.PathGraph(3) sage: g.dominating_set(value_only=True) # needs sage.numerical.mip 1
Cardinality of distance\(k\) dominating sets:
sage: G = graphs.PetersenGraph() sage: [G.dominating_set(k=k, value_only=True) for k in range(G.radius() + 1)] # needs sage.numerical.mip [10, 3, 1] sage: G = graphs.PathGraph(5) sage: [G.dominating_set(k=k, value_only=True) for k in range(G.radius() + 1)] # needs sage.numerical.mip [5, 2, 1]
 dominating_sets(g, k, independent=1, total=False, solver=False, verbose=None, integrality_tolerance=0)#
Return an iterator over the minimum distance\(k\) dominating sets of the graph.
A minimum dominating set \(S\) of a graph \(G\) is a set of its vertices of minimal cardinality such that any vertex of \(G\) is in \(S\) or has one of its neighbors in \(S\). See the Wikipedia article Dominating_set.
A minimum distance\(k\) dominating set is a set \(S\) of vertices of \(G\) of minimal cardinality such that any vertex of \(G\) is in \(S\) or at distance at most \(k\) from a vertex in \(S\). A distance\(0\) dominating set is the set of vertices itself, and when \(k\) is the radius of the graph, any vertex dominates all the other vertices.
As an optimization problem, it can be expressed as follows, where \(N^k(u)\) denotes the set of vertices at distance at most \(k\) from \(u\) (the set of neighbors when \(k=1\)):
\[\begin{split}\mbox{Minimize : }&\sum_{v\in G} b_v\\ \mbox{Such that : }&\forall v \in G, b_v+\sum_{u \in N^k(v)} b_u\geq 1\\ &\forall x\in G, b_x\mbox{ is a binary variable}\end{split}\]We use constraints generation to iterate over the minimum distance\(k\) dominating sets. That is, after reporting a solution, we add a constraint to discard it and solve the problem again until no more solution can be found.
INPUT:
k
– a nonnegative integer (default:1
); the domination distanceindependent
– boolean (default:False
); whenTrue
, computes minimum independent dominating sets, that is minimum dominating sets that are also independent sets (see alsoindependent_set()
)total
– boolean (default:False
); whenTrue
, computes total dominating sets (see the See the Wikipedia article Dominating_set)solver
– string (default:None
); specify a Mixed Integer Linear Programming (MILP) solver to be used. If set toNone
, the default one is used. For more information on MILP solvers and which default solver is used, see the methodsolve
of the classMixedIntegerLinearProgram
.verbose
– integer (default:0
); sets the level of verbosity. Set to 0 by default, which means quiet.integrality_tolerance
– float; parameter for use with MILP solvers over an inexact base ring; seeMixedIntegerLinearProgram.get_values()
.
EXAMPLES:
Number of distance\(k\) dominating sets of a Path graph of order 10:
sage: g = graphs.PathGraph(10) sage: [sum(1 for _ in g.dominating_sets(k=k)) for k in range(11)] # needs sage.numerical.mip [1, 13, 1, 13, 25, 2, 4, 6, 8, 10, 10]
If we build a graph from two disjoint stars, then link their centers we will find a difference between the cardinality of an independent set and a stable independent set:
sage: g = 2 * graphs.StarGraph(5) sage: g.add_edge(0, 6) sage: [sum(1 for _ in g.dominating_sets(k=k)) for k in range(11)] # needs sage.numerical.mip [1, 1, 2, 12, 12, 12, 12, 12, 12, 12, 12]
The total dominating set of the Petersen graph has cardinality 4:
sage: G = graphs.PetersenGraph() sage: G.dominating_set(total=True, value_only=True) # needs sage.numerical.mip 4 sage: sorted(G.dominating_sets(k=1)) # needs sage.numerical.mip [[0, 2, 6], [0, 3, 9], [0, 7, 8], [1, 3, 7], [1, 4, 5], [1, 8, 9], [2, 4, 8], [2, 5, 9], [3, 5, 6], [4, 6, 7]]
Independent distance\(k\) dominating sets of a Path graph:
sage: # needs sage.numerical.mip sage: G = graphs.PathGraph(6) sage: sorted(G.dominating_sets(k=1, independent=True)) [[1, 4]] sage: sorted(G.dominating_sets(k=2, independent=True)) [[0, 3], [0, 4], [0, 5], [1, 3], [1, 4], [1, 5], [2, 4], [2, 5]] sage: sorted(G.dominating_sets(k=3, independent=True)) [[2], [3]]
The dominating set is calculated for both the directed and undirected graphs (modification introduced in github issue #17905):
sage: # needs sage.numerical.mip sage: g = digraphs.Path(3) sage: g.dominating_set(value_only=True) 2 sage: list(g.dominating_sets()) [[0, 1], [0, 2]] sage: list(g.dominating_sets(k=2)) [[0]] sage: g = graphs.PathGraph(3) sage: g.dominating_set(value_only=True) 1 sage: next(g.dominating_sets()) [1]
 dominator_tree(g, root, return_dict=False, reverse=False)#
Use Boost to compute the dominator tree of
g
, rooted atroot
.A node \(d\) dominates a node \(n\) if every path from the entry node
root
to \(n\) must go through \(d\). The immediate dominator of a node \(n\) is the unique node that strictly dominates \(n\) but does not dominate any other node that dominates \(n\). A dominator tree is a tree where each node’s children are those nodes it immediately dominates. For more information, see the Wikipedia article Dominator_(graph_theory).If the graph is connected and undirected, the parent of a vertex \(v\) is:
the root if \(v\) is in the same biconnected component as the root;
the first cut vertex in a path from \(v\) to the root, otherwise.
If the graph is not connected, the dominator tree of the whole graph is equal to the dominator tree of the connected component of the root.
If the graph is directed, computing a dominator tree is more complicated, and it needs time \(O(m\log m)\), where \(m\) is the number of edges. The implementation provided by Boost is the most general one, so it needs time \(O(m\log m)\) even for undirected graphs.
INPUT:
g
– the input Sage (Di)Graphroot
– the root of the dominator treereturn_dict
– boolean (default:False
); ifTrue
, the function returns a dictionary associating to each vertex its parent in the dominator tree. IfFalse
(default), it returns the whole tree, as aGraph
or aDiGraph
.reverse
– boolean (default:False
); when set toTrue
, computes the dominator tree in the reverse graph
OUTPUT:
The dominator tree, as a graph or as a dictionary, depending on the value of
return_dict
. If the output is a dictionary, it will containNone
in correspondence ofroot
and of vertices that are not reachable fromroot
. If the output is a graph, it will not contain vertices that are not reachable fromroot
.EXAMPLES:
An undirected grid is biconnected, and its dominator tree is a star (everyone’s parent is the root):
sage: g = graphs.GridGraph([2,2]).dominator_tree((0,0)) sage: g.to_dictionary() {(0, 0): [(0, 1), (1, 0), (1, 1)], (0, 1): [(0, 0)], (1, 0): [(0, 0)], (1, 1): [(0, 0)]}
If the graph is made by two 3cycles \(C_1,C_2\) connected by an edge \((v,w)\), with \(v \in C_1\), \(w \in C_2\), the cut vertices are \(v\) and \(w\), the biconnected components are \(C_1\), \(C_2\), and the edge \((v,w)\). If the root is in \(C_1\), the parent of each vertex in \(C_1\) is the root, the parent of \(w\) is \(v\), and the parent of each vertex in \(C_2\) is \(w\):
sage: G = 2 * graphs.CycleGraph(3) sage: v = 0 sage: w = 3 sage: G.add_edge(v,w) sage: G.dominator_tree(1, return_dict=True) {0: 1, 1: None, 2: 1, 3: 0, 4: 3, 5: 3}
An example with a directed graph:
sage: g = digraphs.Circuit(10).dominator_tree(5) sage: g.to_dictionary() {0: [1], 1: [2], 2: [3], 3: [4], 4: [], 5: [6], 6: [7], 7: [8], 8: [9], 9: [0]} sage: g = digraphs.Circuit(10).dominator_tree(5, reverse=True) sage: g.to_dictionary() {0: [9], 1: [0], 2: [1], 3: [2], 4: [3], 5: [4], 6: [], 7: [6], 8: [7], 9: [8]}
If the output is a dictionary:
sage: graphs.GridGraph([2,2]).dominator_tree((0,0), return_dict=True) {(0, 0): None, (0, 1): (0, 0), (1, 0): (0, 0), (1, 1): (0, 0)}
 edge_boundary(vertices1, vertices2=None, labels=True, sort=False, key=None)#
Return a list of edges
(u,v,l)
withu
invertices1
andv
invertices2
.If
vertices2
isNone
, then it is set to the complement ofvertices1
.In a digraph, the external boundary of a vertex \(v\) are those vertices \(u\) with an arc \((v, u)\).
INPUT:
labels
– boolean (default:True
); ifFalse
, each edge is a tuple \((u,v)\) of verticessort
– boolean (default:False
); whether to sort the resultkey
– a function (default:None
); a function that takes an edge as its one argument and returns a value that can be used for comparisons in the sorting algorithm (we must havesort=True
)
EXAMPLES:
sage: K = graphs.CompleteBipartiteGraph(9, 3) sage: len(K.edge_boundary([0, 1, 2, 3, 4, 5, 6, 7, 8], [9, 10, 11])) 27 sage: K.size() 27
Note that the edge boundary preserves direction:
sage: K = graphs.CompleteBipartiteGraph(9, 3).to_directed() sage: len(K.edge_boundary([0, 1, 2, 3, 4, 5, 6, 7, 8], [9, 10, 11])) 27 sage: K.size() 54
sage: D = DiGraph({0: [1, 2], 3: [0]}) sage: D.edge_boundary([0], sort=True) [(0, 1, None), (0, 2, None)] sage: D.edge_boundary([0], labels=False, sort=True) [(0, 1), (0, 2)]
Using the
key
argument to order multiple edges of incomparable types (see github issue #35903):sage: G = Graph([(1, 'A', 4), (1, 2, 3)]) sage: G.edge_boundary([1], sort=True) Traceback (most recent call last): ... TypeError: unsupported operand parent(s) for <: 'Integer Ring' and '<class 'str'>' sage: G.edge_boundary([1], sort=True, key=str) [('A', 1, 4), (1, 2, 3)] sage: G.edge_boundary([1], sort=True, key=lambda e:e[2]) [(1, 2, 3), ('A', 1, 4)] sage: G.edge_boundary([1], labels=False, sort=True, key=lambda e:e[2]) Traceback (most recent call last): ... IndexError: tuple index out of range
 edge_connectivity(G, value_only=True, implementation=None, use_edge_labels=False, vertices=False, solver=None, verbose=0, integrality_tolerance=0.001)#
Return the edge connectivity of the graph.
For more information, see the Wikipedia article Connectivity_(graph_theory).
Note
When the graph is a directed graph, this method actually computes the strong connectivity, (i.e. a directed graph is strongly \(k\)connected if there are \(k\) disjoint paths between any two vertices \(u, v\)). If you do not want to consider strong connectivity, the best is probably to convert your
DiGraph
object to aGraph
object, and compute the connectivity of this other graph.INPUT:
G
– the input Sage (Di)Graphvalue_only
– boolean (default:True
)When set to
True
(default), only the value is returned.When set to
False
, both the value and a minimum vertex cut are returned.
implementation
– string (default:None
); selects an implementation:None
(default) – selects the best implementation available"boost"
– use the Boost graph library (which is much more efficient). It is not available whenedge_labels=True
, and it is unreliable for directed graphs (see github issue #18753).
 
"Sage"
– use Sage’s implementation based on integer linear programming
use_edge_labels
– boolean (default:False
)When set to
True
, computes a weighted minimum cut where each edge has a weight defined by its label. (If an edge has no label, \(1\) is assumed.). Impliesboost
=False
.When set to
False
, each edge has weight \(1\).
vertices
– boolean (default:False
)When set to
True
, also returns the two sets of vertices that are disconnected by the cut. Impliesvalue_only=False
.
solver
– string (default:None
); specify a Mixed Integer Linear Programming (MILP) solver to be used. If set toNone
, the default one is used. For more information on MILP solvers and which default solver is used, see the methodsolve
of the classMixedIntegerLinearProgram
.verbose
– integer (default:0
); sets the level of verbosity. Set to 0 by default, which means quiet.integrality_tolerance
– float; parameter for use with MILP solvers over an inexact base ring; seeMixedIntegerLinearProgram.get_values()
.
EXAMPLES:
A basic application on the PappusGraph:
sage: from sage.graphs.connectivity import edge_connectivity sage: g = graphs.PappusGraph() sage: edge_connectivity(g) 3 sage: g.edge_connectivity() 3
The edge connectivity of a complete graph is its minimum degree, and one of the two parts of the bipartition is reduced to only one vertex. The graph of the cut edges is isomorphic to a Star graph:
sage: g = graphs.CompleteGraph(5) sage: [ value, edges, [ setA, setB ]] = edge_connectivity(g,vertices=True) sage: value 4 sage: len(setA) == 1 or len(setB) == 1 True sage: cut = Graph() sage: cut.add_edges(edges) sage: cut.is_isomorphic(graphs.StarGraph(4)) True
Even if obviously in any graph we know that the edge connectivity is less than the minimum degree of the graph:
sage: g = graphs.RandomGNP(10,.3) sage: min(g.degree()) >= edge_connectivity(g) True
If we build a tree then assign to its edges a random value, the minimum cut will be the edge with minimum value:
sage: tree = graphs.RandomTree(10) sage: for u,v in tree.edge_iterator(labels=None): ....: tree.set_edge_label(u, v, random()) sage: minimum = min(tree.edge_labels()) sage: [_, [(_, _, l)]] = edge_connectivity(tree, value_only=False, use_edge_labels=True) sage: l == minimum True
When
value_only=True
andimplementation="sage"
, this function is optimized for small connectivity values and does not need to build a linear program.It is the case for graphs which are not connected
sage: g = 2 * graphs.PetersenGraph() sage: edge_connectivity(g, implementation="sage") 0.0
For directed graphs, the strong connectivity is tested through the dedicated function:
sage: g = digraphs.ButterflyGraph(3) sage: edge_connectivity(g, implementation="sage") 0.0
We check that the result with Boost is the same as the result without Boost:
sage: g = graphs.RandomGNP(15, .3) sage: edge_connectivity(g, implementation="boost") == edge_connectivity(g, implementation="sage") True
Boost interface also works with directed graphs:
sage: edge_connectivity(digraphs.Circuit(10), implementation="boost", vertices=True) [1, [(0, 1)], [{0}, {1, 2, 3, 4, 5, 6, 7, 8, 9}]]
However, the Boost algorithm is not reliable if the input is directed (see github issue #18753):
sage: g = digraphs.Path(3) sage: edge_connectivity(g) 0.0 sage: edge_connectivity(g, implementation="boost") 1 sage: g.add_edge(1, 0) sage: edge_connectivity(g) 0.0 sage: edge_connectivity(g, implementation="boost") 0
 edge_cut(s, t, value_only, use_edge_labels=True, vertices=False, algorithm=False, solver='FF', verbose=None, integrality_tolerance=0)#
Return a minimum edge cut between vertices \(s\) and \(t\).
A minimum edge cut between two vertices \(s\) and \(t\) of self is a set \(A\) of edges of minimum weight such that the graph obtained by removing \(A\) from the graph is disconnected. For more information, see the Wikipedia article Cut_(graph_theory).
INPUT:
s
– source vertext
– sink vertexvalue_only
– boolean (default:True
); whether to return only the weight of a minimum cut (True
) or a list of edges of a minimum cut (False
)use_edge_labels
– boolean (default:False
); whether to compute a weighted minimum edge cut where the weight of an edge is defined by its label (if an edge has no label, \(1\) is assumed), or to compute a cut of minimum cardinality (i.e., edge weights are set to 1)vertices
– boolean (default:False
); whether set toTrue
, return a list of edges in the edge cut and the two sets of vertices that are disconnected by the cutNote:
vertices=True
impliesvalue_only=False
.algorithm
– string (default:'FF'
); algorithm to use:If
algorithm = "FF"
, a Python implementation of the FordFulkerson algorithm is usedIf
algorithm = "LP"
, the problem is solved using Linear Programming.If
algorithm = "igraph"
, the igraph implementation of the GoldbergTarjan algorithm is used (only available whenigraph
is installed)If
algorithm = None
, the problem is solved using the default maximum flow algorithm (seeflow()
)
solver
– string (default:None
); specify a Mixed Integer Linear Programming (MILP) solver to be used. If set toNone
, the default one is used. For more information on MILP solvers and which default solver is used, see the methodsolve
of the classMixedIntegerLinearProgram
.verbose
– integer (default:0
); sets the level of verbosity. Set to 0 by default, which means quiet.integrality_tolerance
– float; parameter for use with MILP solvers over an inexact base ring; seeMixedIntegerLinearProgram.get_values()
.
Note
The use of Linear Programming for noninteger problems may possibly mean the presence of a (slight) numerical noise.
OUTPUT:
Real number or tuple, depending on the given arguments (examples are given below).
EXAMPLES:
A basic application in the Pappus graph:
sage: g = graphs.PappusGraph() sage: g.edge_cut(1, 2, value_only=True) 3
Or on Petersen’s graph, with the corresponding bipartition of the vertex set:
sage: g = graphs.PetersenGraph() sage: g.edge_cut(0, 3, vertices=True) [3, [(0, 1, None), (0, 4, None), (0, 5, None)], [[0], [1, 2, 3, 4, 5, 6, 7, 8, 9]]]
If the graph is a path with randomly weighted edges:
sage: g = graphs.PathGraph(15) sage: for u,v in g.edge_iterator(labels=None): ....: g.set_edge_label(u, v, random())
The edge cut between the two ends is the edge of minimum weight:
sage: minimum = min(g.edge_labels()) sage: minimum == g.edge_cut(0, 14, use_edge_labels=True) True sage: [value, [e]] = g.edge_cut(0, 14, use_edge_labels=True, value_only=False) sage: g.edge_label(e[0], e[1]) == minimum True
The two sides of the edge cut are obviously shorter paths:
sage: value,edges,[set1,set2] = g.edge_cut(0, 14, use_edge_labels=True, vertices=True) sage: g.subgraph(set1).is_isomorphic(graphs.PathGraph(len(set1))) True sage: g.subgraph(set2).is_isomorphic(graphs.PathGraph(len(set2))) True sage: len(set1) + len(set2) == g.order() True
 edge_disjoint_paths(s, t, algorithm, solver='FF', verbose=None, integrality_tolerance=False)#
Return a list of edgedisjoint paths between two vertices.
The edge version of Menger’s theorem asserts that the size of the minimum edge cut between two vertices \(s\) and`t` (the minimum number of edges whose removal disconnects \(s\) and \(t\)) is equal to the maximum number of pairwise edgeindependent paths from \(s\) to \(t\).
This function returns a list of such paths.
INPUT:
algorithm
– string (default:"FF"
); the algorithm to use among:"FF"
, a Python implementation of the FordFulkerson algorithm"LP"
, the flow problem is solved using Linear Programming
solver
– string (default:None
); specify a Mixed Integer Linear Programming (MILP) solver to be used. If set toNone
, the default one is used. For more information on MILP solvers and which default solver is used, see the methodsolve
of the classMixedIntegerLinearProgram
.Only used when \(àlgorithm`\) is
"LP"
.verbose
– integer (default:0
); sets the level of verbosity. Set to 0 by default, which means quiet.Only used when \(àlgorithm`\) is
"LP"
.integrality_tolerance
– float; parameter for use with MILP solvers over an inexact base ring; seeMixedIntegerLinearProgram.get_values()
.Only used when \(àlgorithm`\) is
"LP"
.
Note
This function is topological: it does not take the eventual weights of the edges into account.
EXAMPLES:
In a complete bipartite graph
sage: g = graphs.CompleteBipartiteGraph(2, 3) sage: g.edge_disjoint_paths(0, 1) [[0, 2, 1], [0, 3, 1], [0, 4, 1]]
 edge_disjoint_spanning_trees(k, algorithm, root=None, solver=None, verbose=None, integrality_tolerance=0)#
Return the desired number of edgedisjoint spanning trees/arborescences.
INPUT:
k
– integer; the required number of edgedisjoint spanning trees/arborescencesalgorithm
– string (default:None
); specify the algorithm to use among:"RoskindTarjan"
– use the algorithm proposed by Roskind and Tarjan [RT1985] for finding edgedisjoint spanningtrees in undirected simple graphs in time \(O(m\log{m} + k^2n^2)\)."MILP"
– use a mixed integer linear programming formulation. This is the default method for directed graphs.None
– use"RoskindTarjan"
for undirected graphs and"MILP"
for directed graphs.
root
– vertex (default:None
); root of the disjoint arborescences when the graph is directed. If set toNone
, the first vertex in the graph is picked.solver
– string (default:None
); specify a Mixed Integer Linear Programming (MILP) solver to be used. If set toNone
, the default one is used. For more information on MILP solvers and which default solver is used, see the methodsolve
of the classMixedIntegerLinearProgram
.verbose
– integer (default:0
); sets the level of verbosity. Set to 0 by default, which means quiet.integrality_tolerance
– float; parameter for use with MILP solvers over an inexact base ring; seeMixedIntegerLinearProgram.get_values()
.
ALGORITHM:
Mixed Integer Linear Program.
There are at least two possible rewritings of this method which do not use Linear Programming:
EXAMPLES:
The Petersen Graph does have a spanning tree (it is connected):
sage: g = graphs.PetersenGraph() sage: [T] = g.edge_disjoint_spanning_trees(1) # needs sage.numerical.mip sage: T.is_tree() # needs sage.numerical.mip True
Though, it does not have 2 edgedisjoint trees (as it has less than \(2(V1)\) edges):
sage: g.edge_disjoint_spanning_trees(2) # needs sage.numerical.mip Traceback (most recent call last): ... EmptySetError: this graph does not contain the required number of trees/arborescences
By Edmonds’ theorem, a graph which is \(k\)connected always has \(k\) edgedisjoint arborescences, regardless of the root we pick:
sage: # needs sage.numerical.mip sage: g = digraphs.RandomDirectedGNP(11, .3) # reduced from 30 to 11, cf. #32169 sage: k = Integer(g.edge_connectivity()) sage: while not k: ....: g = digraphs.RandomDirectedGNP(11, .3) ....: k = Integer(g.edge_connectivity()) sage: arborescences = g.edge_disjoint_spanning_trees(k) # long time (up to 15s on sage.math, 2011) sage: all(a.is_directed_acyclic() for a in arborescences) # long time True sage: all(a.is_connected() for a in arborescences) # long time True
In the undirected case, we can only ensure half of it:
sage: # needs sage.numerical.mip sage: g = graphs.RandomGNP(14, .3) # reduced from 30 to 14, see #32169 sage: while not g.is_biconnected(): ....: g = graphs.RandomGNP(14, .3) sage: k = Integer(g.edge_connectivity()) // 2 sage: trees = g.edge_disjoint_spanning_trees(k) sage: all(t.is_tree() for t in trees) True
Check the validity of the algorithms for undirected graphs:
sage: # needs sage.numerical.mip sage: g = graphs.RandomGNP(12, .7) sage: k = Integer(g.edge_connectivity()) // 2 sage: trees = g.edge_disjoint_spanning_trees(k, algorithm="MILP") sage: all(t.is_tree() for t in trees) True sage: all(g.order() == t.size() + 1 for t in trees) True sage: trees = g.edge_disjoint_spanning_trees(k, algorithm="RoskindTarjan") sage: all(t.is_tree() for t in trees) True sage: all(g.order() == t.size() + 1 for t in trees) True
Example of github issue #32169:
sage: d6 = r'[E_S?_hKIH@eos[BSg???Q@FShGC?hTHUGM?IPug?JOEYCdOzdkQGo' sage: d6 += r'@ADA@AAg?GAQW?[aIaSwHYcD@qQb@Dd?\hJTI@OHlJ_?C_OEIKoeC' sage: d6 += r'R@_BC?Q??YBFosqITEA?IvCU_' sage: G = DiGraph(d6, format='dig6') sage: G.edge_connectivity() 5 sage: G.edge_disjoint_spanning_trees(5) # long time # needs sage.numerical.mip [Digraph on 28 vertices, Digraph on 28 vertices, Digraph on 28 vertices, Digraph on 28 vertices, Digraph on 28 vertices]
Small cases:
sage: # needs sage.numerical.mip sage: Graph().edge_disjoint_spanning_trees(0) [] sage: Graph(1).edge_disjoint_spanning_trees(0) [] sage: Graph(2).edge_disjoint_spanning_trees(0) [] sage: Graph([(0, 1)]).edge_disjoint_spanning_trees(0) [] sage: Graph([(0, 1)]).edge_disjoint_spanning_trees(1) [Graph on 2 vertices] sage: Graph([(0, 1)]).edge_disjoint_spanning_trees(2) Traceback (most recent call last): ... EmptySetError: this graph does not contain the required number of trees/arborescences
 edge_iterator(vertices=None, labels=True, ignore_direction=False, sort_vertices=True)#
Return an iterator over edges.
The iterator returned is over the edges incident with any vertex given in the parameter
vertices
. If the graph is directed, iterates over edges going out only. Ifvertices
isNone
, then returns an iterator over all edges. Ifself
is directed, returns outgoing edges only.INPUT:
vertices
– object (default:None
); a vertex, a list of vertices orNone
labels
– boolean (default:True
); ifFalse
, each edge isa tuple \((u,v)\) of vertices
ignore_direction
– boolean (default:False
); only applies todirected graphs. If
True
, searches across edges in either direction.
sort_vertices
– boolean (default:True
); only applies to undirected graphs. IfTrue
, sort the ends of the edges. Not sorting the ends is faster.
Note
It is somewhat safe to modify the graph during iterating.
vertices
must be specified if modifying the vertices.Without multiedges, you can safely use this graph to relabel edges or delete some edges. If you add edges, they might later appear in the iterator or not (depending on the internal order of vertices).
In case of multiedges, all arcs from one vertex to another are internally cached. So the iterator will yield them, even if you delete them all after seeing the first one.
EXAMPLES:
sage: for i in graphs.PetersenGraph().edge_iterator([0]): ....: print(i) (0, 1, None) (0, 4, None) (0, 5, None) sage: D = DiGraph({0: [1, 2], 1: [0]}) sage: for i in D.edge_iterator([0]): ....: print(i) (0, 1, None) (0, 2, None)
sage: G = graphs.TetrahedralGraph() sage: list(G.edge_iterator(labels=False)) [(0, 1), (0, 2), (0, 3), (1, 2), (1, 3), (2, 3)]
sage: G = graphs.TetrahedralGraph() sage: list(G.edge_iterator(labels=False, sort_vertices=False)) [(1, 0), (2, 0), (3, 0), (2, 1), (3, 1), (3, 2)]
sage: D = DiGraph({1: [0], 2: [0]}) sage: list(D.edge_iterator(0)) [] sage: list(D.edge_iterator(0, ignore_direction=True)) [(1, 0, None), (2, 0, None)]
 edge_label(u, v)#
Return the label of an edge.
If the graph allows multiple edges, then the list of labels on the edges is returned.
See also
EXAMPLES:
sage: G = Graph({0: {1: 'edgelabel'}}) sage: G.edge_label(0, 1) 'edgelabel' sage: D = DiGraph({1: {2: 'up'}, 2: {1: 'down'}}) sage: D.edge_label(2, 1) 'down'
sage: G = Graph(multiedges=True) sage: [G.add_edge(0, 1, i) for i in range(1, 6)] [None, None, None, None, None] sage: sorted(G.edge_label(0, 1)) [1, 2, 3, 4, 5]
 edge_labels()#
Return a list of the labels of all edges in
self
.The output list is not sorted.
EXAMPLES:
sage: G = Graph({0: {1: 'x', 2: 'z', 3: 'a'}, 2: {5: 'out'}}, sparse=True) sage: G.edge_labels() ['x', 'z', 'a', 'out'] sage: G = DiGraph({0: {1: 'x', 2: 'z', 3: 'a'}, 2: {5: 'out'}}, sparse=True) sage: G.edge_labels() ['x', 'z', 'a', 'out']
 edge_polytope(backend=None)#
Return the edge polytope of
self
.The edge polytope (EP) of a Graph on \(n\) vertices is the polytope in \(\ZZ^{n}\) defined as the convex hull of \(e_i + e_j\) for each edge \((i, j)\). Here \(e_1, \dots, e_n\) denotes the standard basis.
INPUT:
backend
– string orNone
(default); the backend to use; seesage.geometry.polyhedron.constructor.Polyhedron()
EXAMPLES:
The EP of a \(4\)cycle is a square:
sage: G = graphs.CycleGraph(4) sage: P = G.edge_polytope(); P # needs sage.geometry.polyhedron A 2dimensional polyhedron in ZZ^4 defined as the convex hull of 4 vertices
The EP of a complete graph on \(4\) vertices is cross polytope:
sage: G = graphs.CompleteGraph(4) sage: P = G.edge_polytope(); P # needs sage.geometry.polyhedron A 3dimensional polyhedron in ZZ^4 defined as the convex hull of 6 vertices sage: P.is_combinatorially_isomorphic(polytopes.cross_polytope(3)) # needs sage.geometry.polyhedron True
The EP of a graph is isomorphic to the subdirect sum of its connected components EPs:
sage: n = randint(3, 6) sage: G1 = graphs.RandomGNP(n, 0.2) # needs networkx sage: n = randint(3, 6) sage: G2 = graphs.RandomGNP(n, 0.2) # needs networkx sage: G = G1.disjoint_union(G2) # needs networkx sage: P = G.edge_polytope() # needs networkx sage.geometry.polyhedron sage: P1 = G1.edge_polytope() # needs networkx sage.geometry.polyhedron sage: P2 = G2.edge_polytope() # needs networkx sage.geometry.polyhedron sage: P.is_combinatorially_isomorphic(P1.subdirect_sum(P2)) # needs networkx sage.geometry.polyhedron True
All trees on \(n\) vertices have isomorphic EPs:
sage: n = randint(4, 10) sage: G1 = graphs.RandomTree(n) sage: G2 = graphs.RandomTree(n) sage: P1 = G1.edge_polytope() # needs sage.geometry.polyhedron sage: P2 = G2.edge_polytope() # needs sage.geometry.polyhedron sage: P1.is_combinatorially_isomorphic(P2) # needs sage.geometry.polyhedron True
However, there are still many different EPs:
sage: len(list(graphs(5))) 34 sage: polys = [] sage: for G in graphs(5): # needs sage.geometry.polyhedron ....: P = G.edge_polytope() ....: for P1 in polys: ....: if P.is_combinatorially_isomorphic(P1): ....: break ....: else: ....: polys.append(P) sage: len(polys) # needs sage.geometry.polyhedron 19
 edges(vertices=None, labels=True, sort=False, key=None, ignore_direction=False, sort_vertices=True)#
Return a
EdgesView
of edges.Each edge is a triple
(u, v, l)
whereu
andv
are vertices andl
is a label. If the parameterlabels
isFalse
then a list of couple(u, v)
is returned whereu
andv
are vertices.The returned
EdgesView
is over the edges incident with any vertex given in the parametervertices
(all edges ifNone
). Ifself
is directed, iterates over outgoing edges only, unless parameterignore_direction
isTrue
in which case it searches across edges in either direction.INPUT:
vertices
– object (default:None
); a vertex, a list of vertices orNone
labels
– boolean (default:True
); ifFalse
, each edge is simply a pair(u, v)
of verticessort
– boolean (default:False
); whether to sort edges according the ordering specified with parameterkey
. IfFalse
(default), edges are not sorted. This is the fastest and less memory consuming method for iterating over edges.key
– a function (default:None
); a function that takes an edge (a pair or a triple, according to thelabels
keyword) as its one argument and returns a value that can be used for comparisons in the sorting algorithmignore_direction
– boolean (default:False
); only applies todirected graphs. If
True
, searches across edges in either direction.
sort_vertices
– boolean (default:True
); only applies to undirected graphs. IfTrue
, sort the ends of the edges. Not sorting the ends is faster.
OUTPUT: A
EdgesView
.Warning
Since any object may be a vertex, there is no guarantee that any two vertices will be comparable, and thus no guarantee how two edges may compare. With default objects for vertices (all integers), or when all the vertices are of the same simple type, then there should not be a problem with how the vertices will be sorted. However, if you need to guarantee a total order for the sorting of the edges, use the
key
argument, as illustrated in the examples below.EXAMPLES:
sage: graphs.DodecahedralGraph().edges(sort=True) [(0, 1, None), (0, 10, None), (0, 19, None), (1, 2, None), (1, 8, None), (2, 3, None), (2, 6, None), (3, 4, None), (3, 19, None), (4, 5, None), (4, 17, None), (5, 6, None), (5, 15, None), (6, 7, None), (7, 8, None), (7, 14, None), (8, 9, None), (9, 10, None), (9, 13, None), (10, 11, None), (11, 12, None), (11, 18, None), (12, 13, None), (12, 16, None), (13, 14, None), (14, 15, None), (15, 16, None), (16, 17, None), (17, 18, None), (18, 19, None)]
sage: graphs.DodecahedralGraph().edges(sort=True, labels=False) [(0, 1), (0, 10), (0, 19), (1, 2), (1, 8), (2, 3), (2, 6), (3, 4), (3, 19), (4, 5), (4, 17), (5, 6), (5, 15), (6, 7), (7, 8), (7, 14), (8, 9), (9, 10), (9, 13), (10, 11), (11, 12), (11, 18), (12, 13), (12, 16), (13, 14), (14, 15), (15, 16), (16, 17), (17, 18), (18, 19)]
sage: D = graphs.DodecahedralGraph().to_directed() sage: D.edges(sort=True) [(0, 1, None), (0, 10, None), (0, 19, None), (1, 0, None), (1, 2, None), (1, 8, None), (2, 1, None), (2, 3, None), (2, 6, None), (3, 2, None), (3, 4, None), (3, 19, None), (4, 3, None), (4, 5, None), (4, 17, None), (5, 4, None), (5, 6, None), (5, 15, None), (6, 2, None), (6, 5, None), (6, 7, None), (7, 6, None), (7, 8, None), (7, 14, None), (8, 1, None), (8, 7, None), (8, 9, None), (9, 8, None), (9, 10, None), (9, 13, None), (10, 0, None), (10, 9, None), (10, 11, None), (11, 10, None), (11, 12, None), (11, 18, None), (12, 11, None), (12, 13, None), (12, 16, None), (13, 9, None), (13, 12, None), (13, 14, None), (14, 7, None), (14, 13, None), (14, 15, None), (15, 5, None), (15, 14, None), (15, 16, None), (16, 12, None), (16, 15, None), (16, 17, None), (17, 4, None), (17, 16, None), (17, 18, None), (18, 11, None), (18, 17, None), (18, 19, None), (19, 0, None), (19, 3, None), (19, 18, None)] sage: D.edges(sort=True, labels=False) [(0, 1), (0, 10), (0, 19), (1, 0), (1, 2), (1, 8), (2, 1), (2, 3), (2, 6), (3, 2), (3, 4), (3, 19), (4, 3), (4, 5), (4, 17), (5, 4), (5, 6), (5, 15), (6, 2), (6, 5), (6, 7), (7, 6), (7, 8), (7, 14), (8, 1), (8, 7), (8, 9), (9, 8), (9, 10), (9, 13), (10, 0), (10, 9), (10, 11), (11, 10), (11, 12), (11, 18), (12, 11), (12, 13), (12, 16), (13, 9), (13, 12), (13, 14), (14, 7), (14, 13), (14, 15), (15, 5), (15, 14), (15, 16), (16, 12), (16, 15), (16, 17), (17, 4), (17, 16), (17, 18), (18, 11), (18, 17), (18, 19), (19, 0), (19, 3), (19, 18)]
The default is to sort the returned list in the default fashion, as in the above examples. This can be overridden by specifying a key function. This first example just ignores the labels in the third component of the triple:
sage: G = graphs.CycleGraph(5) sage: G.edges(sort=True, key=lambda x: (x[1], x[0])) [(0, 1, None), (1, 2, None), (2, 3, None), (3, 4, None), (0, 4, None)]
We set the labels to characters and then perform a default sort followed by a sort according to the labels:
sage: G = graphs.CycleGraph(5) sage: for e in G.edges(sort=False): ....: G.set_edge_label(e[0], e[1], chr(ord('A') + e[0] + 5 * e[1])) sage: G.edges(sort=True) [(0, 1, 'F'), (0, 4, 'U'), (1, 2, 'L'), (2, 3, 'R'), (3, 4, 'X')] sage: G.edges(sort=True, key=lambda x: x[2]) [(0, 1, 'F'), (1, 2, 'L'), (2, 3, 'R'), (0, 4, 'U'), (3, 4, 'X')]
We can restrict considered edges to those incident to a given set:
sage: for i in graphs.PetersenGraph().edges(sort=True, vertices=[0]): ....: print(i) (0, 1, None) (0, 4, None) (0, 5, None) sage: D = DiGraph({0: [1, 2], 1: [0]}) sage: for i in D.edges(sort=True, vertices=[0]): ....: print(i) (0, 1, None) (0, 2, None)
Ignoring the direction of edges:
sage: D = DiGraph({1: [0], 2: [0]}) sage: D.edges(sort=True, vertices=0) [] sage: D.edges(sort=True, vertices=0, ignore_direction=True) [(1, 0, None), (2, 0, None)] sage: D.edges(sort=True, vertices=[0], ignore_direction=True) [(1, 0, None), (2, 0, None)]
Not sorting the ends of the edges:
sage: G = Graph() sage: G = Graph() sage: G.add_edges([[1,2], [2,3], [0,3]]) sage: list(G.edge_iterator(sort_vertices=False)) [(3, 0, None), (2, 1, None), (3, 2, None)]
 edges_incident(vertices=None, labels=True, sort=False)#
Return incident edges to some vertices.
If
vertices
is a vertex, then it returns the list of edges incident to that vertex. Ifvertices
is a list of vertices then it returns the list of all edges adjacent to those vertices. Ifvertices
isNone
, it returns a list of all edges in graph. For digraphs, only lists outward edges.INPUT:
vertices
– object (default:None
); a vertex, a list of vertices orNone
labels
– boolean (default:True
); ifFalse
, each edge isa tuple \((u,v)\) of vertices
sort
– boolean (default:False
); ifTrue
the returned list is sorted
EXAMPLES:
sage: graphs.PetersenGraph().edges_incident([0, 9], labels=False) [(0, 1), (0, 4), (0, 5), (4, 9), (6, 9), (7, 9)] sage: D = DiGraph({0: [1]}) sage: D.edges_incident([0]) [(0, 1, None)] sage: D.edges_incident([1]) []
 eigenspaces(laplacian=False)#
Return the right eigenspaces of the adjacency matrix of the graph.
INPUT:
laplacian
– boolean (default:False
); ifTrue
, use theLaplacian matrix (see
kirchhoff_matrix()
)
OUTPUT:
A list of pairs. Each pair is an eigenvalue of the adjacency matrix of the graph, followed by the vector space that is the eigenspace for that eigenvalue, when the eigenvectors are placed on the right of the matrix.
For some graphs, some of the eigenspaces are described exactly by vector spaces over a
NumberField()
. For numerical eigenvectors useeigenvectors()
.EXAMPLES:
sage: P = graphs.PetersenGraph() sage: P.eigenspaces() # needs sage.modules sage.rings.number_field [ (3, Vector space of degree 10 and dimension 1 over Rational Field User basis matrix: [1 1 1 1 1 1 1 1 1 1]), (2, Vector space of degree 10 and dimension 4 over Rational Field User basis matrix: [ 1 0 0 0 1 1 1 0 1 1] [ 0 1 0 0 1 0 2 1 1 2] [ 0 0 1 0 1 1 1 2 0 2] [ 0 0 0 1 1 1 0 1 1 1]), (1, Vector space of degree 10 and dimension 5 over Rational Field User basis matrix: [ 1 0 0 0 0 1 1 0 0 1] [ 0 1 0 0 0 1 1 1 0 0] [ 0 0 1 0 0 0 1 1 1 0] [ 0 0 0 1 0 0 0 1 1 1] [ 0 0 0 0 1 1 0 0 1 1]) ]
Eigenspaces for the Laplacian should be identical since the Petersen graph is regular. However, since the output also contains the eigenvalues, the two outputs are slightly different:
sage: P.eigenspaces(laplacian=True) # needs sage.modules sage.rings.number_field [ (0, Vector space of degree 10 and dimension 1 over Rational Field User basis matrix: [1 1 1 1 1 1 1 1 1 1]), (5, Vector space of degree 10 and dimension 4 over Rational Field User basis matrix: [ 1 0 0 0 1 1 1 0 1 1] [ 0 1 0 0 1 0 2 1 1 2] [ 0 0 1 0 1 1 1 2 0 2] [ 0 0 0 1 1 1 0 1 1 1]), (2, Vector space of degree 10 and dimension 5 over Rational Field User basis matrix: [ 1 0 0 0 0 1 1 0 0 1] [ 0 1 0 0 0 1 1 1 0 0] [ 0 0 1 0 0 0 1 1 1 0] [ 0 0 0 1 0 0 0 1 1 1] [ 0 0 0 0 1 1 0 0 1 1]) ]
Notice how one eigenspace below is described with a square root of 2. For the two possible values (positive and negative) there is a corresponding eigenspace:
sage: C = graphs.CycleGraph(8) sage: C.eigenspaces() # needs sage.modules sage.rings.number_field [ (2, Vector space of degree 8 and dimension 1 over Rational Field User basis matrix: [1 1 1 1 1 1 1 1]), (2, Vector space of degree 8 and dimension 1 over Rational Field User basis matrix: [ 1 1 1 1 1 1 1 1]), (0, Vector space of degree 8 and dimension 2 over Rational Field User basis matrix: [ 1 0 1 0 1 0 1 0] [ 0 1 0 1 0 1 0 1]), (a3, Vector space of degree 8 and dimension 2 over Number Field in a3 with defining polynomial x^2  2 User basis matrix: [ 1 0 1 a3 1 0 1 a3] [ 0 1 a3 1 0 1 a3 1]) ]
A digraph may have complex eigenvalues and eigenvectors. For a 3cycle, we have:
sage: T = DiGraph({0: [1], 1: [2], 2: [0]}) sage: T.eigenspaces() # needs sage.modules sage.rings.number_field [ (1, Vector space of degree 3 and dimension 1 over Rational Field User basis matrix: [1 1 1]), (a1, Vector space of degree 3 and dimension 1 over Number Field in a1 with defining polynomial x^2 + x + 1 User basis matrix: [ 1 a1 a1  1]) ]
 eigenvectors(laplacian=False)#
Return the right eigenvectors of the adjacency matrix of the graph.
INPUT:
laplacian
– boolean (default:False
); ifTrue
, use the Laplacian matrix (seekirchhoff_matrix()
)
OUTPUT:
A list of triples. Each triple begins with an eigenvalue of the adjacency matrix of the graph. This is followed by a list of eigenvectors for the eigenvalue, when the eigenvectors are placed on the right side of the matrix. Together, the eigenvectors form a basis for the eigenspace. The triple concludes with the algebraic multiplicity of the eigenvalue.
For some graphs, the exact eigenspaces provided by
eigenspaces()
provide additional insight into the structure of the eigenspaces.EXAMPLES:
sage: P = graphs.PetersenGraph() sage: P.eigenvectors() # needs sage.modules sage.rings.number_field [(3, [ (1, 1, 1, 1, 1, 1, 1, 1, 1, 1) ], 1), (2, [ (1, 0, 0, 0, 1, 1, 1, 0, 1, 1), (0, 1, 0, 0, 1, 0, 2, 1, 1, 2), (0, 0, 1, 0, 1, 1, 1, 2, 0, 2), (0, 0, 0, 1, 1, 1, 0, 1, 1, 1) ], 4), (1, [ (1, 0, 0, 0, 0, 1, 1, 0, 0, 1), (0, 1, 0, 0, 0, 1, 1, 1, 0, 0), (0, 0, 1, 0, 0, 0, 1, 1, 1, 0), (0, 0, 0, 1, 0, 0, 0, 1, 1, 1), (0, 0, 0, 0, 1, 1, 0, 0, 1, 1) ], 5)]
Eigenspaces for the Laplacian should be identical since the Petersen graph is regular. However, since the output also contains the eigenvalues, the two outputs are slightly different:
sage: P.eigenvectors(laplacian=True) # needs sage.modules sage.rings.number_field [(0, [ (1, 1, 1, 1, 1, 1, 1, 1, 1, 1) ], 1), (5, [ (1, 0, 0, 0, 1, 1, 1, 0, 1, 1), (0, 1, 0, 0, 1, 0, 2, 1, 1, 2), (0, 0, 1, 0, 1, 1, 1, 2, 0, 2), (0, 0, 0, 1, 1, 1, 0, 1, 1, 1) ], 4), (2, [ (1, 0, 0, 0, 0, 1, 1, 0, 0, 1), (0, 1, 0, 0, 0, 1, 1, 1, 0, 0), (0, 0, 1, 0, 0, 0, 1, 1, 1, 0), (0, 0, 0, 1, 0, 0, 0, 1, 1, 1), (0, 0, 0, 0, 1, 1, 0, 0, 1, 1) ], 5)]
sage: C = graphs.CycleGraph(8) sage: C.eigenvectors() # needs sage.modules sage.rings.number_field [(2, [ (1, 1, 1, 1, 1, 1, 1, 1) ], 1), (2, [ (1, 1, 1, 1, 1, 1, 1, 1) ], 1), (0, [ (1, 0, 1, 0, 1, 0, 1, 0), (0, 1, 0, 1, 0, 1, 0, 1) ], 2), (1.4142135623..., [(1, 0, 1, 1.4142135623..., 1, 0, 1, 1.4142135623...), (0, 1, 1.4142135623..., 1, 0, 1, 1.4142135623..., 1)], 2), (1.4142135623..., [(1, 0, 1, 1.4142135623..., 1, 0, 1, 1.4142135623...), (0, 1, 1.4142135623..., 1, 0, 1, 1.4142135623..., 1)], 2)]
A digraph may have complex eigenvalues. Previously, the complex parts of graph eigenvalues were being dropped. For a 3cycle, we have:
sage: T = DiGraph({0:[1], 1:[2], 2:[0]}) sage: T.eigenvectors() # needs sage.modules sage.rings.number_field [(1, [ (1, 1, 1) ], 1), (0.5000000000...  0.8660254037...*I, [(1, 0.5000000000...  0.8660254037...*I, 0.5000000000... + 0.8660254037...*I)], 1), (0.5000000000... + 0.8660254037...*I, [(1, 0.5000000000... + 0.8660254037...*I, 0.5000000000...  0.8660254037...*I)], 1)]
 eulerian_circuit(return_vertices=False, labels=True, path=False)#
Return a list of edges forming an Eulerian circuit if one exists.
If no Eulerian circuit is found, the method returns
False
.This is implemented using Hierholzer’s algorithm.
INPUT:
return_vertices
– boolean (default:False
); optionallyprovide a list of vertices for the path
labels
– boolean (default:True
); whether to return edgeswith labels (3tuples)
path
– boolean (default:False
); find an Eulerian pathinstead
OUTPUT:
either ([edges], [vertices]) or [edges] of an Eulerian circuit (or path)
EXAMPLES:
sage: g = graphs.CycleGraph(5) sage: g.eulerian_circuit() [(0, 4, None), (4, 3, None), (3, 2, None), (2, 1, None), (1, 0, None)] sage: g.eulerian_circuit(labels=False) [(0, 4), (4, 3), (3, 2), (2, 1), (1, 0)]
sage: g = graphs.CompleteGraph(7) sage: edges, vertices = g.eulerian_circuit(return_vertices=True) sage: vertices [0, 6, 5, 4, 6, 3, 5, 2, 4, 3, 2, 6, 1, 5, 0, 4, 1, 3, 0, 2, 1, 0]
sage: graphs.CompleteGraph(4).eulerian_circuit() False
A disconnected graph can be Eulerian:
sage: g = Graph({0: [], 1: [2], 2: [3], 3: [1], 4: []}) sage: g.eulerian_circuit(labels=False) [(1, 3), (3, 2), (2, 1)]
sage: g = DiGraph({0: [1], 1: [2, 4], 2:[3], 3:[1]}) sage: g.eulerian_circuit(labels=False, path=True) [(0, 1), (1, 2), (2, 3), (3, 1), (1, 4)]
sage: g = Graph({0:[1,2,3], 1:[2,3], 2:[3,4], 3:[4]}) sage: g.is_eulerian(path=True) (0, 1) sage: g.eulerian_circuit(labels=False, path=True) [(1, 3), (3, 4), (4, 2), (2, 3), (3, 0), (0, 2), (2, 1), (1, 0)]
 eulerian_orientation()#
Return a DiGraph which is an Eulerian orientation of the current graph.
An Eulerian graph being a graph such that any vertex has an even degree, an Eulerian orientation of a graph is an orientation of its edges such that each vertex \(v\) verifies \(d^+(v)=d^(v)=d(v)/2\), where \(d^+\) and \(d^\) respectively represent the outdegree and the indegree of a vertex.
If the graph is not Eulerian, the orientation verifies for any vertex \(v\) that \( d^+(v)d^(v)  \leq 1\).
ALGORITHM:
This algorithm is a random walk through the edges of the graph, which orients the edges according to the walk. When a vertex is reached which has no nonoriented edge (this vertex must have odd degree), the walk resumes at another vertex of odd degree, if any.
This algorithm has complexity \(O(m)\), where \(m\) is the number of edges in the graph.
EXAMPLES:
The CubeGraph with parameter 4, which is regular of even degree, has an Eulerian orientation such that \(d^+ = d^\):
sage: g = graphs.CubeGraph(4) sage: g.degree() [4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4] sage: o = g.eulerian_orientation() sage: o.in_degree() [2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2] sage: o.out_degree() [2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2]
Secondly, the Petersen Graph, which is 3 regular has an orientation such that the difference between \(d^+\) and \(d^\) is at most 1:
sage: g = graphs.PetersenGraph() sage: o = g.eulerian_orientation() sage: o.in_degree() [2, 2, 2, 2, 2, 1, 1, 1, 1, 1] sage: o.out_degree() [1, 1, 1, 1, 1, 2, 2, 2, 2, 2]
 export_to_file(filename, format=None, **kwds)#
Export the graph to a file.
INPUT:
filename
– string; a file nameformat
– string (default:None
); select the output format explicitly. If set toNone
(default), the format is set to be the file extension offilename
. Admissible formats are:adjlist
,dot
,edgelist
,gexf
,gml
,graphml
,multiline_adjlist
,pajek
,yaml
.All other arguments are forwarded to the subfunction. For more information, see their respective documentation:
See also
save()
– save a Sage object to a ‘sobj’ file (preserves all its attributes)
Note
This functions uses the
write_*
functions defined in NetworkX (see http://networkx.lanl.gov/reference/readwrite.html).EXAMPLES:
sage: g = graphs.PetersenGraph() sage: filename = tmp_filename(ext=".pajek") sage: g.export_to_file(filename) # needs networkx sage: import networkx # needs networkx sage: G_networkx = networkx.read_pajek(filename) # needs networkx sage: Graph(G_networkx).is_isomorphic(g) # needs networkx True sage: filename = tmp_filename(ext=".edgelist") sage: g.export_to_file(filename, data=False) # needs networkx sage: h = Graph(networkx.read_edgelist(filename)) # needs networkx sage: g.is_isomorphic(h) # needs networkx True
 faces(embedding=None)#
Return the faces of an embedded graph.
A combinatorial embedding of a graph is a clockwise ordering of the neighbors of each vertex. From this information one can define the faces of the embedding, which is what this method returns.
If no embedding is provided or stored as
self._embedding
, this method will compute the set of faces from the embedding returned byis_planar()
(if the graph is, of course, planar).Warning
This method is not well defined when the graph is not connected. Indeed, the result may contain several faces corresponding to the external face.
INPUT:
embedding
– dictionary (default:None
); a combinatorial embedding dictionary. Format:{v1: [v2,v3], v2: [v1], v3: [v1]}
(clockwise ordering of neighbors at each vertex). If set toNone
(default) the method will use the embedding stored asself._embedding
. If none is stored, the method will compute the set of faces from the embedding returned byis_planar()
(if the graph is, of course, planar).
Note
embedding
is an ordered list based on the hash order of the vertices of graph. To avoid confusion, it might be best to set the rot_sys based on a ‘nice_copy’ of the graph.EXAMPLES:
Providing an embedding:
sage: T = graphs.TetrahedralGraph() sage: T.faces({0: [1, 3, 2], 1: [0, 2, 3], 2: [0, 3, 1], 3: [0, 1, 2]}) [[(0, 1), (1, 2), (2, 0)], [(0, 2), (2, 3), (3, 0)], [(0, 3), (3, 1), (1, 0)], [(1, 3), (3, 2), (2, 1)]]
With no embedding provided:
sage: graphs.TetrahedralGraph().faces() [[(0, 1), (1, 2), (2, 0)], [(0, 2), (2, 3), (3, 0)], [(0, 3), (3, 1), (1, 0)], [(1, 3), (3, 2), (2, 1)]]
With no embedding provided (nonplanar graph):
sage: graphs.PetersenGraph().faces() Traceback (most recent call last): ... ValueError: no embedding is provided and the graph is not planar
 feedback_vertex_set(value_only, solver=False, verbose=None, constraint_generation=0, integrality_tolerance=True)#
Return the minimum feedback vertex set of a (di)graph.
The minimum feedback vertex set of a (di)graph is a set of vertices that intersect all of its cycles. Equivalently, a minimum feedback vertex set of a (di)graph is a set \(S\) of vertices such that the digraph \(GS\) is acyclic. For more information, see the Wikipedia article Feedback_vertex_set.
INPUT:
value_only
– boolean (default:False
); whether to return only the minimum cardinal of a minimum vertex set, or theSet
of vertices of a minimal feedback vertex setsolver
– string (default:None
); specify a Mixed Integer Linear Programming (MILP) solver to be used. If set toNone
, the default one is used. For more information on MILP solvers and which default solver is used, see the methodsolve
of the classMixedIntegerLinearProgram
.verbose
– integer (default:0
); sets the level of verbosity. Set to 0 by default, which means quiet.constraint_generation
– boolean (default:True
); whether to use constraint generation when solving the Mixed Integer Linear Programintegrality_tolerance
– float; parameter for use with MILP solvers over an inexact base ring; seeMixedIntegerLinearProgram.get_values()
.
ALGORITHMS:
(Constraints generation)
When the parameter
constraint_generation
is enabled (default) the following MILP formulation is used to solve the problem:\[\begin{split}\mbox{Minimize : }&\sum_{v\in G} b_{v}\\ \mbox{Such that : }&\\ &\forall C\text{ circuits }\subseteq G, \sum_{v\in C}b_{v}\geq 1\\\end{split}\]As the number of circuits contained in a graph is exponential, this LP is solved through constraint generation. This means that the solver is sequentially asked to solve the problem, knowing only a portion of the circuits contained in \(G\), each time adding to the list of its constraints the circuit which its last answer had left intact.
(Another formulation based on an ordering of the vertices)
When the graph is directed, a second (and very slow) formulation is available, which should only be used to check the result of the first implementation in case of doubt.
\[\begin{split}\mbox{Minimize : }&\sum_{v\in G} b_v\\ \mbox{Such that : }&\\ &\forall (u,v)\in G, d_ud_v+nb_u+nb_v\geq 0\\ &\forall u\in G, 0\leq d_u\leq G\\\end{split}\]A brief explanation:
An acyclic digraph can be seen as a poset, and every poset has a linear extension. This means that in any acyclic digraph the vertices can be ordered with a total order \(<\) in such a way that if \((u,v)\in G\), then \(u<v\). Thus, this linear program is built in order to assign to each vertex \(v\) a number \(d_v\in [0,\dots,n1]\) such that if there exists an edge \((u,v)\in G\) then either \(d_v<d_u\) or one of \(u\) or \(v\) is removed. The number of vertices removed is then minimized, which is the objective.
EXAMPLES:
The necessary example:
sage: # needs sage.numerical.mip sage: g = graphs.PetersenGraph() sage: fvs = g.feedback_vertex_set() sage: len(fvs) 3 sage: g.delete_vertices(fvs) sage: g.is_forest() True
In a digraph built from a graph, any edge is replaced by arcs going in the two opposite directions, thus creating a cycle of length two. Hence, to remove all the cycles from the graph, each edge must see one of its neighbors removed: a feedback vertex set is in this situation a vertex cover:
sage: # needs sage.numerical.mip sage: cycle = graphs.CycleGraph(5) sage: dcycle = DiGraph(cycle) sage: cycle.vertex_cover(value_only=True) 3 sage: feedback = dcycle.feedback_vertex_set() sage: len(feedback) 3 sage: u,v = next(cycle.edge_iterator(labels=None)) sage: u in feedback or v in feedback True
For a circuit, the minimum feedback arc set is clearly \(1\):
sage: circuit = digraphs.Circuit(5) sage: circuit.feedback_vertex_set(value_only=True) == 1 # needs sage.numerical.mip True
 flow(x, y, value_only, integer=True, use_edge_labels=False, vertex_bound=True, algorithm=False, solver=None, verbose=None, integrality_tolerance=0)#
Return a maximum flow in the graph from
x
toy
.The returned flow is represented by an optimal valuation of the edges. For more information, see the Wikipedia article Max_flow.
As an optimization problem, is can be expressed this way :
\[\begin{split}\mbox{Maximize : }&\sum_{e\in G.edges()} w_e b_e\\ \mbox{Such that : }&\forall v \in G, \sum_{(u,v)\in G.edges()} b_{(u,v)}\leq 1\\ &\forall x\in G, b_x\mbox{ is a binary variable}\end{split}\]Observe that the integrality of the flow variables is automatic for all available solvers when all capacities are integers.
INPUT:
x
– source vertexy
– sink vertexvalue_only
– boolean (default:True
); whether to return only the value of a maximal flow, or to also return a flow graph (a copy of the current graph, such that each edge has the flow using it as a label, the edges without flow being omitted)integer
– boolean (default:True
); whether to compute an optimal solution under the constraint that the flow going through an edge has to be an integer, or without this constraintuse_edge_labels
– boolean (default:False
); whether to compute a maximum flow where each edge has a capacity defined by its label (if an edge has no label, capacity \(1\) is assumed), or to use default edge capacity of \(1\)vertex_bound
– boolean (default:False
); when set toTrue
, sets the maximum flow leaving a vertex different from \(x\) to \(1\) (useful for vertex connectivity parameters)algorithm
– string (default:None
); the algorithm to use among:"FF"
, a Python implementation of the FordFulkerson algorithm (only available whenvertex_bound = False
)"LP"
, the flow problem is solved using Linear Programming"igraph"
, theigraph
implementation of the GoldbergTarjan algorithm is used (only available whenigraph
is installed andvertex_bound = False
)
When
algorithm = None
(default), we useLP
ifvertex_bound = True
, otherwise, we useigraph
if it is available,FF
if it is not available.solver
– string (default:None
); specify a Mixed Integer Linear Programming (MILP) solver to be used. If set toNone
, the default one is used. For more information on MILP solvers and which default solver is used, see the methodsolve
of the classMixedIntegerLinearProgram
.Only useful when algorithm
"LP"
is used to solve the flow problem.verbose
– integer (default:0
); sets the level of verbosity. Set to 0 by default, which means quiet.Only useful when algorithm
"LP"
is used to solve the flow problem.integrality_tolerance
– float; parameter for use with MILP solvers over an inexact base ring; seeMixedIntegerLinearProgram.get_values()
.Only useful when
algorithm == "LP"
andinteger == True
.
Note
Even though the three different implementations are meant to return the same Flow values, they cannot be expected to return the same Flow graphs.
Besides, the use of Linear Programming may possibly mean a (slight) numerical noise.
EXAMPLES:
Two basic applications of the flow method for the
PappusGraph
and theButterflyGraph
with parameter \(2\)sage: g=graphs.PappusGraph() sage: int(g.flow(1,2)) 3
sage: b=digraphs.ButterflyGraph(2) sage: int(b.flow(('00', 1), ('00', 2))) 1
The flow method can be used to compute a matching in a bipartite graph by linking a source \(s\) to all the vertices of the first set and linking a sink \(t\) to all the vertices of the second set, then computing a maximum \(st\) flow
sage: g = DiGraph() sage: g.add_edges(('s', i) for i in range(4)) sage: g.add_edges((i, 4 + j) for i in range(4) for j in range(4)) sage: g.add_edges((4 + i, 't') for i in range(4)) sage: [cardinal, flow_graph] = g.flow('s', 't', integer=True, value_only=False) sage: flow_graph.delete_vertices(['s', 't']) sage: flow_graph.size() 4
The undirected case:
sage: g = Graph() sage: g.add_edges(('s', i) for i in range(4)) sage: g.add_edges((i, 4 + j) for i in range(4) for j in range(4)) sage: g.add_edges((4 + i, 't') for i in range(4)) sage: [cardinal, flow_graph] = g.flow('s', 't', integer=True, value_only=False) sage: flow_graph.delete_vertices(['s', 't']) sage: flow_graph.size() 4
 genus(set_embedding=True, on_embedding=None, minimal=True, maximal=False, circular=None, ordered=True)#
Return the minimal genus of the graph.
The genus of a compact surface is the number of handles it has. The genus of a graph is the minimal genus of the surface it can be embedded into. It can be seen as a measure of nonplanarity; a planar graph has genus zero.
Note
This function uses Euler’s formula and thus it is necessary to consider only connected graphs.
INPUT:
set_embedding
– boolean (default:True
); whether or not to store an embedding attribute of the computed (minimal) genus of the graphon_embedding
– two kinds of input are allowed (default:None
):
a dictionary representing a combinatorial embedding on which the genus should be computed. Note that this must be a valid embedding for the graph. The dictionary structure is given by:
vertex1: [neighbor1, neighbor2, neighbor3], vertex2: [neighbor]
where there is a key for each vertex in the graph and a (clockwise) ordered list of each vertex’s neighbors as values. The value ofon_embedding
takes precedence over a stored_embedding
attribute ifminimal
is set toFalse
.The value
True
, in order to indicate that the embedding stored as_embedding
should be used (see examples).
minimal
– boolean (default:True
); whether or not to compute the minimal genus of the graph (i.e., testing all embeddings). If minimal isFalse
, then eithermaximal
must beTrue
oron_embedding
must not beNone
. Ifon_embedding
is notNone
, it will take priority overminimal
. Similarly, ifmaximal
isTrue
, it will take priority overminimal
.maximal
– boolean (default:False
); whether or not to compute the maximal genus of the graph (i.e., testing all embeddings). Ifmaximal
isFalse
, then eitherminimal
must beTrue
oron_embedding
must not beNone
. Ifon_embedding
is notNone
, it will take priority overmaximal
. However,maximal
takes priority over the defaultminimal
.circular
– list (default:None
); ifcircular
is a list of vertices, the method computes the genus preserving a planar embedding of the this list. Ifcircular
is defined,on_embedding
is not a valid option.ordered
– boolean (default:True
); ifcircular
isTrue
, then whether or not the boundary order may be permuted (default isTrue
, which means the boundary order is preserved)
EXAMPLES:
sage: g = graphs.PetersenGraph() sage: g.genus() # tests for minimal genus by default 1 sage: g.genus(on_embedding=True, maximal=True) # on_embedding overrides minimal and maximal arguments 1 sage: g.genus(maximal=True) # setting maximal to True overrides default minimal=True 3 sage: g.genus(on_embedding=g.get_embedding()) # can also send a valid combinatorial embedding dict 3 sage: (graphs.CubeGraph(3)).genus() 0 sage: K23 = graphs.CompleteBipartiteGraph(2,3) sage: K23.genus() 0 sage: K33 = graphs.CompleteBipartiteGraph(3,3) sage: K33.genus() 1
Using the circular argument, we can compute the minimal genus preserving a planar, ordered boundary:
sage: cube = graphs.CubeGraph(2) sage: cube.genus(circular=['01','10']) 0 sage: cube.is_circular_planar() True sage: cube.genus(circular=['01','10']) 0 sage: cube.genus(circular=['01','10'], on_embedding=True) Traceback (most recent call last): ... ValueError: on_embedding is not a valid option when circular is defined sage: cube.genus(circular=['01','10'], maximal=True) Traceback (most recent call last): ... NotImplementedError: cannot compute the maximal genus of a genus respecting a boundary
Note: not everything works for multigraphs, looped graphs or digraphs. But the minimal genus is ultimately computable for every connected graph – but the embedding we obtain for the simple graph can’t be easily converted to an embedding of a nonsimple graph. Also, the maximal genus of a multigraph does not trivially correspond to that of its simple graph:
sage: G = DiGraph({0: [0, 1, 1, 1], 1: [2, 2, 3, 3], 2: [1, 3, 3], 3: [0, 3]}) sage: G.genus() Traceback (most recent call last): ... NotImplementedError: cannot work with embeddings of nonsimple graphs sage: G.to_simple().genus() 0 sage: G.genus(set_embedding=False) 0 sage: G.genus(maximal=True, set_embedding=False) Traceback (most recent call last): ... NotImplementedError: cannot compute the maximal genus of a graph with loops or multiple edges
We break graphs with cut vertices into their blocks, which greatly speeds up computation of minimal genus. This is not implemented for maximal genus:
sage: G = graphs.RandomBlockGraph(10, 5) sage: G.genus() 10
 get_embedding()#
Return the stored embedding or
None
.If the stored embedding is no longer valid (because of vertex/edge additions) then the stored embedding is discarded and
None
is returned. In case some vertex/edge has been deleted, the stored embedding is updated accordingly.EXAMPLES:
sage: G = graphs.PetersenGraph() sage: G.genus() 1 sage: G.get_embedding() {0: [1, 4, 5], 1: [0, 2, 6], 2: [1, 3, 7], 3: [2, 4, 8], 4: [0, 3, 9], 5: [0, 7, 8], 6: [1, 9, 8], 7: [2, 5, 9], 8: [3, 6, 5], 9: [4, 6, 7]}
Note that the embeddings gets properly modified on vertex or edge deletion:
sage: G.delete_edge(0, 1) sage: G.delete_vertex(3) sage: G.get_embedding() {0: [4, 5], 1: [2, 6], 2: [1, 7], 4: [0, 9], 5: [0, 7, 8], 6: [1, 9, 8], 7: [2, 5, 9], 8: [6, 5], 9: [4, 6, 7]}
But not under edge addition:
sage: G.add_edge(0, 7) sage: G.get_embedding() is None True
 get_pos(dim=2)#
Return the position dictionary.
The position dictionary specifies the coordinates of each vertex.
INPUT:
dim
– integer (default: 2); whether to return the position dictionary in the plane (dim == 2
) or in the 3dimensional space
EXAMPLES:
By default, the position of a graph is None:
sage: G = Graph() sage: G.get_pos() sage: G.get_pos() is None True sage: P = G.plot(save_pos=True) # needs sage.plot sage: G.get_pos() # needs sage.plot {}
Some of the named graphs come with a prespecified positioning:
sage: G = graphs.PetersenGraph() sage: G.get_pos() {0: (0.0, 1.0), ... 9: (0.475..., 0.154...)}
Note that the position dictionary is modified on vertex removal:
sage: G.delete_vertex(0) sage: G.get_pos() {1: (0.951..., 0.309...), ... 9: (0.475..., 0.154...)}
But is deleted on vertex addition:
sage: G.add_vertex(0) sage: G.get_pos() is None True
 get_vertex(vertex)#
Retrieve the object associated with a given vertex.
If no associated object is found,
None
is returned.INPUT:
vertex
– the given vertex
EXAMPLES:
sage: d = {0: graphs.DodecahedralGraph(), 1: graphs.FlowerSnark(), 2: graphs.MoebiusKantorGraph(), 3: graphs.PetersenGraph()} sage: d[2] MoebiusKantor Graph: Graph on 16 vertices sage: T = graphs.TetrahedralGraph() sage: T.vertices(sort=True) [0, 1, 2, 3] sage: T.set_vertices(d) sage: T.get_vertex(1) Flower Snark: Graph on 20 vertices
 get_vertices(verts=None)#
Return a dictionary of the objects associated to each vertex.
INPUT:
verts
– iterable container of vertices
EXAMPLES:
sage: d = {0: graphs.DodecahedralGraph(), 1: graphs.FlowerSnark(), 2: graphs.MoebiusKantorGraph(), 3: graphs.PetersenGraph()} sage: T = graphs.TetrahedralGraph() sage: T.set_vertices(d) sage: T.get_vertices([1, 2]) {1: Flower Snark: Graph on 20 vertices, 2: MoebiusKantor Graph: Graph on 16 vertices}
 girth(certificate=False)#
Return the girth of the graph.
The girth is the length of the shortest cycle in the graph (directed cycle if the graph is directed). Graphs without (directed) cycles have infinite girth.
INPUT:
certificate
– boolean (default:False
); whether to return(g, c)
, whereg
is the girth andc
is a list of vertices of a (directed) cycle of lengthg
in the graph, thus providing a certificate that the girth is at mostg
, orNone
ifg
infinite
EXAMPLES:
sage: graphs.TetrahedralGraph().girth() 3 sage: graphs.CubeGraph(3).girth() 4 sage: graphs.PetersenGraph().girth(certificate=True) # random (5, [4, 3, 2, 1, 0]) sage: graphs.HeawoodGraph().girth() 6 sage: next(graphs.trees(9)).girth() +Infinity
See also
odd_girth()
– return the odd girth of the graph.
 graphics_array_defaults = {'graph_border': True, 'layout': 'circular', 'vertex_labels': False, 'vertex_size': 50}#
 graphplot(**options)#
Return a
GraphPlot
object.See
GraphPlot
for more details.INPUT:
**options
– parameters for theGraphPlot
constructor
EXAMPLES:
Creating a
GraphPlot
object uses the same options asplot()
:sage: g = Graph({}, loops=True, multiedges=True, sparse=True) sage: g.add_edges([(0,0,'a'),(0,0,'b'),(0,1,'c'),(0,1,'d'), ....: (0,1,'e'),(0,1,'f'),(0,1,'f'),(2,1,'g'),(2,2,'h')]) sage: GP = g.graphplot(edge_labels=True, color_by_label=True, # needs sage.plot ....: edge_style='dashed') sage: GP.plot() # needs sage.plot Graphics object consisting of 22 graphics primitives
We can modify the
GraphPlot
object. Notice that the changes are cumulative:sage: # needs sage.plot sage: GP.set_edges(edge_style='solid') sage: GP.plot() Graphics object consisting of 22 graphics primitives sage: GP.set_vertices(talk=True) sage: GP.plot() Graphics object consisting of 22 graphics primitives
 graphviz_string(labels='string', vertex_labels=True, edge_labels=False, edge_color=None, edge_colors=None, edge_options=(), color_by_label=False, rankdir='down', subgraph_clusters=[], **options)#
Return a representation in the
dot
language.The
dot
language is a text based format for graphs. It is used by the software suitegraphviz
. The specifications of the language are available on the web (see the reference [dotspec]).INPUT:
labels
– string (default:"string"
); either"string"
or"latex"
. If labels is"string"
, latex commands are not interpreted. This option stands for both vertex labels and edge labels.vertex_labels
– boolean (default:True
); whether to add the labels on verticesedge_labels
– boolean (default:False
); whether to add the labels on edgesedge_color
– (default:None
); specify a default color for the edges. The color could be one ofa name given as a string such as
"blue"
or"orchid"
a HSV sequence in a string such as
".52,.386,.22"
an hexadecimal code such as
"#DA3305"
a 3tuple of floating point (to be interpreted as RGB tuple). In this case the 3tuple is converted in hexadecimal code.
edge_colors
– dictionary (default:None
); a dictionary whose keys are colors and values are list of edges. The list of edges need not to be complete in which case the default color is used. See the optionedge_color
for a description of valid color formats.color_by_label
– a boolean or dictionary or function (default:False
); whether to color each edge with a different color according to its label; the colors are chosen along a rainbow, unless they are specified by a function or dictionary mapping labels to colors; this option is incompatible withedge_color
andedge_colors
. See the optionedge_color
for a description of valid color formats.edge_options
– a function (or tuple thereof) mapping edges to a dictionary of options for this edgerankdir
–'left'
,'right'
,'up'
, or'down'
(default:'down'
, for consistency withgraphviz
): the preferred ranking direction for acyclic layouts; see therankdir
option ofgraphviz
.subgraph_clusters
– a list of lists of vertices (default:[]
); From [dotspec]: “If supported, the layout engine will do the layout so that the nodes belonging to the cluster are drawn together, with the entire drawing of the cluster contained within a bounding rectangle. Note that, for good and bad, cluster subgraphs are not part of thedot
language, but solely a syntactic convention adhered to by certain of the layout engines.”
EXAMPLES:
sage: G = Graph({0: {1: None, 2: None}, 1: {0: None, 2: None}, ....: 2: {0: None, 1: None, 3: 'foo'}, 3: {2: 'foo'}}, ....: sparse=True) sage: print(G.graphviz_string(edge_labels=True)) graph { node_0 [label="0"]; node_1 [label="1"]; node_2 [label="2"]; node_3 [label="3"]; node_0  node_1; node_0  node_2; node_1  node_2; node_2  node_3 [label="foo"]; }
A variant, with the labels in latex, for postprocessing with
dot2tex
:sage: print(G.graphviz_string(edge_labels=True, labels="latex")) graph { node [shape="plaintext"]; node_0 [label=" ", texlbl="$0$"]; node_1 [label=" ", texlbl="$1$"]; node_2 [label=" ", texlbl="$2$"]; node_3 [label=" ", texlbl="$3$"]; node_0  node_1; node_0  node_2; node_1  node_2; node_2  node_3 [label=" ", texlbl="$\text{\texttt{foo}}$"]; }
Same, with a digraph and a color for edges:
sage: G = DiGraph({0: {1: None, 2: None}, 1: {2: None}, 2: {3: 'foo'}, 3: {}}, ....: sparse=True) sage: print(G.graphviz_string(edge_color="red")) digraph { node_0 [label="0"]; node_1 [label="1"]; node_2 [label="2"]; node_3 [label="3"]; edge [color="red"]; node_0 > node_1; node_0 > node_2; node_1 > node_2; node_2 > node_3; }
A digraph using latex labels for vertices and edges:
sage: # needs sage.symbolic sage: f(x) = 1 / x sage: g(x) = 1 / (x + 1) sage: G = DiGraph() sage: G.add_edges((i, f(i), f) for i in (1, 2, 1/2, 1/4)) sage: G.add_edges((i, g(i), g) for i in (1, 2, 1/2, 1/4)) sage: print(G.graphviz_string(labels="latex", # random ....: edge_labels=True)) digraph { node [shape="plaintext"]; node_10 [label=" ", texlbl="$1$"]; node_11 [label=" ", texlbl="$2$"]; node_3 [label=" ", texlbl="$\frac{1}{2}$"]; node_6 [label=" ", texlbl="$\frac{1}{2}$"]; node_7 [label=" ", texlbl="$\frac{1}{2}$"]; node_5 [label=" ", texlbl="$\frac{1}{3}$"]; node_8 [label=" ", texlbl="$\frac{2}{3}$"]; node_4 [label=" ", texlbl="$\frac{1}{4}$"]; node_1 [label=" ", texlbl="$2$"]; node_9 [label=" ", texlbl="$\frac{4}{5}$"]; node_0 [label=" ", texlbl="$4$"]; node_2 [label=" ", texlbl="$1$"]; node_10 > node_2 [label=" ", texlbl="$x \ {\mapsto}\ \frac{1}{x}$"]; node_10 > node_6 [label=" ", texlbl="$x \ {\mapsto}\ \frac{1}{x + 1}$"]; node_11 > node_3 [label=" ", texlbl="$x \ {\mapsto}\ \frac{1}{x}$"]; node_11 > node_5 [label=" ", texlbl="$x \ {\mapsto}\ \frac{1}{x + 1}$"]; node_7 > node_1 [label=" ", texlbl="$x \ {\mapsto}\ \frac{1}{x}$"]; node_7 > node_8 [label=" ", texlbl="$x \ {\mapsto}\ \frac{1}{x + 1}$"]; node_4 > node_0 [label=" ", texlbl="$x \ {\mapsto}\ \frac{1}{x}$"]; node_4 > node_9 [label=" ", texlbl="$x \ {\mapsto}\ \frac{1}{x + 1}$"]; } sage: print(G.graphviz_string(labels="latex", # random # needs sage.symbolic ....: color_by_label=True)) digraph { node [shape="plaintext"]; node_10 [label=" ", texlbl="$1$"]; node_11 [label=" ", texlbl="$2$"]; node_3 [label=" ", texlbl="$\frac{1}{2}$"]; node_6 [label=" ", texlbl="$\frac{1}{2}$"]; node_7 [label=" ", texlbl="$\frac{1}{2}$"]; node_5 [label=" ", texlbl="$\frac{1}{3}$"]; node_8 [label=" ", texlbl="$\frac{2}{3}$"]; node_4 [label=" ", texlbl="$\frac{1}{4}$"]; node_1 [label=" ", texlbl="$2$"]; node_9 [label=" ", texlbl="$\frac{4}{5}$"]; node_0 [label=" ", texlbl="$4$"]; node_2 [label=" ", texlbl="$1$"]; node_10 > node_2 [color = "#ff0000"]; node_10 > node_6 [color = "#00ffff"]; node_11 > node_3 [color = "#ff0000"]; node_11 > node_5 [color = "#00ffff"]; node_7 > node_1 [color = "#ff0000"]; node_7 > node_8 [color = "#00ffff"]; node_4 > node_0 [color = "#ff0000"]; node_4 > node_9 [color = "#00ffff"]; } sage: print(G.graphviz_string(labels="latex", # random # needs sage.symbolic ....: color_by_label={f: "red", g: "blue"})) digraph { node [shape="plaintext"]; node_10 [label=" ", texlbl="$1$"]; node_11 [label=" ", texlbl="$2$"]; node_3 [label=" ", texlbl="$\frac{1}{2}$"]; node_6 [label=" ", texlbl="$\frac{1}{2}$"]; node_7 [label=" ", texlbl="$\frac{1}{2}$"]; node_5 [label=" ", texlbl="$\frac{1}{3}$"]; node_8 [label=" ", texlbl="$\frac{2}{3}$"]; node_4 [label=" ", texlbl="$\frac{1}{4}$"]; node_1 [label=" ", texlbl="$2$"]; node_9 [label=" ", texlbl="$\frac{4}{5}$"]; node_0 [label=" ", texlbl="$4$"]; node_2 [label=" ", texlbl="$1$"]; node_10 > node_2 [color = "red"]; node_10 > node_6 [color = "blue"]; node_11 > node_3 [color = "red"]; node_11 > node_5 [color = "blue"]; node_7 > node_1 [color = "red"]; node_7 > node_8 [color = "blue"]; node_4 > node_0 [color = "red"]; node_4 > node_9 [color = "blue"]; }
By default
graphviz
renders digraphs using a hierarchical layout, ranking the vertices down from top to bottom. Here we specify alternative ranking directions for this layout:sage: D = DiGraph([(1, 2)]) sage: print(D.graphviz_string(rankdir="up")) digraph { rankdir=BT node_0 [label="1"]; node_1 [label="2"]; node_0 > node_1; } sage: print(D.graphviz_string(rankdir="down")) digraph { node_0 [label="1"]; node_1 [label="2"]; node_0 > node_1; } sage: print(D.graphviz_string(rankdir="left")) digraph { rankdir=RL node_0 [label="1"]; node_1 [label="2"]; node_0 > node_1; } sage: print(D.graphviz_string(rankdir="right")) digraph { rankdir=LR node_0 [label="1"]; node_1 [label="2"]; node_0 > node_1; }
Edgespecific options can also be specified by providing a function (or tuple thereof) which maps each edge to a dictionary of options. Valid options are
"color"
"dot"
(a string containing a sequence of options indot
format)"label"
(a string)"label_style"
("string"
or"latex"
)"edge_string"
(""
or">"
)"dir"
("forward"
,"back"
,"both"
or"none"
)"backward"
(boolean), instead of defining the edge in the graphviz string asu > v
it draws it asv > u [dir=back]
and instead ofu > v [dir=back]
it draws it asv > u
, this changes the way it is drawn by Graphviz’s dot program: vertexv
will be above vertexu
instead of below.
Here we state that the graph should be laid out so that edges starting from
1
are going backward (e.g. going up instead of down):sage: def edge_options(data): ....: u, v, label = data ....: return {"dir":"back"} if u == 1 else {} sage: print(G.graphviz_string(edge_options=edge_options)) # random # needs sage.symbolic digraph { node_0 [label="1"]; node_1 [label="1/2"]; node_2 [label="1/2"]; node_3 [label="2"]; node_4 [label="1/4"]; node_5 [label="4"]; node_6 [label="1/3"]; node_7 [label="2/3"]; node_8 [label="4/5"]; node_9 [label="1"]; node_10 [label="2"]; node_2 > node_3; node_2 > node_7; node_4 > node_5; node_4 > node_8; node_9 > node_0 [dir=back]; node_9 > node_2 [dir=back]; node_10 > node_1; node_10 > node_6; }
We now test all options:
sage: def edge_options(data): ....: u, v, label = data ....: options = {"color": {f: "red", g: "blue"}[label]} ....: if (u,v) == (1/2, 2): options["label"] = "coucou"; options["label_style"] = "string" ....: if (u,v) == (1/2,2/3): options["dot"] = "x=1,y=2" ....: if (u,v) == (1, 1): options["label_style"] = "latex" ....: if (u,v) == (1, 1/2): options["dir"] = "back" ....: return options sage: print(G.graphviz_string(edge_options=edge_options)) # random # needs sage.symbolic digraph { node_0 [label="1"]; node_1 [label="1/2"]; node_2 [label="1/2"]; node_3 [label="2"]; node_4 [label="1/4"]; node_5 [label="4"]; node_6 [label="1/3"]; node_7 [label="2/3"]; node_8 [label="4/5"]; node_9 [label="1"]; node_10 [label="2"]; node_2 > node_3 [label="coucou", color = "red"]; node_2 > node_7 [x=1,y=2, color = "blue"]; node_4 > node_5 [color = "red"]; node_4 > node_8 [color = "blue"]; node_9 > node_0 [label=" ", texlbl="$x \ {\mapsto}\ \frac{1}{x}$", color = "red"]; node_9 > node_2 [color = "blue", dir=back]; node_10 > node_1 [color = "red"]; node_10 > node_6 [color = "blue"]; }
We test the possible values of the
'dir'
edge option:sage: edges = [(0,1,'a'), (1,2,'b'), (2,3,'c'), (3,4,'d')] sage: G = DiGraph(edges) sage: def edge_options(data): ....: u,v,label = data ....: if label == 'a': return {'dir':'forward'} ....: if label == 'b': return {'dir':'back'} ....: if label == 'c': return {'dir':'none'} ....: if label == 'd': return {'dir':'both'} sage: print(G.graphviz_string(edge_options=edge_options)) digraph { node_0 [label="0"]; node_1 [label="1"]; node_2 [label="2"]; node_3 [label="3"]; node_4 [label="4"]; node_0 > node_1; node_1 > node_2 [dir=back]; node_2 > node_3 [dir=none]; node_3 > node_4 [dir=both]; }
We test the same graph and
'dir'
edge options but withbackward=True
, which reverses the natural direction each edge wants to be pointing for the layout:sage: def edge_options(data): ....: u,v,label = data ....: if label == 'a': return {'dir':'forward', 'backward':True} ....: if label == 'b': return {'dir':'back', 'backward':True} ....: if label == 'c': return {'dir':'none', 'backward':True} ....: if label == 'd': return {'dir':'both', 'backward':True} sage: print(G.graphviz_string(edge_options=edge_options)) digraph { node_0 [label="0"]; node_1 [label="1"]; node_2 [label="2"]; node_3 [label="3"]; node_4 [label="4"]; node_1 > node_0 [dir=back]; node_2 > node_1; node_3 > node_2 [dir=none]; node_4 > node_3 [dir=both]; }
 graphviz_to_file_named(filename, **options)#
Write a representation in the
dot
language in a file.The
dot
language is a plaintext format for graph structures. See the documentation ofgraphviz_string()
for available options.INPUT:
filename
– the name of the file to write in**options
– options for the graphviz string
EXAMPLES:
sage: G = Graph({0: {1: None, 2: None}, 1: {0: None, 2: None}, ....: 2: {0: None, 1: None, 3: 'foo'}, 3: {2: 'foo'}}, ....: sparse=True) sage: import tempfile sage: with tempfile.NamedTemporaryFile(mode="a+t") as f: ....: G.graphviz_to_file_named(f.name, edge_labels=True) ....: print(f.read()) graph { node_0 [label="0"]; node_1 [label="1"]; node_2 [label="2"]; node_3 [label="3"]; node_0  node_1; node_0  node_2; node_1  node_2; node_2  node_3 [label="foo"]; }
 greedy_dominating_set(G, k=1, vertices=None, ordering=None, return_sets=False, closest=False)#
Return a greedy distance\(k\) dominating set of the graph.
A distance\(k\) dominating set \(S\) of a graph \(G\) is a set of its vertices of minimal cardinality such that any vertex of \(G\) is in \(S\) or is at distance at most \(k\) from a vertex in \(S\). See the Wikipedia article Dominating_set.
When \(G\) is directed, vertex \(u\) can be a dominator of vertex \(v\) if there is a directed path of length at most \(k\) from \(u\) to \(v\).
This method implements a greedy heuristic to find a minimal dominatic set.
INPUT:
G
– a Graphk
– integer (default:1
); the domination distance to considervertices
– iterable container of vertices (default:None
); when specified, return a dominating set of the specified vertices onlyordering
– string (default:None
); specify the order in which to consider the verticesNone
– ifvertices
isNone
, then consider the vertices in the order given bylist(G)
. Otherwise, consider the vertices in the order of iteration ofvertices
."degree_min"
– consider the vertices by increasing degree"degree_max"
– consider the vertices by decreasing degree
return_sets
– boolean (default:False
); whether to return the vertices of the dominating set only (default), or a dictionary mapping each vertex of the dominating set to the set of vertices it dominates.closest
– boolean (default:False
); whether to attach a vertex to its closest dominator or not. This parameter is use only whenreturn_sets
isTrue
.
EXAMPLES:
Dominating sets of a path:
sage: from sage.graphs.domination import greedy_dominating_set sage: G = graphs.PathGraph(5) sage: sorted(greedy_dominating_set(G, ordering=None)) [0, 2, 4] sage: sorted(greedy_dominating_set(G, ordering="degree_min")) [0, 2, 4] sage: sorted(greedy_dominating_set(G, ordering="degree_max")) [1, 3] sage: sorted(greedy_dominating_set(G, k=2, ordering=None)) [0, 3] sage: sorted(greedy_dominating_set(G, k=2, ordering="degree_min")) [0, 4] sage: sorted(greedy_dominating_set(G, k=2, ordering="degree_max")) [1, 4] sage: greedy_dominating_set(G, k=3, ordering="degree_min", return_sets=True, closest=False) {0: {0, 1, 2, 3}, 4: {4}} sage: greedy_dominating_set(G, k=3, ordering="degree_min", return_sets=True, closest=True) {0: {0, 2, 3}, 4: {1, 4}}
Asking for a dominating set of a subset of vertices:
sage: from sage.graphs.domination import greedy_dominating_set sage: from sage.graphs.domination import is_dominating sage: G = graphs.PetersenGraph() sage: vertices = {0, 1, 2, 3, 4, 5} sage: dom = greedy_dominating_set(G, vertices=vertices, return_sets=True) sage: sorted(dom) [0, 2] sage: is_dominating(G, dom, focus=vertices) True sage: is_dominating(G, dom) False sage: dominated = [u for v in dom for u in dom[v]] sage: sorted(dominated) == sorted(vertices) True
Influence of the ordering of the vertices on the result:
sage: from sage.graphs.domination import greedy_dominating_set sage: G = graphs.StarGraph(4) sage: greedy_dominating_set(G, vertices=[0, 1, 2, 3, 4]) [0] sage: sorted(greedy_dominating_set(G, vertices=[1, 2, 3, 4, 0])) [1, 2, 3, 4]
Dominating set of a directed graph:
sage: from sage.graphs.domination import greedy_dominating_set sage: D = digraphs.Path(3) sage: sorted(greedy_dominating_set(D, vertices=[0, 1, 2])) [0, 2]
 hamiltonian_cycle(algorithm, solver='tsp', constraint_generation=None, verbose=None, verbose_constraints=0, integrality_tolerance=False)#
Return a Hamiltonian cycle/circuit of the current graph/digraph.
A graph (resp. digraph) is said to be Hamiltonian if it contains as a subgraph a cycle (resp. a circuit) going through all the vertices.
Computing a Hamiltonian cycle/circuit being NPComplete, this algorithm could run for some time depending on the instance.
ALGORITHM:
See
traveling_salesman_problem()
for ‘tsp’ algorithm andfind_hamiltonian()
fromsage.graphs.generic_graph_pyx
for ‘backtrack’ algorithm.INPUT:
algorithm
– string (default:'tsp'
); one of ‘tsp’ or ‘backtrack’solver
– string (default:None
); specify a Mixed Integer Linear Programming (MILP) solver to be used. If set toNone
, the default one is used. For more information on MILP solvers and which default solver is used, see the methodsolve
of the classMixedIntegerLinearProgram
.constraint_generation
– boolean (default:None
); whether to use constraint generation when solving the Mixed Integer Linear Program.When
constraint_generation = None
, constraint generation is used whenever the graph has a density larger than 70%.verbose
– integer (default:0
); sets the level of verbosity. Set to 0 by default, which means quiet.verbose_constraints
– boolean (default:False
); whether to display which constraints are being generatedintegrality_tolerance
– float; parameter for use with MILP solvers over an inexact base ring; seeMixedIntegerLinearProgram.get_values()
.
OUTPUT:
If using the
'tsp'
algorithm, returns a Hamiltonian cycle/circuit if it exists; otherwise, raises aEmptySetError
exception. If using the'backtrack'
algorithm, returns a pair(B, P)
. IfB
isTrue
thenP
is a Hamiltonian cycle and ifB
isFalse
,P
is a longest path found by the algorithm. Observe that ifB
isFalse
, the graph may still be Hamiltonian. The'backtrack'
algorithm is only implemented for undirected graphs.Warning
The
'backtrack'
algorithm may loop endlessly on graphs with vertices of degree 1.NOTE:
This function, as
is_hamiltonian()
, computes a Hamiltonian cycle if it exists: the user should NOT test for Hamiltonicity usingis_hamiltonian()
before calling this function, as it would result in computing it twice.The backtrack algorithm is only implemented for undirected graphs.
EXAMPLES:
The Heawood Graph is known to be Hamiltonian
sage: g = graphs.HeawoodGraph() sage: g.hamiltonian_cycle() # needs sage.numerical.mip TSP from Heawood graph: Graph on 14 vertices
The Petersen Graph, though, is not
sage: g = graphs.PetersenGraph() sage: g.hamiltonian_cycle() # needs sage.numerical.mip Traceback (most recent call last): ... EmptySetError: the given graph is not Hamiltonian
Now, using the backtrack algorithm in the Heawood graph
sage: G=graphs.HeawoodGraph() sage: G.hamiltonian_cycle(algorithm='backtrack') (True, [...])
And now in the Petersen graph
sage: G=graphs.PetersenGraph() sage: B, P = G.hamiltonian_cycle(algorithm='backtrack') sage: B False sage: len(P) 10 sage: G.has_edge(P[0], P[1]) False
Finally, we test the algorithm in a cube graph, which is Hamiltonian
sage: G=graphs.CubeGraph(3) sage: G.hamiltonian_cycle(algorithm='backtrack') (True, [...])
 hamiltonian_path(s, t=None, use_edge_labels=None, maximize=False, algorithm=False, solver='MILP', verbose=None, integrality_tolerance=0)#
Return a Hamiltonian path of the current graph/digraph.
A path is Hamiltonian if it goes through all the vertices exactly once. Computing a Hamiltonian path being NPComplete, this algorithm could run for some time depending on the instance.
When
use_edge_labels == True
, this method returns either a minimum weight hamiltonian path or a maximum weight Hamiltonian path (ifmaximize == True
).See also
INPUT:
s
– vertex (default:None
); if specified, then forces the source of the path (the method then returns a Hamiltonian path starting ats
)t
– vertex (default:None
); if specified, then forces the destination of the path (the method then returns a Hamiltonian path ending att
)use_edge_labels
– boolean (default:False
); whether to compute a weighted hamiltonian path where the weight of an edge is defined by its label (a label set toNone
or{}
being considered as a weight of \(1\)), or a nonweighted hamiltonian pathmaximize
– boolean (default:False
); whether to compute a minimum (default) or a maximum (whenmaximize == True
) weight hamiltonian path. This parameter is considered only ifuse_edge_labels == True
.algorithm
– string (default:"MILP"
); the algorithm the use among"MILP"
and"backtrack"
; two remarks on this respect:While the MILP formulation returns an exact answer, the backtrack algorithm is a randomized heuristic.
The backtrack algorithm does not support edge weighting, so setting
use_edge_labels=True
will force the use of the MILP algorithm.
solver
– string (default:None
); specify a Mixed Integer Linear Programming (MILP) solver to be used. If set toNone
, the default one is used. For more information on MILP solvers and which default solver is used, see the methodsolve
of the classMixedIntegerLinearProgram
.verbose
– integer (default:0
); sets the level of verbosity. Set to 0 by default, which means quiet.integrality_tolerance
– float; parameter for use with MILP solvers over an inexact base ring; seeMixedIntegerLinearProgram.get_values()
.
OUTPUT:
A subgraph of
self
corresponding to a (directed ifself
is directed) hamiltonian path. If no hamiltonian path is found, returnNone
. Ifuse_edge_labels == True
, a pairweight, path
is returned.EXAMPLES:
The \(3 \times 3\)grid has an Hamiltonian path, an hamiltonian path starting from vertex \((0, 0)\) and ending at vertex \((2, 2)\), but no Hamiltonian path starting from \((0, 0)\) and ending at \((0, 1)\):
sage: # needs sage.numerical.mip sage: g = graphs.Grid2dGraph(3, 3) sage: g.hamiltonian_path() Hamiltonian path from 2D Grid Graph for [3, 3]: Graph on 9 vertices sage: g.hamiltonian_path(s=(0, 0), t=(2, 2)) Hamiltonian path from 2D Grid Graph for [3, 3]: Graph on 9 vertices sage: g.hamiltonian_path(s=(0, 0), t=(2, 2), use_edge_labels=True) (8, Hamiltonian path from 2D Grid Graph for [3, 3]: Graph on 9 vertices) sage: g.hamiltonian_path(s=(0, 0), t=(0, 1)) is None True sage: g.hamiltonian_path(s=(0, 0), t=(0, 1), use_edge_labels=True) (0, None)
 has_edge(u, v=None, label=None)#
Check whether
(u, v)
is an edge of the (di)graph.INPUT: The following forms are accepted:
G.has_edge( 1, 2 )
G.has_edge( (1, 2) )
G.has_edge( 1, 2, ‘label’ )
G.has_edge( (1, 2, ‘label’) )
EXAMPLES:
sage: graphs.EmptyGraph().has_edge(9, 2) False sage: DiGraph().has_edge(9, 2) False sage: G = Graph(sparse=True) sage: G.add_edge(0, 1, "label") sage: G.has_edge(0, 1, "different label") False sage: G.has_edge(0, 1, "label") True
 has_loops()#
Return whether there are loops in the (di)graph
EXAMPLES:
sage: G = Graph(loops=True); G Looped graph on 0 vertices sage: G.has_loops() False sage: G.allows_loops() True sage: G.add_edge((0, 0)) sage: G.has_loops() True sage: G.loops() [(0, 0, None)] sage: G.allow_loops(False); G Graph on 1 vertex sage: G.has_loops() False sage: G.edges(sort=True) [] sage: D = DiGraph(loops=True); D Looped digraph on 0 vertices sage: D.has_loops() False sage: D.allows_loops() True sage: D.add_edge((0, 0)) sage: D.has_loops() True sage: D.loops() [(0, 0, None)] sage: D.allow_loops(False); D Digraph on 1 vertex sage: D.has_loops() False sage: D.edges(sort=True) []
 has_multiple_edges(to_undirected=False)#
Return whether there are multiple edges in the (di)graph.
INPUT:
to_undirected
– (default:False)
; ifTrue
, runs the test on the undirected version of a DiGraph. Otherwise, treats DiGraph edges(u, v)
and(v, u)
as unique individual edges.
EXAMPLES:
sage: G = Graph(multiedges=True, sparse=True); G Multigraph on 0 vertices sage: G.has_multiple_edges() False sage: G.allows_multiple_edges() True sage: G.add_edges([(0, 1)] * 3) sage: G.has_multiple_edges() True sage: G.multiple_edges() [(0, 1, None), (0, 1, None), (0, 1, None)] sage: G.allow_multiple_edges(False); G Graph on 2 vertices sage: G.has_multiple_edges() False sage: G.edges(sort=True) [(0, 1, None)] sage: D = DiGraph(multiedges=True, sparse=True); D Multidigraph on 0 vertices sage: D.has_multiple_edges() False sage: D.allows_multiple_edges() True sage: D.add_edges([(0, 1)] * 3) sage: D.has_multiple_edges() True sage: D.multiple_edges() [(0, 1, None), (0, 1, None), (0, 1, None)] sage: D.allow_multiple_edges(False); D Digraph on 2 vertices sage: D.has_multiple_edges() False sage: D.edges(sort=True) [(0, 1, None)] sage: G = DiGraph({1: {2: 'h'}, 2: {1: 'g'}}, sparse=True) sage: G.has_multiple_edges() False sage: G.has_multiple_edges(to_undirected=True) True sage: G.multiple_edges() [] sage: G.multiple_edges(to_undirected=True) [(1, 2, 'h'), (2, 1, 'g')]
A loop is not a multiedge:
sage: g = Graph(loops=True, multiedges=True) sage: g.add_edge(0, 0) sage: g.has_multiple_edges() False
 has_vertex(vertex)#
Check if
vertex
is one of the vertices of this graph.INPUT:
vertex
– the name of a vertex (seeadd_vertex()
)
EXAMPLES:
sage: g = Graph({0: [1, 2, 3], 2: [4]}); g Graph on 5 vertices sage: 2 in g True sage: 10 in g False sage: graphs.PetersenGraph().has_vertex(99) False
 igraph_graph(vertex_list=None, vertex_attrs={}, edge_attrs={})#
Return an
igraph
graph from the Sage graph.Optionally, it is possible to add vertex attributes and edge attributes to the output graph.
Note
This routine needs the optional package igraph to be installed: to do so, it is enough to run
sage i python_igraph
. For more information on the Python version of igraph, see http://igraph.org/python/.INPUT:
vertex_list
– list (default:None
); defines a mapping from the vertices of the graph to consecutive integers in(0, \ldots, n1)`. Otherwise, the result of :meth:`vertices` will be used instead. Because :meth:`vertices` only works if the vertices can be sorted, using ``vertex_list
is useful when working with possibly nonsortable objects in Python 3.vertex_attrs
– dictionary (default:{}
); a dictionary where the key is a string (the attribute name), and the value is an iterable containing in position \(i\) the label of the \(i\)th vertex in the listvertex_list
if it is given or invertices()
whenvertex_list == None
(see http://igraph.org/python/doc/igraph.Graphclass.html#__init__ for more information)edge_attrs
– dictionary (default:{}
); a dictionary where the key is a string (the attribute name), and the value is an iterable containing in position \(i\) the label of the \(i\)th edge in the list outputted byedge_iterator()
(see http://igraph.org/python/doc/igraph.Graphclass.html#__init__ for more information)
Note
In
igraph
, a graph is weighted if the edge labels have attributeweight
. Hence, to create a weighted graph, it is enough to add this attribute.Note
Often, Sage uses its own defined types for integer/floats. These types may not be igraphcompatible (see example below).
EXAMPLES:
Standard conversion:
sage: G = graphs.TetrahedralGraph() sage: H = G.igraph_graph() # optional  python_igraph sage: H.summary() # optional  python_igraph 'IGRAPH U 4 6  ' sage: G = digraphs.Path(3) sage: H = G.igraph_graph() # optional  python_igraph sage: H.summary() # optional  python_igraph 'IGRAPH D 3 2  '
Adding edge attributes:
sage: G = Graph([(1, 2, 'a'), (2, 3, 'b')]) sage: E = list(G.edge_iterator()) sage: H = G.igraph_graph(edge_attrs={'label': [e[2] for e in E]}) # optional  python_igraph sage: H.es['label'] # optional  python_igraph ['a', 'b']
If edges have an attribute
weight
, the igraph graph is considered weighted:sage: G = Graph([(1, 2, {'weight': 1}), (2, 3, {'weight': 2})]) sage: E = list(G.edge_iterator()) sage: H = G.igraph_graph(edge_attrs={'weight': [e[2]['weight'] for e in E]}) # optional  python_igraph sage: H.is_weighted() # optional  python_igraph True sage: H.es['weight'] # optional  python_igraph [1, 2]
Adding vertex attributes:
sage: G = graphs.GridGraph([2, 2]) sage: H = G.igraph_graph(vertex_attrs={'name': G.vertices(sort=True)}) # optional  python_igraph sage: H.vs()['name'] # optional  python_igraph [(0, 0), (0, 1), (1, 0), (1, 1)]
Providing a mapping from vertices to consecutive integers:
sage: G = graphs.GridGraph([2, 2]) sage: V = list(G) sage: H = G.igraph_graph(vertex_list=V, vertex_attrs={'name': V}) # optional  python_igraph sage: H.vs()['name'] == V # optional  python_igraph True
Sometimes, Sage integer/floats are not compatible with igraph:
sage: G = Graph([(0, 1, 2)]) sage: E = list(G.edge_iterator()) sage: H = G.igraph_graph(edge_attrs={'capacity': [e[2] for e in E]}) # optional  python_igraph sage: H.maxflow_value(0, 1, 'capacity') # optional  python_igraph 1.0 sage: H = G.igraph_graph(edge_attrs={'capacity': [float(e[2]) for e in E]}) # optional  python_igraph sage: H.maxflow_value(0, 1, 'capacity') # optional  python_igraph 2.0
 incidence_matrix(oriented, sparse=None, vertices=True, edges=None, base_ring=None, **kwds)#
Return the incidence matrix of the (di)graph.
Each row is a vertex, and each column is an edge. The vertices are ordered as obtained by the method
vertices()
, except when parametervertices
is given (see below), and the edges as obtained by the methodedge_iterator()
.If the graph is not directed, then return a matrix with entries in \(\{0,1,2\}\). Each column will either contain two \(1\) (at the position of the endpoint of the edge), or one \(2\) (if the corresponding edge is a loop).
If the graph is directed return a matrix in \(\{1,0,1\}\) where \(1\) and \(+1\) correspond respectively to the source and the target of the edge. A loop will correspond to a zero column. In particular, it is not possible to recover the loops of an oriented graph from its incidence matrix.
See the Wikipedia article Incidence_matrix for more information.
INPUT:
oriented
– boolean (default:None
); when set toTrue
, the matrix will be oriented (i.e. with entries in \(1\), \(0\), \(1\)) and if set toFalse
the matrix will be not oriented (i.e. with entries in \(0\), \(1\), \(2\)). By default, this argument is inferred from the graph type. Note that in the case the graph is not directed and with the optiondirected=True
, a somewhat random direction is chosen for each edge.sparse
– boolean (default:True
); whether to use a sparse or a dense matrixvertices
– list (default:None
); when specified, the \(i\)th row of the matrix corresponds to the \(i\)th vertex in the ordering ofvertices
, otherwise, the \(i\)th row of the matrix corresponds to the \(i\)th vertex in the ordering given by methodvertices()
.edges
– list (default:None
); when specified, the \(i\)th column of the matrix corresponds to the \(i\)th edge in the ordering ofedges
, otherwise, the \(i\)th column of the matrix corresponds to the \(i\)th edge in the ordering given by methodedge_iterator()
.base_ring
– a ring (default:ZZ
); the base ring of the matrix space to use.**kwds
– other keywords to pass tomatrix()
.
EXAMPLES:
sage: G = graphs.PetersenGraph() sage: G.incidence_matrix() # needs sage.modules [1 1 1 0 0 0 0 0 0 0 0 0 0 0 0] [1 0 0 1 1 0 0 0 0 0 0 0 0 0 0] [0 0 0 1 0 1 1 0 0 0 0 0 0 0 0] [0 0 0 0 0 1 0 1 1 0 0 0 0 0 0] [0 1 0 0 0 0 0 1 0 1 0 0 0 0 0] [0 0 1 0 0 0 0 0 0 0 1 1 0 0 0] [0 0 0 0 1 0 0 0 0 0 0 0 1 1 0] [0 0 0 0 0 0 1 0 0 0 1 0 0 0 1] [0 0 0 0 0 0 0 0 1 0 0 1 1 0 0] [0 0 0 0 0 0 0 0 0 1 0 0 0 1 1] sage: G.incidence_matrix(oriented=True) # needs sage.modules [1 1 1 0 0 0 0 0 0 0 0 0 0 0 0] [ 1 0 0 1 1 0 0 0 0 0 0 0 0 0 0] [ 0 0 0 1 0 1 1 0 0 0 0 0 0 0 0] [ 0 0 0 0 0 1 0 1 1 0 0 0 0 0 0] [ 0 1 0 0 0 0 0 1 0 1 0 0 0 0 0] [ 0 0 1 0 0 0 0 0 0 0 1 1 0 0 0] [ 0 0 0 0 1 0 0 0 0 0 0 0 1 1 0] [ 0 0 0 0 0 0 1 0 0 0 1 0 0 0 1] [ 0 0 0 0 0 0 0 0 1 0 0 1 1 0 0] [ 0 0 0 0 0 0 0 0 0 1 0 0 0 1 1] sage: G = digraphs.Circulant(4, [1, 3]) sage: G.incidence_matrix() # needs sage.modules [1 1 1 0 0 0 1 0] [ 1 0 1 1 1 0 0 0] [ 0 0 0 1 1 1 0 1] [ 0 1 0 0 0 1 1 1] sage: graphs.CompleteGraph(3).incidence_matrix() # needs sage.modules [1 1 0] [1 0 1] [0 1 1] sage: G = Graph([(0, 0), (0, 1), (0, 1)], loops=True, multiedges=True) sage: G.incidence_matrix(oriented=False) # needs sage.modules [2 1 1] [0 1 1]
A well known result states that the product of the (oriented) incidence matrix with its transpose of a (nonoriented graph) is in fact the Kirchhoff matrix:
sage: G = graphs.PetersenGraph() sage: m = G.incidence_matrix(oriented=True) # needs sage.modules sage: m * m.transpose() == G.kirchhoff_matrix() # needs sage.modules True sage: K = graphs.CompleteGraph(3) sage: m = K.incidence_matrix(oriented=True) # needs sage.modules sage: m * m.transpose() == K.kirchhoff_matrix() # needs sage.modules True sage: H = Graph([(0, 0), (0, 1), (0, 1)], loops=True, multiedges=True) sage: m = H.incidence_matrix(oriented=True) # needs sage.modules sage: m * m.transpose() == H.kirchhoff_matrix() # needs sage.modules True
A different ordering of the vertices:
sage: P5 = graphs.PathGraph(5) sage: P5.incidence_matrix() # needs sage.modules [1 0 0 0] [1 1 0 0] [0 1 1 0] [0 0 1 1] [0 0 0 1] sage: P5.incidence_matrix(vertices=[2, 4, 1, 3, 0]) # needs sage.modules [0 1 1 0] [0 0 0 1] [1 1 0 0] [0 0 1 1] [1 0 0 0]
A different ordering of the edges:
sage: E = list(P5.edge_iterator(labels=False)) sage: P5.incidence_matrix(edges=E[::1]) # needs sage.modules [0 0 0 1] [0 0 1 1] [0 1 1 0] [1 1 0 0] [1 0 0 0] sage: P5.incidence_matrix(vertices=[2, 4, 1, 3, 0], edges=E[::1]) # needs sage.modules [0 1 1 0] [1 0 0 0] [0 0 1 1] [1 1 0 0] [0 0 0 1]
A different base ring:
sage: P5.incidence_matrix(base_ring=RDF) # needs sage.modules [1.0 0.0 0.0 0.0] [1.0 1.0 0.0 0.0] [0.0 1.0 1.0 0.0] [0.0 0.0 1.0 1.0] [0.0 0.0 0.0 1.0]
Creating an immutable matrix:
sage: m = P5.incidence_matrix(immutable=True); m # needs sage.modules [1 0 0 0] [1 1 0 0] [0 1 1 0] [0 0 1 1] [0 0 0 1] sage: m[1,2] = 1 # needs sage.modules Traceback (most recent call last): ... ValueError: matrix is immutable; please change a copy instead (i.e., use copy(M) to change a copy of M).
 is_bipartite(certificate=False)#
Check whether the graph is bipartite.
Traverse the graph \(G\) with breadthfirstsearch and color nodes.
INPUT:
certificate
– boolean (default:False
); whether to return a certificate. If set toTrue
, the certificate returned is a proper 2coloring when \(G\) is bipartite, and an odd cycle otherwise.
EXAMPLES:
sage: graphs.CycleGraph(4).is_bipartite() True sage: graphs.CycleGraph(5).is_bipartite() False sage: graphs.RandomBipartite(10, 10, 0.7).is_bipartite() # needs numpy True
A random graph is very rarely bipartite:
sage: g = graphs.PetersenGraph() sage: g.is_bipartite() False sage: false, oddcycle = g.is_bipartite(certificate=True) sage: len(oddcycle) % 2 1
The method works identically with oriented graphs:
sage: g = DiGraph({0: [1, 2, 3], 2: [1], 3: [4]}) sage: g.is_bipartite() False sage: false, oddcycle = g.is_bipartite(certificate=True) sage: len(oddcycle) % 2 1 sage: graphs.CycleGraph(4).random_orientation().is_bipartite() True sage: graphs.CycleGraph(5).random_orientation().is_bipartite() False
 is_cayley(return_group=False, mapping=False, generators=False, allow_disconnected=False)#
Check whether the graph is a Cayley graph.
If none of the parameters are
True
, return a boolean indicating whether the graph is a Cayley graph. Otherwise, return a tuple containing said boolean and the requested data. If the graph is not a Cayley graph, each of the data will beNone
.The empty graph is defined to be not a Cayley graph.
Note
For this routine to work on all graphs, the optional package
gap_packages
needs to be installed: to do so, it is enough to runsage i gap_packages
.INPUT:
return_group
(boolean;False
) – IfTrue
, return a group for which the graph is a Cayley graph.mapping
(boolean;False
) – IfTrue
, return a mapping from vertices to group elements.generators
(boolean;False
) – IfTrue
, return the generating set of the Cayley graph.allow_disconnected
(boolean;False
) – IfTrue
, disconnected graphs are considered Cayley if they can be obtained from the Cayley construction with a generating set that does not generate the group.
ALGORITHM:
For connected graphs, find a regular subgroup of the automorphism group. For disconnected graphs, check that the graph is vertextransitive and perform the check on one of its connected components. If a simple graph has density over 1/2, perform the check on its complement as its disconnectedness may increase performance.
EXAMPLES:
A Petersen Graph is not a Cayley graph:
sage: g = graphs.PetersenGraph() sage: g.is_cayley() # needs sage.groups False
A Cayley digraph is a Cayley graph:
sage: C7 = groups.permutation.Cyclic(7) # needs sage.groups sage: S = [(1,2,3,4,5,6,7), (1,3,5,7,2,4,6), (1,5,2,6,3,7,4)] sage: d = C7.cayley_graph(generators=S) # needs sage.groups sage: d.is_cayley() # needs sage.groups True
Graphs with loops and multiedges will have identity and repeated elements, respectively, among the generators:
sage: # needs sage.rings.finite_rings sage: g = Graph(graphs.PaleyGraph(9), loops=True, multiedges=True) sage: g.add_edges([(u, u) for u in g]) sage: g.add_edges([(u, u+1) for u in g]) sage: _, S = g.is_cayley(generators=True) # needs sage.groups sage: S # random # needs sage.groups [(), (0,2,1)(a,a + 2,a + 1)(2*a,2*a + 2,2*a + 1), (0,2,1)(a,a + 2,a + 1)(2*a,2*a + 2,2*a + 1), (0,1,2)(a,a + 1,a + 2)(2*a,2*a + 1,2*a + 2), (0,1,2)(a,a + 1,a + 2)(2*a,2*a + 1,2*a + 2), (0,2*a + 2,a + 1)(1,2*a,a + 2)(2,2*a + 1,a), (0,a + 1,2*a + 2)(1,a + 2,2*a)(2,a,2*a + 1)]
 is_chordal(certificate=False, algorithm='B')#
Check whether the given graph is chordal.
A Graph \(G\) is said to be chordal if it contains no induced hole (a cycle of length at least 4).
Alternatively, chordality can be defined using a Perfect Elimination Order:
A Perfect Elimination Order of a graph \(G\) is an ordering \(v_1,...,v_n\) of its vertex set such that for all \(i\), the neighbors of \(v_i\) whose index is greater that \(i\) induce a complete subgraph in \(G\). Hence, the graph \(G\) can be totally erased by successively removing vertices whose neighborhood is a clique (also called simplicial vertices) [FG1965].
(It can be seen that if \(G\) contains an induced hole, then it cannot have a perfe