# Directed graphs¶

This module implements functions and operations involving directed graphs. Here is what they can do

Graph basic operations:

 layout_acyclic_dummy() Compute a (dummy) ranked layout so that all edges point upward. layout_acyclic() Compute a ranked layout so that all edges point upward. reverse() Return a copy of digraph with edges reversed in direction. reverse_edge() Reverse an edge. reverse_edges() Reverse a list of edges. out_degree_sequence() Return the outdegree sequence. out_degree_iterator() Same as degree_iterator, but for out degree. out_degree() Same as degree, but for out degree. in_degree_sequence() Return the indegree sequence of this digraph. in_degree_iterator() Same as degree_iterator, but for in degree. in_degree() Same as degree, but for in-degree. neighbors_out() Return the list of the out-neighbors of a given vertex. neighbor_out_iterator() Return an iterator over the out-neighbors of a given vertex. neighbors_in() Return the list of the in-neighbors of a given vertex. neighbor_in_iterator() Return an iterator over the in-neighbors of vertex. outgoing_edges() Return a list of edges departing from vertices. outgoing_edge_iterator() Return an iterator over all departing edges from vertices incoming_edges() Return a list of edges arriving at vertices. incoming_edge_iterator() Return an iterator over all arriving edges from vertices sources() Return the list of all sources (vertices without incoming edges) of this digraph. sinks() Return the list of all sinks (vertices without outgoing edges) of this digraph. to_undirected() Return an undirected version of the graph. to_directed() Since the graph is already directed, simply returns a copy of itself. is_directed() Since digraph is directed, returns True. dig6_string() Return the dig6 representation of the digraph as an ASCII string.

Paths and cycles:

 all_paths_iterator() Return an iterator over the paths of self. all_simple_paths() Return a list of all the simple paths of self starting with one of the given vertices. all_cycles_iterator() Return an iterator over all the cycles of self starting with one of the given vertices. all_simple_cycles() Return a list of all simple cycles of self.

Representation theory:

 path_semigroup() Return the (partial) semigroup formed by the paths of the digraph.

Connectivity:

 is_strongly_connected() Check whether the current DiGraph is strongly connected. strongly_connected_components_digraph() Return the digraph of the strongly connected components strongly_connected_components_subgraphs() Return the strongly connected components as a list of subgraphs. strongly_connected_component_containing_vertex() Return the strongly connected component containing a given vertex strongly_connected_components() Return the list of strongly connected components. immediate_dominators() Return the immediate dominators of all vertices reachable from $$root$$. strong_articulation_points() Return the strong articulation points of this digraph.

Acyclicity:

 is_directed_acyclic() Check whether the digraph is acyclic or not. is_transitive() Check whether the digraph is transitive or not. is_aperiodic() Check whether the digraph is aperiodic or not. is_tournament() Check whether the digraph is a tournament. period() Return the period of the digraph. level_sets() Return the level set decomposition of the digraph. topological_sort_generator() Return a list of all topological sorts of the digraph if it is acyclic topological_sort() Return a topological sort of the digraph if it is acyclic

Hard stuff:

 feedback_edge_set() Compute the minimum feedback edge (arc) set of a digraph

Miscellanous:

 flow_polytope() Compute the flow polytope of a digraph degree_polynomial() Return the generating polynomial of degrees of vertices in self.

## Methods¶

class sage.graphs.digraph.DiGraph(data=None, pos=None, loops=None, format=None, weighted=None, data_structure='sparse', vertex_labels=True, name=None, multiedges=None, convert_empty_dict_labels_to_None=None, sparse=True, immutable=False)

Directed graph.

A digraph or directed graph is a set of vertices connected by oriented edges. See also the Wikipedia article Directed_graph. For a collection of pre-defined digraphs, see the digraph_generators module.

A DiGraph object has many methods whose list can be obtained by typing g.<tab> (i.e. hit the ‘tab’ key) or by reading the documentation of digraph, generic_graph, and graph.

INPUT:

By default, a DiGraph object is simple (i.e. no loops nor multiple edges) and unweighted. This can be easily tuned with the appropriate flags (see below).

• data – can be any of the following (see the format argument):

1. DiGraph() – build a digraph on 0 vertices

2. DiGraph(5) – return an edgeless digraph on the 5 vertices 0,…,4

3. DiGraph([list_of_vertices, list_of_edges]) – return a digraph with given vertices/edges

To bypass auto-detection, prefer the more explicit DiGraph([V, E], format='vertices_and_edges').

4. DiGraph(list_of_edges) – return a digraph with a given list of edges (see documentation of add_edges()).

To bypass auto-detection, prefer the more explicit DiGraph(L, format='list_of_edges').

5. DiGraph({1: [2,3,4], 3: [4]}) – return a digraph by associating to each vertex the list of its out-neighbors.

To bypass auto-detection, prefer the more explicit DiGraph(D, format='dict_of_lists').

6. DiGraph({1: {2: 'a', 3: 'b'}, 3: {2: 'c'}}) – return a digraph by associating a list of out-neighbors to each vertex and providing its edge label.

To bypass auto-detection, prefer the more explicit DiGraph(D, format='dict_of_dicts').

For digraphs with multiple edges, you can provide a list of labels instead, e.g.: DiGraph({1: {2: ['a1', 'a2'], 3:['b']}, 3:{2:['c']}}).

7. DiGraph(a_matrix) – return a digraph with given (weighted) adjacency matrix (see documentation of adjacency_matrix()).

To bypass auto-detection, prefer the more explicit DiGraph(M, format='adjacency_matrix'). To take weights into account, use format='weighted_adjacency_matrix' instead.

8. DiGraph(a_nonsquare_matrix) – return a digraph with given incidence matrix (see documentation of incidence_matrix()).

To bypass auto-detection, prefer the more explicit DiGraph(M, format='incidence_matrix').

9. DiGraph([V, f]) – return a digraph with a vertex set V and an edge $$u,v$$ whenever $$f(u, v)$$ is True. Example: DiGraph([ [1..10], lambda x,y: abs(x - y).is_square()])

10. DiGraph('FOC@?OC@_?') – return a digraph from a dig6 string (see documentation of dig6_string()).

11. DiGraph(another_digraph) – return a digraph from a Sage (di)graph, pygraphviz digraph, NetworkX digraph, or igraph digraph.

• pos – dict (default: None); a positioning dictionary. For example, the spring layout from NetworkX for the 5-cycle is:

{0: [-0.91679746, 0.88169588],
1: [ 0.47294849, 1.125     ],
2: [ 1.125     ,-0.12867615],
3: [ 0.12743933,-1.125     ],
4: [-1.125     ,-0.50118505]}

• name – string (default: None); gives the graph a name (e.g., name=”complete”)

• loops – boolean (default: None); whether to allow loops (ignored if data is an instance of the DiGraph class)

• multiedges – boolean (default: None); whether to allow multiple edges (ignored if data is an instance of the DiGraph class)

• weighted – boolean (default: None); whether digraph thinks of itself as weighted or not. See self.weighted()

• format – string (default: None); if set to None, DiGraph tries to guess input’s format. To avoid this possibly time-consuming step, one of the following values can be specified (see description above): "int", "dig6", "rule", "list_of_edges", "dict_of_lists", "dict_of_dicts", "adjacency_matrix", "weighted_adjacency_matrix", "incidence_matrix", "NX", "igraph".

• sparse – boolean (default: True); sparse=True is an alias for data_structure="sparse", and sparse=False is an alias for data_structure="dense"

• data_structure – string (default: "sparse"); one of the following (for more information, see overview):

• immutable – boolean (default: False); whether to create a immutable digraph. Note that immutable=True is actually a shortcut for data_structure='static_sparse'.

• vertex_labels – boolean (default: True); whether to allow any object as a vertex (slower), or only the integers $$0,...,n-1$$, where $$n$$ is the number of vertices.

• convert_empty_dict_labels_to_None – boolean (default: None); this arguments sets the default edge labels used by NetworkX (empty dictionaries) to be replaced by None, the default Sage edge label. It is set to True iff a NetworkX graph is on the input.

EXAMPLES:

1. A dictionary of dictionaries:

sage: g = DiGraph({0: {1: 'x', 2: 'z', 3: 'a'}, 2: {5: 'out'}}); g
Digraph on 5 vertices


The labels (‘x’, ‘z’, ‘a’, ‘out’) are labels for edges. For example, ‘out’ is the label for the edge from 2 to 5. Labels can be used as weights, if all the labels share some common parent.

2. A dictionary of lists (or iterables):

sage: g = DiGraph({0: [1, 2, 3], 2: [4]}); g
Digraph on 5 vertices
sage: g = DiGraph({0: (1, 2, 3), 2: (4,)}); g
Digraph on 5 vertices

3. A list of vertices and a function describing adjacencies. Note that the list of vertices and the function must be enclosed in a list (i.e., [list of vertices, function]).

We construct a graph on the integers 1 through 12 such that there is a directed edge from $$i$$ to $$j$$ if and only if $$i$$ divides $$j$$:

sage: g = DiGraph([[1..12], lambda i,j: i != j and i.divides(j)])
sage: g.vertices()
[1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12]
[0 1 1 1 1 1 1 1 1 1 1 1]
[0 0 0 1 0 1 0 1 0 1 0 1]
[0 0 0 0 0 1 0 0 1 0 0 1]
[0 0 0 0 0 0 0 1 0 0 0 1]
[0 0 0 0 0 0 0 0 0 1 0 0]
[0 0 0 0 0 0 0 0 0 0 0 1]
[0 0 0 0 0 0 0 0 0 0 0 0]
[0 0 0 0 0 0 0 0 0 0 0 0]
[0 0 0 0 0 0 0 0 0 0 0 0]
[0 0 0 0 0 0 0 0 0 0 0 0]
[0 0 0 0 0 0 0 0 0 0 0 0]
[0 0 0 0 0 0 0 0 0 0 0 0]

4. A Sage matrix: Note: If format is not specified, then Sage assumes a square matrix is an adjacency matrix, and a nonsquare matrix is an incidence matrix.

sage: M = Matrix([[0, 1, 1, 1, 0],[0, 0, 0, 0, 0],[0, 0, 0, 0, 1],[0, 0, 0, 0, 0],[0, 0, 0, 0, 0]]); M
[0 1 1 1 0]
[0 0 0 0 0]
[0 0 0 0 1]
[0 0 0 0 0]
[0 0 0 0 0]
sage: DiGraph(M)
Digraph on 5 vertices

sage: M = Matrix([[0,1,-1],[-1,0,-1/2],[1,1/2,0]]); M
[   0    1   -1]
[  -1    0 -1/2]
[   1  1/2    0]
sage: G = DiGraph(M,sparse=True,weighted=True); G
Digraph on 3 vertices
sage: G.weighted()
True

• an incidence matrix:

sage: M = Matrix(6, [-1,0,0,0,1, 1,-1,0,0,0, 0,1,-1,0,0, 0,0,1,-1,0, 0,0,0,1,-1, 0,0,0,0,0]); M
[-1  0  0  0  1]
[ 1 -1  0  0  0]
[ 0  1 -1  0  0]
[ 0  0  1 -1  0]
[ 0  0  0  1 -1]
[ 0  0  0  0  0]
sage: DiGraph(M)
Digraph on 6 vertices

5. A dig6 string: Sage automatically recognizes whether a string is in dig6 format, which is a directed version of graph6:

sage: D = DiGraph('IRAaDCIIOWEOKcPWAo')
sage: D
Digraph on 10 vertices

Traceback (most recent call last):
...
RuntimeError: the string (IRAaDCIIOEOKcPWAo) seems corrupt: for n = 10, the string is too short

Traceback (most recent call last):
...
RuntimeError: the string seems corrupt: valid characters are
[email protected][\]^_abcdefghijklmnopqrstuvwxyz{|}~

6. A NetworkX XDiGraph:

sage: import networkx
sage: g = networkx.MultiDiGraph({0: [1, 2, 3], 2: [4]})
sage: DiGraph(g)
Digraph on 5 vertices

7. A NetworkX digraph:

sage: import networkx
sage: g = networkx.DiGraph({0: [1, 2, 3], 2: [4]})
sage: DiGraph(g)
Digraph on 5 vertices

8. An igraph directed Graph (see also igraph_graph()):

sage: import igraph                                  # optional - python_igraph
sage: g = igraph.Graph([(0,1),(0,2)], directed=True) # optional - python_igraph
sage: DiGraph(g)                                     # optional - python_igraph
Digraph on 3 vertices


If vertex_labels is True, the names of the vertices are given by the vertex attribute 'name', if available:

sage: g = igraph.Graph([(0,1),(0,2)], directed=True, vertex_attrs={'name':['a','b','c']})  # optional - python_igraph
sage: DiGraph(g).vertices()                                                                # optional - python_igraph
['a', 'b', 'c']
sage: g = igraph.Graph([(0,1),(0,2)], directed=True, vertex_attrs={'label':['a','b','c']}) # optional - python_igraph
sage: DiGraph(g).vertices()                                                                # optional - python_igraph
[0, 1, 2]


If the igraph Graph has edge attributes, they are used as edge labels:

sage: g = igraph.Graph([(0,1),(0,2)], directed=True, edge_attrs={'name':['a','b'], 'weight':[1,3]}) # optional - python_igraph
sage: DiGraph(g).edges()                                                                            # optional - python_igraph
[(0, 1, {'name': 'a', 'weight': 1}), (0, 2, {'name': 'b', 'weight': 3})]

all_cycles_iterator(starting_vertices=None, simple=False, rooted=False, max_length=None, trivial=False)

Return an iterator over all the cycles of self starting with one of the given vertices.

The cycles are enumerated in increasing length order.

INPUT:

• starting_vertices – iterable (default: None); vertices from which the cycles must start. If None, then all vertices of the graph can be starting points. This argument is necessary if rooted is set to True.
• simple – boolean (default: False); if set to True, then only simple cycles are considered. A cycle is simple if the only vertex occuring twice in it is the starting and ending one.
• rooted – boolean (default: False); if set to False, then cycles differing only by their starting vertex are considered the same (e.g. ['a', 'b', 'c', 'a'] and ['b', 'c', 'a', 'b']). Otherwise, all cycles are enumerated.
• max_length – non negative integer (default: None); the maximum length of the enumerated paths. If set to None, then all lengths are allowed.
• trivial - boolean (default: False); if set to True, then the empty paths are also enumerated.

OUTPUT:

iterator

AUTHOR:

Alexandre Blondin Masse

EXAMPLES:

sage: g = DiGraph({'a': ['a', 'b'], 'b': ['c'], 'c': ['d'], 'd': ['c']}, loops=True)
sage: it = g.all_cycles_iterator()
sage: for _ in range(7): print(next(it))
['a', 'a']
['a', 'a', 'a']
['c', 'd', 'c']
['a', 'a', 'a', 'a']
['a', 'a', 'a', 'a', 'a']
['c', 'd', 'c', 'd', 'c']
['a', 'a', 'a', 'a', 'a', 'a']


There are no cycles in the empty graph and in acyclic graphs:

sage: g = DiGraph()
sage: it = g.all_cycles_iterator()
sage: list(it)
[]
sage: g = DiGraph({0:[1]})
sage: it = g.all_cycles_iterator()
sage: list(it)
[]


It is possible to restrict the starting vertices of the cycles:

sage: g = DiGraph({'a': ['a', 'b'], 'b': ['c'], 'c': ['d'], 'd': ['c']}, loops=True)
sage: it = g.all_cycles_iterator(starting_vertices=['b', 'c'])
sage: for _ in range(3): print(next(it))
['c', 'd', 'c']
['c', 'd', 'c', 'd', 'c']
['c', 'd', 'c', 'd', 'c', 'd', 'c']


Also, one can bound the length of the cycles:

sage: it = g.all_cycles_iterator(max_length=3)
sage: list(it)
[['a', 'a'], ['a', 'a', 'a'], ['c', 'd', 'c'],
['a', 'a', 'a', 'a']]


By default, cycles differing only by their starting point are not all enumerated, but this may be parametrized:

sage: it = g.all_cycles_iterator(max_length=3, rooted=False)
sage: list(it)
[['a', 'a'], ['a', 'a', 'a'], ['c', 'd', 'c'],
['a', 'a', 'a', 'a']]
sage: it = g.all_cycles_iterator(max_length=3, rooted=True)
sage: list(it)
[['a', 'a'], ['a', 'a', 'a'], ['c', 'd', 'c'], ['d', 'c', 'd'],
['a', 'a', 'a', 'a']]


One may prefer to enumerate simple cycles, i.e. cycles such that the only vertex occuring twice in it is the starting and ending one (see also all_simple_cycles()):

sage: it = g.all_cycles_iterator(simple=True)
sage: list(it)
[['a', 'a'], ['c', 'd', 'c']]
sage: g = digraphs.Circuit(4)
sage: list(g.all_cycles_iterator(simple=True))
[[0, 1, 2, 3, 0]]

all_paths_iterator(starting_vertices=None, ending_vertices=None, simple=False, max_length=None, trivial=False, use_multiedges=False, report_edges=False, labels=False)

Return an iterator over the paths of self.

The paths are enumerated in increasing length order.

INPUT:

• starting_vertices – iterable (default: None); vertices from which the paths must start. If None, then all vertices of the graph can be starting points.

• ending_vertices – iterable (default: None); allowed ending vertices of the paths. If None, then all vertices are allowed.

• simple – boolean (default: False); if set to True, then only simple paths are considered. Simple paths are paths in which no two arcs share a head or share a tail, i.e. every vertex in the path is entered at most once and exited at most once.

• max_length – non negative integer (default: None); the maximum length of the enumerated paths. If set to None, then all lengths are allowed.

• trivial - boolean (default: False); if set to True, then the empty paths are also enumerated.

• use_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 in sage.graphs.generic_graph.GenericGraph.to_simple() if report_edges is True
• If True, a path will be reported as many times as the edges multiplicities along that path (when report_edges = False or labels = False), or with all possible combinations of edge labels (when report_edges = True and labels = True)
• report_edges – boolean (default: False); whether to report paths as list of vertices (default) or list of edges, if False then labels parameter is ignored

• labels – boolean (default: False); if False, 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.

OUTPUT:

iterator

AUTHOR:

Alexandre Blondin Masse

EXAMPLES:

sage: g = DiGraph({'a': ['a', 'b'], 'b': ['c'], 'c': ['d'], 'd': ['c']}, loops=True)
sage: pi = g.all_paths_iterator(starting_vertices=['a'], ending_vertices=['d'], report_edges=True, simple=True)
sage: list(pi)
[[('a', 'b'), ('b', 'c'), ('c', 'd')]]

sage: g = DiGraph([(0, 1, 'a'), (0, 1, 'b'), (1, 2,'c'), (1, 2,'d')], multiedges=True)
sage: pi =  g.all_paths_iterator(starting_vertices=[0], use_multiedges=True)
sage: for _ in range(6):
....:     print(next(pi))
[0, 1]
[0, 1]
[0, 1, 2]
[0, 1, 2]
[0, 1, 2]
[0, 1, 2]
sage: pi =  g.all_paths_iterator(starting_vertices=[0], use_multiedges=True, report_edges=True, labels=True)
sage: for _ in range(6):
....:     print(next(pi))
[(0, 1, 'b')]
[(0, 1, 'a')]
[(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: list(g.all_paths_iterator(starting_vertices=[0, 1], ending_vertices=[2], use_multiedges=False, report_edges=True, labels=True, simple=True))
[[(1, 2, 'd')], [(0, 1, 'b'), (1, 2, 'd')]]
sage: list(g.all_paths_iterator(starting_vertices=[0, 1], ending_vertices=[2], use_multiedges=False, report_edges=False, labels=True))
[[1, 2], [0, 1, 2]]
sage: list(g.all_paths_iterator(use_multiedges=True, report_edges=False, labels=True, max_length=1))
[[1, 2], [1, 2], [0, 1], [0, 1]]
sage: list(g.all_paths_iterator(use_multiedges=True, report_edges=True, labels=True, max_length=1))
[[(1, 2, 'd')], [(1, 2, 'c')], [(0, 1, 'b')], [(0, 1, 'a')]]

sage: g = DiGraph({'a': ['a', 'b'], 'b': ['c'], 'c': ['d'], 'd': ['c']}, loops=True)
sage: pi = g.all_paths_iterator()
sage: for _ in range(7):   # py2
....:     print(next(pi))  # py2
['d', 'c']
['b', 'c']
['c', 'd']
['a', 'a']
['a', 'b']
['a', 'a', 'a']
['a', 'a', 'b']
sage: pi = g.all_paths_iterator()
sage: [len(next(pi)) - 1 for _ in range(7)]
[1, 1, 1, 1, 1, 2, 2]


It is possible to precise the allowed starting and/or ending vertices:

sage: pi = g.all_paths_iterator(starting_vertices=['a'])
sage: for _ in range(5):   # py2
....:     print(next(pi))  # py2
['a', 'a']
['a', 'b']
['a', 'a', 'a']
['a', 'a', 'b']
['a', 'b', 'c']
sage: pi = g.all_paths_iterator(starting_vertices=['a'])
sage: [len(next(pi)) - 1 for _ in range(5)]
[1, 1, 2, 2, 2]
sage: pi = g.all_paths_iterator(starting_vertices=['a'], ending_vertices=['b'])
sage: for _ in range(5):
....:     print(next(pi))
['a', 'b']
['a', 'a', 'b']
['a', 'a', 'a', 'b']
['a', 'a', 'a', 'a', 'b']
['a', 'a', 'a', 'a', 'a', 'b']


One may prefer to enumerate only simple paths (see all_simple_paths()):

sage: pi = g.all_paths_iterator(simple=True)
sage: sorted(list(pi), key=lambda x:(len(x), x))
[['a', 'a'], ['a', 'b'], ['b', 'c'], ['c', 'd'], ['d', 'c'],
['a', 'b', 'c'], ['b', 'c', 'd'], ['c', 'd', 'c'], ['d', 'c', 'd'],
['a', 'b', 'c', 'd']]
sage: pi = g.all_paths_iterator(simple=True)
sage: [len(p) - 1 for p in pi]
[1, 1, 1, 1, 1, 2, 2, 2, 2, 3]


Or simply bound the length of the enumerated paths:

sage: pi = g.all_paths_iterator(starting_vertices=['a'], ending_vertices=['b', 'c'], max_length=6)
sage: sorted(list(pi), key=lambda x:(len(x), x))
[['a', 'b'], ['a', 'a', 'b'], ['a', 'b', 'c'], ['a', 'a', 'a', 'b'],
['a', 'a', 'b', 'c'], ['a', 'a', 'a', 'a', 'b'],
['a', 'a', 'a', 'b', 'c'], ['a', 'b', 'c', 'd', 'c'],
['a', 'a', 'a', 'a', 'a', 'b'], ['a', 'a', 'a', 'a', 'b', 'c'],
['a', 'a', 'b', 'c', 'd', 'c'],
['a', 'a', 'a', 'a', 'a', 'a', 'b'],
['a', 'a', 'a', 'a', 'a', 'b', 'c'],
['a', 'a', 'a', 'b', 'c', 'd', 'c'],
['a', 'b', 'c', 'd', 'c', 'd', 'c']]
sage: pi = g.all_paths_iterator(starting_vertices=['a'], ending_vertices=['b', 'c'], max_length=6)
sage: [len(p) - 1 for p in pi]
[1, 2, 2, 3, 3, 4, 4, 4, 5, 5, 5, 6, 6, 6, 6]


By default, empty paths are not enumerated, but it may be parametrized:

sage: pi = g.all_paths_iterator(simple=True, trivial=True)
sage: sorted(list(pi), key=lambda x:(len(x), x))
[['a'], ['b'], ['c'], ['d'], ['a', 'a'], ['a', 'b'], ['b', 'c'],
['c', 'd'], ['d', 'c'], ['a', 'b', 'c'], ['b', 'c', 'd'],
['c', 'd', 'c'], ['d', 'c', 'd'], ['a', 'b', 'c', 'd']]
sage: pi = g.all_paths_iterator(simple=True, trivial=True)
sage: [len(p) - 1 for p in pi]
[0, 0, 0, 0, 1, 1, 1, 1, 1, 2, 2, 2, 2, 3]
sage: pi = g.all_paths_iterator(simple=True, trivial=False)
sage: sorted(list(pi), key=lambda x:(len(x), x))
[['a', 'a'], ['a', 'b'], ['b', 'c'], ['c', 'd'], ['d', 'c'],
['a', 'b', 'c'], ['b', 'c', 'd'], ['c', 'd', 'c'], ['d', 'c', 'd'],
['a', 'b', 'c', 'd']]
sage: pi = g.all_paths_iterator(simple=True, trivial=False)
sage: [len(p) - 1 for p in pi]
[1, 1, 1, 1, 1, 2, 2, 2, 2, 3]

all_simple_cycles(starting_vertices=None, rooted=False, max_length=None, trivial=False)

Return a list of all simple cycles of self.

INPUT:

• starting_vertices – iterable (default: None); vertices from which the cycles must start. If None, then all vertices of the graph can be starting points. This argument is necessary if rooted is set to True.
• rooted – boolean (default: False); if set to False, then cycles differing only by their starting vertex are considered the same (e.g. ['a', 'b', 'c', 'a'] and ['b', 'c', 'a', 'b']). Otherwise, all cycles are enumerated.
• max_length – non negative integer (default: None); the maximum length of the enumerated paths. If set to None, then all lengths are allowed.
• trivial - boolean (default: False); if set to True, then the empty paths are also enumerated.

OUTPUT:

list

Note

Although the number of simple cycles of a finite graph is always finite, computing all its cycles may take a very long time.

EXAMPLES:

sage: g = DiGraph({'a': ['a', 'b'], 'b': ['c'], 'c': ['d'], 'd': ['c']}, loops=True)
sage: g.all_simple_cycles()
[['a', 'a'], ['c', 'd', 'c']]


The directed version of the Petersen graph:

sage: g = graphs.PetersenGraph().to_directed()
sage: g.all_simple_cycles(max_length=4)
[[0, 1, 0], [0, 4, 0], [0, 5, 0], [1, 2, 1], [1, 6, 1], [2, 3, 2],
[2, 7, 2], [3, 8, 3], [3, 4, 3], [4, 9, 4], [5, 8, 5], [5, 7, 5],
[6, 8, 6], [6, 9, 6], [7, 9, 7]]
sage: g.all_simple_cycles(max_length=6)
[[0, 1, 0], [0, 4, 0], [0, 5, 0], [1, 2, 1], [1, 6, 1], [2, 3, 2],
[2, 7, 2], [3, 8, 3], [3, 4, 3], [4, 9, 4], [5, 8, 5], [5, 7, 5],
[6, 8, 6], [6, 9, 6], [7, 9, 7], [0, 1, 2, 3, 4, 0],
[0, 1, 2, 7, 5, 0], [0, 1, 6, 8, 5, 0], [0, 1, 6, 9, 4, 0],
[0, 4, 9, 6, 1, 0], [0, 4, 9, 7, 5, 0], [0, 4, 3, 8, 5, 0],
[0, 4, 3, 2, 1, 0], [0, 5, 8, 3, 4, 0], [0, 5, 8, 6, 1, 0],
[0, 5, 7, 9, 4, 0], [0, 5, 7, 2, 1, 0], [1, 2, 3, 8, 6, 1],
[1, 2, 7, 9, 6, 1], [1, 6, 8, 3, 2, 1], [1, 6, 9, 7, 2, 1],
[2, 3, 8, 5, 7, 2], [2, 3, 4, 9, 7, 2], [2, 7, 9, 4, 3, 2],
[2, 7, 5, 8, 3, 2], [3, 8, 6, 9, 4, 3], [3, 4, 9, 6, 8, 3],
[5, 8, 6, 9, 7, 5], [5, 7, 9, 6, 8, 5], [0, 1, 2, 3, 8, 5, 0],
[0, 1, 2, 7, 9, 4, 0], [0, 1, 6, 8, 3, 4, 0],
[0, 1, 6, 9, 7, 5, 0], [0, 4, 9, 6, 8, 5, 0],
[0, 4, 9, 7, 2, 1, 0], [0, 4, 3, 8, 6, 1, 0],
[0, 4, 3, 2, 7, 5, 0], [0, 5, 8, 3, 2, 1, 0],
[0, 5, 8, 6, 9, 4, 0], [0, 5, 7, 9, 6, 1, 0],
[0, 5, 7, 2, 3, 4, 0], [1, 2, 3, 4, 9, 6, 1],
[1, 2, 7, 5, 8, 6, 1], [1, 6, 8, 5, 7, 2, 1],
[1, 6, 9, 4, 3, 2, 1], [2, 3, 8, 6, 9, 7, 2],
[2, 7, 9, 6, 8, 3, 2], [3, 8, 5, 7, 9, 4, 3],
[3, 4, 9, 7, 5, 8, 3]]


The complete graph (without loops) on $$4$$ vertices:

sage: g = graphs.CompleteGraph(4).to_directed()
sage: g.all_simple_cycles()
[[0, 1, 0], [0, 2, 0], [0, 3, 0], [1, 2, 1], [1, 3, 1], [2, 3, 2],
[0, 1, 2, 0], [0, 1, 3, 0], [0, 2, 1, 0], [0, 2, 3, 0],
[0, 3, 1, 0], [0, 3, 2, 0], [1, 2, 3, 1], [1, 3, 2, 1],
[0, 1, 2, 3, 0], [0, 1, 3, 2, 0], [0, 2, 1, 3, 0],
[0, 2, 3, 1, 0], [0, 3, 1, 2, 0], [0, 3, 2, 1, 0]]


If the graph contains a large number of cycles, one can bound the length of the cycles, or simply restrict the possible starting vertices of the cycles:

sage: g = graphs.CompleteGraph(20).to_directed()
sage: g.all_simple_cycles(max_length=2)
[[0, 1, 0], [0, 2, 0], [0, 3, 0], [0, 4, 0], [0, 5, 0], [0, 6, 0],
[0, 7, 0], [0, 8, 0], [0, 9, 0], [0, 10, 0], [0, 11, 0],
[0, 12, 0], [0, 13, 0], [0, 14, 0], [0, 15, 0], [0, 16, 0],
[0, 17, 0], [0, 18, 0], [0, 19, 0], [1, 2, 1], [1, 3, 1],
[1, 4, 1], [1, 5, 1], [1, 6, 1], [1, 7, 1], [1, 8, 1], [1, 9, 1],
[1, 10, 1], [1, 11, 1], [1, 12, 1], [1, 13, 1], [1, 14, 1],
[1, 15, 1], [1, 16, 1], [1, 17, 1], [1, 18, 1], [1, 19, 1],
[2, 3, 2], [2, 4, 2], [2, 5, 2], [2, 6, 2], [2, 7, 2], [2, 8, 2],
[2, 9, 2], [2, 10, 2], [2, 11, 2], [2, 12, 2], [2, 13, 2],
[2, 14, 2], [2, 15, 2], [2, 16, 2], [2, 17, 2], [2, 18, 2],
[2, 19, 2], [3, 4, 3], [3, 5, 3], [3, 6, 3], [3, 7, 3], [3, 8, 3],
[3, 9, 3], [3, 10, 3], [3, 11, 3], [3, 12, 3], [3, 13, 3],
[3, 14, 3], [3, 15, 3], [3, 16, 3], [3, 17, 3], [3, 18, 3],
[3, 19, 3], [4, 5, 4], [4, 6, 4], [4, 7, 4], [4, 8, 4], [4, 9, 4],
[4, 10, 4], [4, 11, 4], [4, 12, 4], [4, 13, 4], [4, 14, 4],
[4, 15, 4], [4, 16, 4], [4, 17, 4], [4, 18, 4], [4, 19, 4],
[5, 6, 5], [5, 7, 5], [5, 8, 5], [5, 9, 5], [5, 10, 5],
[5, 11, 5], [5, 12, 5], [5, 13, 5], [5, 14, 5], [5, 15, 5],
[5, 16, 5], [5, 17, 5], [5, 18, 5], [5, 19, 5], [6, 7, 6],
[6, 8, 6], [6, 9, 6], [6, 10, 6], [6, 11, 6], [6, 12, 6],
[6, 13, 6], [6, 14, 6], [6, 15, 6], [6, 16, 6], [6, 17, 6],
[6, 18, 6], [6, 19, 6], [7, 8, 7], [7, 9, 7], [7, 10, 7],
[7, 11, 7], [7, 12, 7], [7, 13, 7], [7, 14, 7], [7, 15, 7],
[7, 16, 7], [7, 17, 7], [7, 18, 7], [7, 19, 7], [8, 9, 8],
[8, 10, 8], [8, 11, 8], [8, 12, 8], [8, 13, 8], [8, 14, 8],
[8, 15, 8], [8, 16, 8], [8, 17, 8], [8, 18, 8], [8, 19, 8],
[9, 10, 9], [9, 11, 9], [9, 12, 9], [9, 13, 9], [9, 14, 9],
[9, 15, 9], [9, 16, 9], [9, 17, 9], [9, 18, 9], [9, 19, 9],
[10, 11, 10], [10, 12, 10], [10, 13, 10], [10, 14, 10],
[10, 15, 10], [10, 16, 10], [10, 17, 10], [10, 18, 10],
[10, 19, 10], [11, 12, 11], [11, 13, 11], [11, 14, 11],
[11, 15, 11], [11, 16, 11], [11, 17, 11], [11, 18, 11],
[11, 19, 11], [12, 13, 12], [12, 14, 12], [12, 15, 12],
[12, 16, 12], [12, 17, 12], [12, 18, 12], [12, 19, 12],
[13, 14, 13], [13, 15, 13], [13, 16, 13], [13, 17, 13],
[13, 18, 13], [13, 19, 13], [14, 15, 14], [14, 16, 14],
[14, 17, 14], [14, 18, 14], [14, 19, 14], [15, 16, 15],
[15, 17, 15], [15, 18, 15], [15, 19, 15], [16, 17, 16],
[16, 18, 16], [16, 19, 16], [17, 18, 17], [17, 19, 17],
[18, 19, 18]]
sage: g = graphs.CompleteGraph(20).to_directed()
sage: g.all_simple_cycles(max_length=2, starting_vertices=[0])
[[0, 1, 0], [0, 2, 0], [0, 3, 0], [0, 4, 0], [0, 5, 0], [0, 6, 0],
[0, 7, 0], [0, 8, 0], [0, 9, 0], [0, 10, 0], [0, 11, 0],
[0, 12, 0], [0, 13, 0], [0, 14, 0], [0, 15, 0], [0, 16, 0],
[0, 17, 0], [0, 18, 0], [0, 19, 0]]


One may prefer to distinguish equivalent cycles having distinct starting vertices (compare the following examples):

sage: g = graphs.CompleteGraph(4).to_directed()
sage: g.all_simple_cycles(max_length=2, rooted=False)
[[0, 1, 0], [0, 2, 0], [0, 3, 0], [1, 2, 1], [1, 3, 1], [2, 3, 2]]
sage: g.all_simple_cycles(max_length=2, rooted=True)
[[0, 1, 0], [0, 2, 0], [0, 3, 0], [1, 0, 1], [1, 2, 1], [1, 3, 1],
[2, 0, 2], [2, 1, 2], [2, 3, 2], [3, 0, 3], [3, 1, 3], [3, 2, 3]]

all_simple_paths(starting_vertices=None, ending_vertices=None, max_length=None, trivial=False, use_multiedges=False, report_edges=False, labels=False)

Return a list of all the simple paths of self starting with one of the given vertices.

Simple paths are paths in which no two arcs share a head or share a tail, i.e. every vertex in the path is entered at most once and exited at most once.

INPUT:

• starting_vertices – list (default: None); vertices from which the paths must start. If None, then all vertices of the graph can be starting points.

• ending_vertices – iterable (default: None); allowed ending vertices of the paths. If None, then all vertices are allowed.

• max_length – non negative integer (default: None); the maximum length of the enumerated paths. If set to None, then all lengths are allowed.

• trivial - boolean (default: False); if set to True, then the empty paths are also enumerated.

• use_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 in sage.graphs.generic_graph.GenericGraph.to_simple() if report_edges is True
• If True, a path will be reported as many times as the edges multiplicities along that path (when report_edges = False or labels = False), or with all possible combinations of edge labels (when report_edges = True and labels = True)
• report_edges – boolean (default: False); whether to report paths as list of vertices (default) or list of edges, if False then labels parameter is ignored

• labels – boolean (default: False); if False, 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.

OUTPUT:

list

Note

Although the number of simple paths of a finite graph is always finite, computing all its paths may take a very long time.

EXAMPLES:

sage: g = DiGraph({0: [0, 1], 1: [2], 2: [3], 3: [2]}, loops=True)
sage: g.all_simple_paths()
[[3, 2],
[2, 3],
[1, 2],
[0, 0],
[0, 1],
[0, 1, 2],
[1, 2, 3],
[2, 3, 2],
[3, 2, 3],
[0, 1, 2, 3]]

sage: g = DiGraph([(0, 1, 'a'), (0, 1, 'b'), (1, 2,'c'), (1, 2,'d')], multiedges=True)
sage: g.all_simple_paths(starting_vertices=[0], ending_vertices=[2], use_multiedges=False)
[[0, 1, 2]]
sage: g.all_simple_paths(starting_vertices=[0], ending_vertices=[2], use_multiedges=True)
[[0, 1, 2], [0, 1, 2], [0, 1, 2], [0, 1, 2]]
sage: g.all_simple_paths(starting_vertices=[0], ending_vertices=[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_simple_paths(starting_vertices=[0], ending_vertices=[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_simple_paths(starting_vertices=[0, 1], ending_vertices=[2], use_multiedges=False, report_edges=True, labels=True)
[[(1, 2, 'd')], [(0, 1, 'b'), (1, 2, 'd')]]
sage: g.all_simple_paths(starting_vertices=[0, 1], ending_vertices=[2], use_multiedges=False, report_edges=False, labels=True)
[[1, 2], [0, 1, 2]]
sage: g.all_simple_paths(use_multiedges=True, report_edges=False, labels=True)
[[1, 2], [1, 2], [0, 1], [0, 1], [0, 1, 2], [0, 1, 2], [0, 1, 2], [0, 1, 2]]
sage: g.all_simple_paths(starting_vertices=[0, 1], ending_vertices=[2], use_multiedges=False, report_edges=True, labels=True, trivial=True)
[[(1, 2, 'd')], [(0, 1, 'b'), (1, 2, 'd')]]


One may compute all paths having specific starting and/or ending vertices:

sage: g = DiGraph({'a': ['a', 'b'], 'b': ['c'], 'c': ['d'], 'd': ['c']}, loops=True)
sage: g.all_simple_paths(starting_vertices=['a'])
[['a', 'a'], ['a', 'b'], ['a', 'b', 'c'], ['a', 'b', 'c', 'd']]
sage: g.all_simple_paths(starting_vertices=['a'], ending_vertices=['c'])
[['a', 'b', 'c']]
sage: g.all_simple_paths(starting_vertices=['a'], ending_vertices=['b', 'c'])
[['a', 'b'], ['a', 'b', 'c']]


It is also possible to bound the length of the paths:

sage: g = DiGraph({0: [0, 1], 1: [2], 2: [3], 3: [2]}, loops=True)
sage: g.all_simple_paths(max_length=2)
[[3, 2],
[2, 3],
[1, 2],
[0, 0],
[0, 1],
[0, 1, 2],
[1, 2, 3],
[2, 3, 2],
[3, 2, 3]]


By default, empty paths are not enumerated, but this can be parametrized:

sage: g = DiGraph({'a': ['a', 'b'], 'b': ['c'], 'c': ['d'], 'd': ['c']}, loops=True)
sage: g.all_simple_paths(starting_vertices=['a'], trivial=True)
[['a'], ['a', 'a'], ['a', 'b'], ['a', 'b', 'c'],
['a', 'b', 'c', 'd']]
sage: g.all_simple_paths(starting_vertices=['a'], trivial=False)
[['a', 'a'], ['a', 'b'], ['a', 'b', 'c'], ['a', 'b', 'c', 'd']]

degree_polynomial()

Return the generating polynomial of degrees of vertices in self.

This is the sum

$\sum_{v \in G} x^{\operatorname{in}(v)} y^{\operatorname{out}(v)},$

where in(v) and out(v) are the number of incoming and outgoing edges at vertex $$v$$ in the digraph $$G$$.

Because this polynomial is multiplicative for Cartesian product of digraphs, it is useful to help see if the digraph can be isomorphic to a Cartesian product.

num_verts() for the value at $$(x, y) = (1, 1)$$

EXAMPLES:

sage: G = posets.PentagonPoset().hasse_diagram()
sage: G.degree_polynomial()
x^2 + 3*x*y + y^2

sage: G = posets.BooleanLattice(4).hasse_diagram()
sage: G.degree_polynomial().factor()
(x + y)^4

dig6_string()

Return the dig6 representation of the digraph as an ASCII string.

This is only valid for single (no multiple edges) digraphs on at most $$2^{18} - 1 = 262143$$ vertices.

Note

As the dig6 format only handles graphs with vertex set $$\{0, \ldots, n-1\}$$, a relabelled copy will be encoded, if necessary.

EXAMPLES:

sage: D = DiGraph({0: [1, 2], 1: [2], 2: [3], 3: [0]})
sage: D.dig6_string()
'CW_'

feedback_edge_set(constraint_generation=True, value_only=False, solver=None, verbose=0)

Compute the minimum feedback edge set of a digraph (also called feedback arc set).

The minimum feedback edge set of a digraph is a set of edges that intersect all the circuits of the digraph. Equivalently, a minimum feedback arc set of a DiGraph is a set $$S$$ of arcs such that the digraph $$G - S$$ is acyclic. For more information, see the Wikipedia article Feedback_arc_set.

INPUT:

• value_only – boolean (default: False)
• When set to True, only the minimum cardinal of a minimum edge set is returned.
• When set to False, the Set of edges of a minimal edge set is returned.
• constraint_generation – boolean (default: True); whether to use constraint generation when solving the Mixed Integer Linear Program.
• solver – string (default: None); specify a Linear Program (LP) solver to be used. If set to None, the default one is used. For more information on LP solvers and which default solver is used, see the method solve of the class MixedIntegerLinearProgram.
• verbose – integer (default: 0); sets the level of verbosity. Set to 0 by default, which means quiet.

ALGORITHM:

This problem is solved using Linear Programming, in two different ways. The first one is to solve the following formulation:

$\begin{split}\mbox{Minimize : }&\sum_{(u,v)\in G} b_{(u,v)}\\ \mbox{Such that : }&\\ &\forall (u,v)\in G, d_u-d_v+ n \cdot b_{(u,v)}\geq 0\\ &\forall u\in G, 0\leq d_u\leq |G|\\\end{split}$

An 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 $$d_v \in [0,\dots,n-1]$$ such that if there exists an edge $$(u, v) \in G$$ such that $$d_v < d_u$$, then the edge $$(u,v)$$ is removed.

The number of edges removed is then minimized, which is the objective.

(Constraint Generation)

If the parameter constraint_generation is enabled, a more efficient formulation is used :

$\begin{split}\mbox{Minimize : }&\sum_{(u,v)\in G} b_{(u,v)}\\ \mbox{Such that : }&\\ &\forall C\text{ circuits }\subseteq G, \sum_{uv\in C}b_{(u,v)}\geq 1\\\end{split}$

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 solved 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.

EXAMPLES:

If the digraph is created from a graph, and hence is symmetric (if $$uv$$ is an edge, then $$vu$$ is an edge too), then obviously the cardinality of its feedback arc set is the number of edges in the first graph:

sage: cycle=graphs.CycleGraph(5)
sage: dcycle=DiGraph(cycle)
sage: cycle.size()
5
sage: dcycle.feedback_edge_set(value_only=True)
5


And in this situation, for any edge $$uv$$ of the first graph, $$uv$$ of $$vu$$ is in the returned feedback arc set:

sage: g = graphs.RandomGNP(5,.3)
sage: dg = DiGraph(g)
sage: feedback = dg.feedback_edge_set()
sage: u,v,l = next(g.edge_iterator())
sage: (u,v) in feedback or (v,u) in feedback
True

flow_polytope(edges=None, ends=None, backend=None)

Return the flow polytope of a digraph.

The flow polytope of a directed graph is the polytope consisting of all nonnegative flows on the graph with a given set $$S$$ of sources and a given set $$T$$ of sinks.

A flow on a directed graph $$G$$ with a given set $$S$$ of sources and a given set $$T$$ of sinks means an assignment of a nonnegative real to each edge of $$G$$ such that the flow is conserved in each vertex outside of $$S$$ and $$T$$, and there is a unit of flow entering each vertex in $$S$$ and a unit of flow leaving each vertex in $$T$$. These flows clearly form a polytope in the space of all assignments of reals to the edges of $$G$$.

The polytope is empty unless the sets $$S$$ and $$T$$ are equinumerous.

By default, $$S$$ is taken to be the set of all sources (i.e., vertices of indegree $$0$$) of $$G$$, and $$T$$ is taken to be the set of all sinks (i.e., vertices of outdegree $$0$$) of $$G$$. If a different choice of $$S$$ and $$T$$ is desired, it can be specified using the optional ends parameter.

The polytope is returned as a polytope in $$\RR^m$$, where $$m$$ is the number of edges of the digraph self. The $$k$$-th coordinate of a point in the polytope is the real assigned to the $$k$$-th edge of self. The order of the edges is the one returned by self.edges(). If a different order is desired, it can be specified using the optional edges parameter.

The faces and volume of these polytopes are of interest. Examples of these polytopes are the Chan-Robbins-Yuen polytope and the Pitman-Stanley polytope [PS2002].

INPUT:

• edges – (optional, default: self.edges()) a list or tuple of all edges of self (each only once). This determines which coordinate of a point in the polytope will correspond to which edge of self. It is also possible to specify a list which contains not all edges of self; this results in a polytope corresponding to the flows which are $$0$$ on all remaining edges. Notice that the edges entered here must be in the precisely same format as outputted by self.edges(); so, if self.edges() outputs an edge in the form (1, 3, None), then (1, 3) will not do!
• ends – (optional, default: (self.sources(), self.sinks())) a pair $$(S, T)$$ of an iterable $$S$$ and an iterable $$T$$.
• backend – string or None (default); the backend to use; see sage.geometry.polyhedron.constructor.Polyhedron()

Note

Flow polytopes can also be built through the polytopes.<tab> object:

sage: polytopes.flow_polytope(digraphs.Path(5))
A 0-dimensional polyhedron in QQ^4 defined as the convex hull of 1 vertex


EXAMPLES:

A commutative square:

sage: G = DiGraph({1: [2, 3], 2: [4], 3: [4]})
sage: fl = G.flow_polytope(); fl
A 1-dimensional polyhedron in QQ^4 defined as the convex hull
of 2 vertices
sage: fl.vertices()
(A vertex at (0, 1, 0, 1), A vertex at (1, 0, 1, 0))


Using a different order for the edges of the graph:

sage: fl = G.flow_polytope(edges=G.edges(key=lambda x: x[0] - x[1])); fl
A 1-dimensional polyhedron in QQ^4 defined as the convex hull of 2 vertices
sage: fl.vertices()
(A vertex at (0, 1, 1, 0), A vertex at (1, 0, 0, 1))


A tournament on 4 vertices:

sage: H = digraphs.TransitiveTournament(4)
sage: fl = H.flow_polytope(); fl
A 3-dimensional polyhedron in QQ^6 defined as the convex hull
of 4 vertices
sage: fl.vertices()
(A vertex at (0, 0, 1, 0, 0, 0),
A vertex at (0, 1, 0, 0, 0, 1),
A vertex at (1, 0, 0, 0, 1, 0),
A vertex at (1, 0, 0, 1, 0, 1))


Restricting to a subset of the edges:

sage: fl = H.flow_polytope(edges=[(0, 1, None), (1, 2, None),
....:                             (2, 3, None), (0, 3, None)])
sage: fl
A 1-dimensional polyhedron in QQ^4 defined as the convex hull
of 2 vertices
sage: fl.vertices()
(A vertex at (0, 0, 0, 1), A vertex at (1, 1, 1, 0))


Using a different choice of sources and sinks:

sage: fl = H.flow_polytope(ends=([1], [3])); fl
A 1-dimensional polyhedron in QQ^6 defined as the convex hull
of 2 vertices
sage: fl.vertices()
(A vertex at (0, 0, 0, 1, 0, 1), A vertex at (0, 0, 0, 0, 1, 0))
sage: fl = H.flow_polytope(ends=([0, 1], [3])); fl
The empty polyhedron in QQ^6
sage: fl = H.flow_polytope(ends=([3], [0])); fl
The empty polyhedron in QQ^6
sage: fl = H.flow_polytope(ends=([0, 1], [2, 3])); fl
A 3-dimensional polyhedron in QQ^6 defined as the convex hull
of 5 vertices
sage: fl.vertices()
(A vertex at (0, 0, 1, 1, 0, 0),
A vertex at (0, 1, 0, 0, 1, 0),
A vertex at (1, 0, 0, 2, 0, 1),
A vertex at (1, 0, 0, 1, 1, 0),
A vertex at (0, 1, 0, 1, 0, 1))
sage: fl = H.flow_polytope(edges=[(0, 1, None), (1, 2, None),
....:                             (2, 3, None), (0, 2, None),
....:                             (1, 3, None)],
....:                      ends=([0, 1], [2, 3])); fl
A 2-dimensional polyhedron in QQ^5 defined as the convex hull
of 4 vertices
sage: fl.vertices()
(A vertex at (0, 0, 0, 1, 1),
A vertex at (1, 2, 1, 0, 0),
A vertex at (1, 1, 0, 0, 1),
A vertex at (0, 1, 1, 1, 0))


A digraph with one source and two sinks:

sage: Y = DiGraph({1: [2], 2: [3, 4]})
sage: Y.flow_polytope()
The empty polyhedron in QQ^3


A digraph with one vertex and no edge:

sage: Z = DiGraph({1: []})
sage: Z.flow_polytope()
A 0-dimensional polyhedron in QQ^0 defined as the convex hull
of 1 vertex

in_degree(vertices=None, labels=False)

Same as degree, but for in degree.

EXAMPLES:

sage: D = DiGraph({0: [1,2,3], 1: [0,2], 2: [3], 3: [4], 4: [0,5], 5: [1]})
sage: D.in_degree(vertices=[0, 1, 2], labels=True)
{0: 2, 1: 2, 2: 2}
sage: D.in_degree()
[2, 2, 2, 2, 1, 1]
sage: G = graphs.PetersenGraph().to_directed()
sage: G.in_degree(0)
3

in_degree_iterator(vertices=None, labels=False)

Same as degree_iterator, but for in degree.

EXAMPLES:

sage: D = graphs.Grid2dGraph(2,4).to_directed()
sage: sorted(D.in_degree_iterator())
[2, 2, 2, 2, 3, 3, 3, 3]
sage: sorted(D.in_degree_iterator(labels=True))
[((0, 0), 2),
((0, 1), 3),
((0, 2), 3),
((0, 3), 2),
((1, 0), 2),
((1, 1), 3),
((1, 2), 3),
((1, 3), 2)]

in_degree_sequence()

Return the in-degree sequence.

EXAMPLES:

The in-degree sequences of two digraphs:

sage: g = DiGraph({1: [2, 5, 6], 2: [3, 6], 3: [4, 6], 4: [6], 5: [4, 6]})
sage: g.in_degree_sequence()
[5, 2, 1, 1, 1, 0]

sage: V = [2, 3, 5, 7, 8, 9, 10, 11]
sage: E = [[], [8, 10], [11], [8, 11], [9], [], [], [2, 9, 10]]
sage: g = DiGraph(dict(zip(V, E)))
sage: g.in_degree_sequence()
[2, 2, 2, 2, 1, 0, 0, 0]

incoming_edge_iterator(vertices, labels=True)

Return an iterator over all arriving edges from vertices.

INPUT:

• vertices – a vertex or a list of vertices
• labels – boolean (default: True); whether to return edges as pairs of vertices, or as triples containing the labels

EXAMPLES:

sage: D = DiGraph({0: [1,2,3], 1: [0,2], 2: [3], 3: [4], 4: [0,5], 5: [1]})
sage: for a in D.incoming_edge_iterator([0]):
....:     print(a)
(1, 0, None)
(4, 0, None)

incoming_edges(vertices, labels=True)

Return a list of edges arriving at vertices.

INPUT:

• vertices – a vertex or a list of vertices
• labels – boolean (default: True); whether to return edges as pairs of vertices, or as triples containing the labels.

EXAMPLES:

sage: D = DiGraph({0: [1,2,3], 1: [0,2], 2: [3], 3: [4], 4: [0,5], 5: [1]})
sage: D.incoming_edges([0])
[(1, 0, None), (4, 0, None)]

is_aperiodic()

Return whether the current DiGraph is aperiodic.

A directed graph is aperiodic if there is no integer $$k > 1$$ that divides the length of every cycle in the graph. See the Wikipedia article Aperiodic_graph for more information.

EXAMPLES:

The following graph has period 2, so it is not aperiodic:

sage: g = DiGraph({0: [1], 1: [0]})
sage: g.is_aperiodic()
False


The following graph has a cycle of length 2 and a cycle of length 3, so it is aperiodic:

sage: g = DiGraph({0: [1, 4], 1: [2], 2: [0], 4: [0]})
sage: g.is_aperiodic()
True

is_directed()

Since digraph is directed, return True.

EXAMPLES:

sage: DiGraph().is_directed()
True

is_directed_acyclic(certificate=False)

Return whether the digraph is acyclic or not.

A directed graph is acyclic if for any vertex $$v$$, there is no directed path that starts and ends at $$v$$. Every directed acyclic graph (DAG) corresponds to a partial ordering of its vertices, however multiple dags may lead to the same partial ordering.

INPUT:

• certificate – boolean (default: False); whether to return a certificate

OUTPUT:

• When certificate=False, returns a boolean value.
• When certificate=True:
• If the graph is acyclic, returns a pair (True, ordering) where ordering is a list of the vertices such that u appears before v in ordering if u, v is an edge.
• Else, returns a pair (False, cycle) where cycle is a list of vertices representing a circuit in the graph.

EXAMPLES:

At first, the following graph is acyclic:

sage: D = DiGraph({0:[1, 2, 3], 4:[2, 5], 1:[8], 2:[7], 3:[7], 5:[6,7], 7:[8], 6:[9], 8:[10], 9:[10]})
sage: D.plot(layout='circular').show()
sage: D.is_directed_acyclic()
True


Adding an edge from $$9$$ to $$7$$ does not change it:

sage: D.add_edge(9, 7)
sage: D.is_directed_acyclic()
True


We can obtain as a proof an ordering of the vertices such that $$u$$ appears before $$v$$ if $$uv$$ is an edge of the graph:

sage: D.is_directed_acyclic(certificate=True)
(True, [4, 5, 6, 9, 0, 1, 2, 3, 7, 8, 10])


Adding an edge from 7 to 4, though, makes a difference:

sage: D.add_edge(7, 4)
sage: D.is_directed_acyclic()
False


Indeed, it creates a circuit $$7, 4, 5$$:

sage: D.is_directed_acyclic(certificate=True)
(False, [7, 4, 5])


Checking acyclic graphs are indeed acyclic

sage: def random_acyclic(n, p):
....:  g = graphs.RandomGNP(n, p)
....:  h = DiGraph()
....:  h.add_edges(((u, v) if u < v else (v, u)) for u, v in g.edge_iterator(labels=False))
....:  return h
...
sage: all(random_acyclic(100, .2).is_directed_acyclic()    # long time
....:      for i in range(50))                             # long time
True

is_strongly_connected(G)

Check whether the current DiGraph is strongly connected.

EXAMPLES:

The circuit is obviously strongly connected:

sage: from sage.graphs.connectivity import is_strongly_connected
sage: g = digraphs.Circuit(5)
sage: is_strongly_connected(g)
True
sage: g.is_strongly_connected()
True


But a transitive triangle is not:

sage: g = DiGraph({0: [1, 2], 1: [2]})
sage: is_strongly_connected(g)
False

is_tournament()

Check whether the digraph is a tournament.

A tournament is a digraph in which each pair of distinct vertices is connected by a single arc.

EXAMPLES:

sage: g = digraphs.RandomTournament(6)
sage: g.is_tournament()
True
sage: u,v = next(g.edge_iterator(labels=False))
sage: g.is_tournament()
False
sage: g.is_tournament()
False

is_transitive(g, certificate=False)

Tests whether the digraph is transitive.

A digraph is transitive if for any pair of vertices $$u,v\in G$$ linked by a $$uv$$-path the edge $$uv$$ belongs to $$G$$.

INPUT:

• certificate – whether to return a certificate for negative answers.
• If certificate = False (default), this method returns True or False according to the graph.
• If certificate = True, this method either returns True answers or yield a pair of vertices $$uv$$ such that there exists a $$uv$$-path in $$G$$ but $$uv\not\in G$$.

EXAMPLES:

sage: digraphs.Circuit(4).is_transitive()
False
sage: digraphs.Circuit(4).is_transitive(certificate=True)
(0, 2)
sage: digraphs.RandomDirectedGNP(30,.2).is_transitive()
False
sage: D = digraphs.DeBruijn(5, 2)
sage: D.is_transitive()
False
sage: cert = D.is_transitive(certificate=True)
sage: D.has_edge(*cert)
False
sage: D.shortest_path(*cert) != []
True
sage: digraphs.RandomDirectedGNP(20,.2).transitive_closure().is_transitive()
True

layout_acyclic(rankdir='up', **options)

Return a ranked layout so that all edges point upward.

To this end, the heights of the vertices are set according to the level set decomposition of the graph (see level_sets()).

This is achieved by calling graphviz and dot2tex if available (see layout_graphviz()), and using a spring layout with fixed vertical placement of the vertices otherwise (see layout_acyclic_dummy() and layout_ranked()).

Non acyclic graphs are partially supported by graphviz, which then chooses some edges to point down.

INPUT:

• rankdir – string (default: 'up'); indicates which direction the edges should point toward among 'up', 'down', 'left', or 'right'
• **options – passed down to layout_ranked() or layout_graphviz()

EXAMPLES:

sage: H = DiGraph({0: [1, 2], 1: [3], 2: [3], 3: [], 5: [1, 6], 6: [2, 3]})


The actual layout computed depends on whether dot2tex and graphviz are installed, so we don’t test its relative values:

sage: H.layout_acyclic()
{0: [..., ...], 1: [..., ...], 2: [..., ...], 3: [..., ...], 5: [..., ...], 6: [..., ...]}

sage: H = DiGraph({0: [1]})
sage: pos = H.layout_acyclic(rankdir='up')
sage: pos[1][1] > pos[0][1] + .5
True
sage: pos = H.layout_acyclic(rankdir='down')
sage: pos[1][1] < pos[0][1] - .5
True
sage: pos = H.layout_acyclic(rankdir='right')
sage: pos[1][0] > pos[0][0] + .5
True
sage: pos = H.layout_acyclic(rankdir='left')
sage: pos[1][0] < pos[0][0] - .5
True

layout_acyclic_dummy(heights=None, rankdir='up', **options)

Return a ranked layout so that all edges point upward.

To this end, the heights of the vertices are set according to the level set decomposition of the graph (see level_sets()). This is achieved by a spring layout with fixed vertical placement of the vertices otherwise (see layout_acyclic_dummy() and layout_ranked()).

INPUT:

• rankdir – string (default: 'up'); indicates which direction the edges should point toward among 'up', 'down', 'left', or 'right'
• **options – passed down to layout_ranked()

EXAMPLES:

sage: H = DiGraph({0: [1, 2], 1: [3], 2: [3], 3: [], 5: [1, 6], 6: [2, 3]})
sage: H.layout_acyclic_dummy()
{0: [1.00..., 0], 1: [1.00..., 1], 2: [1.51..., 2], 3: [1.50..., 3], 5: [2.01..., 0], 6: [2.00..., 1]}

sage: H = DiGraph({0: [1]})
sage: H.layout_acyclic_dummy(rankdir='up')
{0: [0.5..., 0], 1: [0.5..., 1]}
sage: H.layout_acyclic_dummy(rankdir='down')
{0: [0.5..., 1], 1: [0.5..., 0]}
sage: H.layout_acyclic_dummy(rankdir='left')
{0: [1, 0.5...], 1: [0, 0.5...]}
sage: H.layout_acyclic_dummy(rankdir='right')
{0: [0, 0.5...], 1: [1, 0.5...]}
sage: H = DiGraph({0: [1, 2], 1: [3], 2: [3], 3: [1], 5: [1, 6], 6: [2, 3]})
sage: H.layout_acyclic_dummy()
Traceback (most recent call last):
...
ValueError: self should be an acyclic graph

level_sets()

Return the level set decomposition of the digraph.

OUTPUT:

• a list of non empty lists of vertices of this graph

The level set decomposition of the digraph is a list $$l$$ such that the level $$l[i]$$ contains all the vertices having all their predecessors in the levels $$l[j]$$ for $$j < i$$, and at least one in level $$l[i-1]$$ (unless $$i = 0$$).

The level decomposition contains exactly the vertices not occuring in any cycle of the graph. In particular, the graph is acyclic if and only if the decomposition forms a set partition of its vertices, and we recover the usual level set decomposition of the corresponding poset.

EXAMPLES:

sage: H = DiGraph({0: [1, 2], 1: [3], 2: [3], 3: [], 5: [1, 6], 6: [2, 3]})
sage: H.level_sets()
[[0, 5], [1, 6], [2], [3]]

sage: H = DiGraph({0: [1, 2], 1: [3], 2: [3], 3: [1], 5: [1, 6], 6: [2, 3]})
sage: H.level_sets()
[[0, 5], [6], [2]]


This routine is mostly used for Hasse diagrams of posets:

sage: from sage.combinat.posets.hasse_diagram import HasseDiagram
sage: H = HasseDiagram({0: [1, 2], 1: [3], 2: [3], 3: []})
sage: [len(x) for x in H.level_sets()]
[1, 2, 1]

sage: from sage.combinat.posets.hasse_diagram import HasseDiagram
sage: H = HasseDiagram({0: [1, 2], 1: [3], 2: [4], 3: [4]})
sage: [len(x) for x in H.level_sets()]
[1, 2, 1, 1]


Complexity: $$O(n+m)$$ in time and $$O(n)$$ in memory (besides the storage of the graph itself), where $$n$$ and $$m$$ are respectively the number of vertices and edges (assuming that appending to a list is constant time, which it is not quite).

neighbor_in_iterator(vertex)

Return an iterator over the in-neighbors of vertex.

An vertex $$u$$ is an in-neighbor of a vertex $$v$$ if $$uv$$ in an edge.

EXAMPLES:

sage: D = DiGraph({0: [1,2,3], 1: [0,2], 2: [3], 3: [4], 4: [0,5], 5: [1]})
sage: for a in D.neighbor_in_iterator(0):
....:     print(a)
1
4

neighbor_out_iterator(vertex)

Return an iterator over the out-neighbors of a given vertex.

A vertex $$u$$ is an out-neighbor of a vertex $$v$$ if $$vu$$ in an edge.

EXAMPLES:

sage: D = DiGraph({0: [1,2,3], 1: [0,2], 2: [3], 3: [4], 4: [0,5], 5: [1]})
sage: for a in D.neighbor_out_iterator(0):
....:     print(a)
1
2
3

neighbors_in(vertex)

Return the list of the in-neighbors of a given vertex.

A vertex $$u$$ is an in-neighbor of a vertex $$v$$ if $$uv$$ in an edge.

EXAMPLES:

sage: D = DiGraph({0: [1,2,3], 1: [0,2], 2: [3], 3: [4], 4: [0,5], 5: [1]})
sage: D.neighbors_in(0)
[1, 4]

neighbors_out(vertex)

Return the list of the out-neighbors of a given vertex.

A vertex $$u$$ is an out-neighbor of a vertex $$v$$ if $$vu$$ in an edge.

EXAMPLES:

sage: D = DiGraph({0: [1,2,3], 1: [0,2], 2: [3], 3: [4], 4: [0,5], 5: [1]})
sage: D.neighbors_out(0)
[1, 2, 3]

out_degree(vertices=None, labels=False)

Same as degree, but for out degree.

EXAMPLES:

sage: D = DiGraph({0: [1,2,3], 1: [0,2], 2: [3], 3: [4], 4: [0,5], 5: [1]})
sage: D.out_degree(vertices=[0, 1 ,2], labels=True)
{0: 3, 1: 2, 2: 1}
sage: D.out_degree()
[3, 2, 1, 1, 2, 1]
sage: D.out_degree(2)
1

out_degree_iterator(vertices=None, labels=False)

Same as degree_iterator, but for out degree.

EXAMPLES:

sage: D = graphs.Grid2dGraph(2,4).to_directed()
sage: sorted(D.out_degree_iterator())
[2, 2, 2, 2, 3, 3, 3, 3]
sage: sorted(D.out_degree_iterator(labels=True))
[((0, 0), 2),
((0, 1), 3),
((0, 2), 3),
((0, 3), 2),
((1, 0), 2),
((1, 1), 3),
((1, 2), 3),
((1, 3), 2)]

out_degree_sequence()

Return the outdegree sequence of this digraph.

EXAMPLES:

The outdegree sequences of two digraphs:

sage: g = DiGraph({1: [2, 5, 6], 2: [3, 6], 3: [4, 6], 4: [6], 5: [4, 6]})
sage: g.out_degree_sequence()
[3, 2, 2, 2, 1, 0]

sage: V = [2, 3, 5, 7, 8, 9, 10, 11]
sage: E = [[], [8, 10], [11], [8, 11], [9], [], [], [2, 9, 10]]
sage: g = DiGraph(dict(zip(V, E)))
sage: g.out_degree_sequence()
[3, 2, 2, 1, 1, 0, 0, 0]

outgoing_edge_iterator(vertices, labels=True)

Return an iterator over all departing edges from vertices.

INPUT:

• vertices – a vertex or a list of vertices
• labels – boolean (default: True); whether to return edges as pairs of vertices, or as triples containing the labels.

EXAMPLES:

sage: D = DiGraph({0: [1,2,3], 1: [0,2], 2: [3], 3: [4], 4: [0,5], 5: [1]})
sage: for a in D.outgoing_edge_iterator([0]):
....:     print(a)
(0, 1, None)
(0, 2, None)
(0, 3, None)

outgoing_edges(vertices, labels=True)

Return a list of edges departing from vertices.

INPUT:

• vertices – a vertex or a list of vertices
• labels – boolean (default: True); whether to return edges as pairs of vertices, or as triples containing the labels.

EXAMPLES:

sage: D = DiGraph({0: [1,2,3], 1: [0,2], 2: [3], 3: [4], 4: [0,5], 5: [1]})
sage: D.outgoing_edges([0])
[(0, 1, None), (0, 2, None), (0, 3, None)]

path_semigroup()

The partial semigroup formed by the paths of this quiver.

EXAMPLES:

sage: Q = DiGraph({1: {2: ['a', 'c']}, 2: {3: ['b']}})
sage: F = Q.path_semigroup(); F
Partial semigroup formed by the directed paths of Multi-digraph on 3 vertices
sage: list(F)
[e_1, e_2, e_3, a, c, b, a*b, c*b]

period()

Return the period of the current DiGraph.

The period of a directed graph is the largest integer that divides the length of every cycle in the graph. See the Wikipedia article Aperiodic_graph for more information.

EXAMPLES:

The following graph has period 2:

sage: g = DiGraph({0: [1], 1: [0]})
sage: g.period()
2


The following graph has a cycle of length 2 and a cycle of length 3, so it has period 1:

sage: g = DiGraph({0: [1, 4], 1: [2], 2: [0], 4: [0]})
sage: g.period()
1


Here is an example of computing the period of a digraph which is not strongly connected. By definition, it is the gcd() of the periods of its strongly connected components:

sage: g = DiGraph({-1: [-2], -2: [-3], -3: [-1],
....:     1: [2], 2: [1]})
sage: g.period()
1
sage: sorted([s.period() for s
....:         in g.strongly_connected_components_subgraphs()])
[2, 3]


ALGORITHM:

See the networkX implementation of is_aperiodic, that is based on breadth first search.

reverse()

Return a copy of digraph with edges reversed in direction.

EXAMPLES:

sage: D = DiGraph({0: [1,2,3], 1: [0,2], 2: [3], 3: [4], 4: [0,5], 5: [1]})
sage: D.reverse()
Reverse of (): Digraph on 6 vertices

reverse_edge(u, v=None, label=None, inplace=True, multiedges=None)

Reverse the edge from $$u$$ to $$v$$.

INPUT:

• inplace – boolean (default: True); if False, a new digraph is created and returned as output, otherwise self is modified.

• multiedges – boolean (default: None); how to decide what should be done in case of doubt (for instance when edge $$(1,2)$$ is to be reversed in a graph while $$(2,1)$$ already exists):

• If set to True, input graph will be forced to allow parallel edges if necessary and edge $$(1,2)$$ will appear twice in the graph.
• If set to False, only one edge $$(1,2)$$ will remain in the graph after $$(2,1)$$ is reversed. Besides, the label of edge $$(1,2)$$ will be overwritten with the label of edge $$(2,1)$$.

The default behaviour (multiedges = None) will raise an exception each time a subjective decision (setting multiedges to True or False) is necessary to perform the operation.

The following forms are all accepted:

• D.reverse_edge( 1, 2 )
• D.reverse_edge( (1, 2) )
• D.reverse_edge( [1, 2] )
• D.reverse_edge( 1, 2, ‘label’ )
• D.reverse_edge( ( 1, 2, ‘label’) )
• D.reverse_edge( [1, 2, ‘label’] )
• D.reverse_edge( ( 1, 2), label=’label’ )

EXAMPLES:

If inplace is True (default value), self is modified:

sage: D = DiGraph([(0, 1 ,2)])
sage: D.reverse_edge(0, 1)
sage: D.edges()
[(1, 0, 2)]


If inplace is False, self is not modified and a new digraph is returned:

sage: D = DiGraph([(0, 1, 2)])
sage: re = D.reverse_edge(0, 1, inplace=False)
sage: re.edges()
[(1, 0, 2)]
sage: D.edges()
[(0, 1, 2)]


If multiedges is True, self will be forced to allow parallel edges when and only when it is necessary:

sage: D = DiGraph([(1, 2, 'A'), (2, 1, 'A'), (2, 3, None)])
sage: D.reverse_edge(1, 2, multiedges=True)
sage: D.edges()
[(2, 1, 'A'), (2, 1, 'A'), (2, 3, None)]
sage: D.allows_multiple_edges()
True


Even if multiedges is True, self will not be forced to allow parallel edges when it is not necessary:

sage: D = DiGraph( [(1, 2, 'A'), (2, 1, 'A'), (2, 3, None)] )
sage: D.reverse_edge(2, 3, multiedges=True)
sage: D.edges()
[(1, 2, 'A'), (2, 1, 'A'), (3, 2, None)]
sage: D.allows_multiple_edges()
False


If user specifies multiedges = False, self will not be forced to allow parallel edges and a parallel edge will get deleted:

sage: D = DiGraph( [(1, 2, 'A'), (2, 1, 'A'), (2, 3, None)] )
sage: D.edges()
[(1, 2, 'A'), (2, 1, 'A'), (2, 3, None)]
sage: D.reverse_edge(1, 2, multiedges=False)
sage: D.edges()
[(2, 1, 'A'), (2, 3, None)]


Note that in the following graph, specifying multiedges = False will result in overwriting the label of $$(1, 2)$$ with the label of $$(2, 1)$$:

sage: D = DiGraph( [(1, 2, 'B'), (2, 1, 'A'), (2, 3, None)] )
sage: D.edges()
[(1, 2, 'B'), (2, 1, 'A'), (2, 3, None)]
sage: D.reverse_edge(2, 1, multiedges=False)
sage: D.edges()
[(1, 2, 'A'), (2, 3, None)]


If input edge in digraph has weight/label, then the weight/label should be preserved in the output digraph. User does not need to specify the weight/label when calling function:

sage: D = DiGraph([[0, 1, 2], [1, 2, 1]], weighted=True)
sage: D.reverse_edge(0, 1)
sage: D.edges()
[(1, 0, 2), (1, 2, 1)]
sage: re = D.reverse_edge([1, 2], inplace=False)
sage: re.edges()
[(1, 0, 2), (2, 1, 1)]


If self has multiple copies (parallel edges) of the input edge, only 1 of the parallel edges is reversed:

sage: D = DiGraph([(0, 1, '01'), (0, 1, '01'), (0, 1, 'cat'), (1, 2, '12')], weighted=True, multiedges=True)
sage: re = D.reverse_edge([0, 1, '01'], inplace=False)
sage: re.edges()
[(0, 1, '01'), (0, 1, 'cat'), (1, 0, '01'), (1, 2, '12')]


If self has multiple copies (parallel edges) of the input edge but with distinct labels and no input label is specified, only 1 of the parallel edges is reversed (the edge that is labeled by the first label on the list returned by edge_label()):

sage: D = DiGraph([(0, 1, 'A'), (0, 1, 'B'), (0, 1, 'mouse'), (0, 1, 'cat')], multiedges=true)
sage: D.edge_label(0, 1)
['cat', 'mouse', 'B', 'A']
sage: D.reverse_edge(0, 1)
sage: D.edges()
[(0, 1, 'A'), (0, 1, 'B'), (0, 1, 'mouse'), (1, 0, 'cat')]


Finally, an exception is raised when Sage does not know how to choose between allowing multiple edges and losing some data:

sage: D = DiGraph([(0, 1, 'A'), (1, 0, 'B')])
sage: D.reverse_edge(0, 1)
Traceback (most recent call last):
...
ValueError: reversing the given edge is about to create two parallel
edges but input digraph doesn't allow them - User needs to specify
multiedges is True or False.


The following syntax is supported, but note that you must use the label keyword:

sage: D = DiGraph()
sage: D.edges()
[(1, 2, 'label')]
sage: D.reverse_edge((1, 2), label='label')
sage: D.edges()
[(2, 1, 'label')]
sage: D.edges(sort=False)
[((1, 2), 'label', None), (2, 1, 'label')]
sage: D.reverse_edge((1, 2), 'label')
sage: D.edges(sort=False)
[('label', (1, 2), None), (2, 1, 'label')]

reverse_edges(edges, inplace=True, multiedges=None)

Reverse a list of edges.

INPUT:

• edges – a list of edges in the DiGraph.
• inplace – boolean (default: True); if False, a new digraph is created and returned as output, otherwise self is modified.
• multiedges – boolean (default: None); if True, input graph will be forced to allow parallel edges when necessary (for more information see the documentation of reverse_edge())

reverse_edge() - Reverses a single edge.

EXAMPLES:

If inplace is True (default value), self is modified:

sage: D = DiGraph({ 0: [1, 1, 3], 2: [3, 3], 4: [1, 5]}, multiedges=true)
sage: D.reverse_edges([[0, 1], [0, 3]])
sage: D.reverse_edges([(2, 3), (4, 5)])
sage: D.edges()
[(0, 1, None), (1, 0, None), (2, 3, None), (3, 0, None),
(3, 2, None), (4, 1, None), (5, 4, None)]


If inplace is False, self is not modified and a new digraph is returned:

sage: D = DiGraph([(0, 1, 'A'), (1, 0, 'B'), (1, 2, 'C')])
sage: re = D.reverse_edges([(0, 1), (1, 2)],
....:                       inplace=False,
....:                       multiedges=True)
sage: re.edges()
[(1, 0, 'A'), (1, 0, 'B'), (2, 1, 'C')]
sage: D.edges()
[(0, 1, 'A'), (1, 0, 'B'), (1, 2, 'C')]
sage: D.allows_multiple_edges()
False
sage: re.allows_multiple_edges()
True


If multiedges is True, self will be forced to allow parallel edges when and only when it is necessary:

sage: D = DiGraph([(1, 2, 'A'), (2, 1, 'A'), (2, 3, None)])
sage: D.reverse_edges([(1, 2), (2, 3)], multiedges=True)
sage: D.edges()
[(2, 1, 'A'), (2, 1, 'A'), (3, 2, None)]
sage: D.allows_multiple_edges()
True


Even if multiedges is True, self will not be forced to allow parallel edges when it is not necessary:

sage: D = DiGraph([(1, 2, 'A'), (2, 1, 'A'), (2, 3, None)])
sage: D.reverse_edges([(2, 3)], multiedges=True)
sage: D.edges()
[(1, 2, 'A'), (2, 1, 'A'), (3, 2, None)]
sage: D.allows_multiple_edges()
False


If multiedges is False, self will not be forced to allow parallel edges and an edge will get deleted:

sage: D = DiGraph([(1, 2), (2, 1)])
sage: D.edges()
[(1, 2, None), (2, 1, None)]
sage: D.reverse_edges([(1, 2)], multiedges=False)
sage: D.edges()
[(2, 1, None)]


If input edge in digraph has weight/label, then the weight/label should be preserved in the output digraph. User does not need to specify the weight/label when calling function:

sage: D = DiGraph([(0, 1, '01'), (1, 2, 1), (2, 3, '23')], weighted=True)
sage: D.reverse_edges([(0, 1, '01'), (1, 2), (2, 3)])
sage: D.edges()
[(1, 0, '01'), (2, 1, 1), (3, 2, '23')]

sinks()

Return a list of sinks of the digraph.

OUTPUT:

• list of the vertices of the digraph that have no edges beginning at them

EXAMPLES:

sage: G = DiGraph({1: {3: ['a']}, 2: {3: ['b']}})
sage: G.sinks()
[3]
sage: T = DiGraph({1: {}})
sage: T.sinks()
[1]

sources()

Return a list of sources of the digraph.

OUTPUT:

• list of the vertices of the digraph that have no edges going into them

EXAMPLES:

sage: G = DiGraph({1: {3: ['a']}, 2: {3: ['b']}})
sage: G.sources()
[1, 2]
sage: T = DiGraph({1: {}})
sage: T.sources()
[1]

strong_articulation_points(G)

Return the strong articulation points of this digraph.

A vertex is a strong articulation point if its deletion increases the number of strongly connected components. This method implements the algorithm described in [ILS2012]. The time complexity is dominated by the time complexity of the immediate dominators finding algorithm.

OUTPUT: The list of strong articulation points.

EXAMPLES:

Two cliques sharing a vertex:

sage: from sage.graphs.connectivity import strong_articulation_points
sage: D = digraphs.Complete(4)
sage: strong_articulation_points(D)
[3]
sage: D.strong_articulation_points()
[3]


Two cliques connected by some arcs:

sage: D = digraphs.Complete(4) * 2
sage: sorted(strong_articulation_points(D))
[0, 3, 4, 7]
sage: sorted(strong_articulation_points(D))
[3, 7]
sage: strong_articulation_points(D)
[]

strongly_connected_component_containing_vertex(G, v)

Return the strongly connected component containing a given vertex

INPUT:

• G – the input DiGraph
• v – a vertex

EXAMPLES:

In the symmetric digraph of a graph, the strongly connected components are the connected components:

sage: from sage.graphs.connectivity import strongly_connected_component_containing_vertex
sage: g = graphs.PetersenGraph()
sage: d = DiGraph(g)
sage: strongly_connected_component_containing_vertex(d, 0)
[0, 1, 2, 3, 4, 5, 6, 7, 8, 9]
sage: d.strongly_connected_component_containing_vertex(0)
[0, 1, 2, 3, 4, 5, 6, 7, 8, 9]

sage: g = DiGraph([(0, 1), (1, 0), (1, 2), (2, 3), (3, 2)])
sage: strongly_connected_component_containing_vertex(g, 0)
[0, 1]

strongly_connected_components(G)

Return the lists of vertices in each strongly connected components (SCCs).

This method implements the Tarjan algorithm to compute the strongly connected components of the digraph. It returns a list of lists of vertices, each list of vertices representing a strongly connected component.

The basic idea of the algorithm is this: a depth-first search (DFS) begins from an arbitrary start node (and subsequent DFSes are conducted on any nodes that have not yet been found). As usual with DFSes, the search visits every node of the graph exactly once, declining to revisit any node that has already been explored. Thus, the collection of search trees is a spanning forest of the graph. The strongly connected components correspond to the subtrees of this spanning forest that have no edge directed outside the subtree.

To recover these components, during the DFS, we keep the index of a node, that is, the position in the DFS tree, and the lowlink: as soon as the subtree rooted at $$v$$ has been fully explored, the lowlink of $$v$$ is the smallest index reachable from $$v$$ passing from descendants of $$v$$. If the subtree rooted at $$v$$ has been fully explored, and the index of $$v$$ equals the lowlink of $$v$$, that whole subtree is a new SCC.

EXAMPLES:

sage: from sage.graphs.base.static_sparse_graph import tarjan_strongly_connected_components
sage: tarjan_strongly_connected_components(digraphs.Path(3))
[[2], [1], [0]]
sage: D = DiGraph( { 0 : [1, 3], 1 : [2], 2 : [3], 4 : [5, 6], 5 : [6] } )
sage: D.connected_components()
[[0, 1, 2, 3], [4, 5, 6]]
sage: D = DiGraph( { 0 : [1, 3], 1 : [2], 2 : [3], 4 : [5, 6], 5 : [6] } )
sage: D.strongly_connected_components()
[[3], [2], [1], [0], [6], [5], [4]]
sage: D.strongly_connected_components()
[[3], [0, 1, 2], [6], [5], [4]]
sage: D = DiGraph([('a','b'), ('b','c'), ('c', 'd'), ('d', 'b'), ('c', 'e')])
sage: [sorted(scc) for scc in D.strongly_connected_components()]
[['e'], ['b', 'c', 'd'], ['a']]

strongly_connected_components_digraph(G, keep_labels=False)

Return the digraph of the strongly connected components

The digraph of the strongly connected components of a graph $$G$$ has a vertex per strongly connected component included in $$G$$. There is an edge from a component $$C_1$$ to a component $$C_2$$ if there is an edge in $$G$$ from a vertex $$u_1 \in C_1$$ to a vertex $$u_2 \in C_2$$.

INPUT:

• G – the input DiGraph
• keep_labels – boolean (default: False); when keep_labels=True, the resulting digraph has an edge from a component $$C_i$$ to a component $$C_j$$ for each edge in $$G$$ from a vertex $$u_i \in C_i$$ to a vertex $$u_j \in C_j$$. Hence the resulting digraph may have loops and multiple edges. However, edges in the result with same source, target, and label are not duplicated (see examples below). When keep_labels=False, the return digraph is simple, so without loops nor multiple edges, and edges are unlabelled.

EXAMPLES:

Such a digraph is always acyclic:

sage: from sage.graphs.connectivity import strongly_connected_components_digraph
sage: g = digraphs.RandomDirectedGNP(15, .1)
sage: scc_digraph = strongly_connected_components_digraph(g)
sage: scc_digraph.is_directed_acyclic()
True
sage: scc_digraph = g.strongly_connected_components_digraph()
sage: scc_digraph.is_directed_acyclic()
True


The vertices of the digraph of strongly connected components are exactly the strongly connected components:

sage: g = digraphs.ButterflyGraph(2)
sage: scc_digraph = strongly_connected_components_digraph(g)
sage: g.is_directed_acyclic()
True
sage: V_scc = list(scc_digraph)
sage: all(Set(scc) in V_scc for scc in g.strongly_connected_components())
True


The following digraph has three strongly connected components, and the digraph of those is a TransitiveTournament():

sage: g = DiGraph({0: {1: "01", 2: "02", 3: "03"}, 1: {2: "12"}, 2:{1: "21", 3: "23"}})
sage: scc_digraph = strongly_connected_components_digraph(g)
sage: scc_digraph.is_isomorphic(digraphs.TransitiveTournament(3))
True


By default, the labels are discarded, and the result has no loops nor multiple edges. If keep_labels is True, then the labels are kept, and the result is a multi digraph, possibly with multiple edges and loops. However, edges in the result with same source, target, and label are not duplicated (see the edges from 0 to the strongly connected component $$\{1,2\}$$ below):

sage: g = DiGraph({0: {1: "0-12", 2: "0-12", 3: "0-3"}, 1: {2: "1-2", 3: "1-3"}, 2: {1: "2-1", 3: "2-3"}})
sage: g.order(), g.size()
(4, 7)
sage: scc_digraph = strongly_connected_components_digraph(g, keep_labels=True)
sage: (scc_digraph.order(), scc_digraph.size())
(3, 6)
sage: set(g.edge_labels()) == set(scc_digraph.edge_labels())
True

strongly_connected_components_subgraphs(G)

Return the strongly connected components as a list of subgraphs.

EXAMPLES:

In the symmetric digraph of a graph, the strongly connected components are the connected components:

sage: from sage.graphs.connectivity import strongly_connected_components_subgraphs
sage: g = graphs.PetersenGraph()
sage: d = DiGraph(g)
sage: strongly_connected_components_subgraphs(d)
[Subgraph of (Petersen graph): Digraph on 10 vertices]
sage: d.strongly_connected_components_subgraphs()
[Subgraph of (Petersen graph): Digraph on 10 vertices]

sage: g = DiGraph([(0, 1), (1, 0), (1, 2), (2, 3), (3, 2)])
sage: strongly_connected_components_subgraphs(g)
[Subgraph of (): Digraph on 2 vertices, Subgraph of (): Digraph on 2 vertices]

to_directed()

Since the graph is already directed, simply returns a copy of itself.

EXAMPLES:

sage: DiGraph({0: [1, 2, 3], 4: [5, 1]}).to_directed()
Digraph on 6 vertices

to_undirected(data_structure=None, sparse=None)

Return an undirected version of the graph.

Every directed edge becomes an edge.

INPUT:

• data_structure – string (default: None); one of "sparse", "static_sparse", or "dense". See the documentation of Graph or DiGraph.
• sparse – boolean (default: None); sparse=True is an alias for data_structure="sparse", and sparse=False is an alias for data_structure="dense".

EXAMPLES:

sage: D = DiGraph({0: [1, 2], 1: [0]})
sage: G = D.to_undirected()
sage: D.edges(labels=False)
[(0, 1), (0, 2), (1, 0)]
sage: G.edges(labels=False)
[(0, 1), (0, 2)]

topological_sort(implementation='default')

Return a topological sort of the digraph if it is acyclic.

If the digraph contains a directed cycle, a TypeError is raised. As topological sorts are not necessarily unique, different implementations may yield different results.

A topological sort is an ordering of the vertices of the digraph such that each vertex comes before all of its successors. That is, if $$u$$ comes before $$v$$ in the sort, then there may be a directed path from $$u$$ to $$v$$, but there will be no directed path from $$v$$ to $$u$$.

INPUT:

• implementation – string (default: "default"); either use the default Cython implementation, or the default NetworkX library (implementation = "NetworkX")

EXAMPLES:

sage: D = DiGraph({0: [1, 2, 3], 4: [2, 5], 1: [8], 2: [7], 3: [7],
....:   5: [6, 7], 7: [8], 6: [9], 8: [10], 9: [10]})
sage: D.plot(layout='circular').show()
sage: D.topological_sort()
[4, 5, 6, 9, 0, 1, 2, 3, 7, 8, 10]

sage: D.add_edge(9, 7)
sage: D.topological_sort()
[4, 5, 6, 9, 0, 1, 2, 3, 7, 8, 10]


Using the NetworkX implementation

sage: list(D.topological_sort(implementation="NetworkX"))
[4, 5, 6, 9, 0, 3, 2, 7, 1, 8, 10]

sage: D.add_edge(7, 4)
sage: D.topological_sort()
Traceback (most recent call last):
...
TypeError: digraph is not acyclic; there is no topological sort

topological_sort_generator()

Return an iterator over all topological sorts of the digraph if it is acyclic.

If the digraph contains a directed cycle, a TypeError is raised.

A topological sort is an ordering of the vertices of the digraph such that each vertex comes before all of its successors. That is, if u comes before v in the sort, then there may be a directed path from u to v, but there will be no directed path from v to u. See also topological_sort().

AUTHORS:

• Mike Hansen - original implementation
• Robert L. Miller: wrapping, documentation

REFERENCE:

• [1] Pruesse, Gara and Ruskey, Frank. Generating Linear Extensions Fast. SIAM J. Comput., Vol. 23 (1994), no. 2, pp. 373-386.

EXAMPLES:

sage: D = DiGraph({0: [1, 2], 1: [3], 2: [3, 4]})
sage: D.plot(layout='circular').show()
sage: list(D.topological_sort_generator())
[[0, 1, 2, 3, 4], [0, 2, 1, 3, 4], [0, 2, 1, 4, 3], [0, 2, 4, 1, 3], [0, 1, 2, 4, 3]]

sage: for sort in D.topological_sort_generator():
....:     for u, v in D.edge_iterator(labels=False):
....:         if sort.index(u) > sort.index(v):
....:             print("this should never happen")