Graphic Matroids#
Let \(G = (V,E)\) be a graph and let \(C\) be the collection of the edge sets of cycles in \(G\). The corresponding graphic matroid \(M(G)\) has ground set \(E\) and circuits \(C\).
Construction#
The recommended way to create a graphic matroid is by using the
Matroid()
function, with a
graph \(G\) as input. This function can accept many different kinds of input
to get a graphic matroid if the graph
keyword is used, similar to the
Graph()
constructor. However,
invoking the class directly is possible too. To get access to it, type:
sage: from sage.matroids.advanced import *
See also sage.matroids.advanced
.
Graphic matroids do not have a representation matrix or any of the
functionality of regular matroids. It is possible to get an instance of the
RegularMatroid
class
by using the regular
keyword when constructing the matroid.
It is also possible to cast a GraphicMatroid as a RegularMatroid with the
regular_matroid()
method:
sage: M1 = Matroid(graphs.DiamondGraph(), regular=True)
sage: M2 = Matroid(graphs.DiamondGraph())
sage: M3 = M2.regular_matroid()
Below are some examples of constructing a graphic matroid.
sage: from sage.matroids.advanced import *
sage: edgelist = [(0, 1, 'a'), (0, 2, 'b'), (1, 2, 'c')]
sage: G = Graph(edgelist)
sage: M1 = Matroid(G)
sage: M2 = Matroid(graph=edgelist)
sage: M3 = Matroid(graphs.CycleGraph(3))
sage: M1 == M3
False
sage: M1.is_isomorphic(M3)
True
sage: M1.equals(M2)
True
sage: M1 == M2
True
sage: isinstance(M1, GraphicMatroid)
True
sage: isinstance(M1, RegularMatroid)
False
Note that if there is not a complete set of unique edge labels, and there are no parallel edges, then vertex tuples will be used for the ground set. The user may wish to override this by specifying the ground set, as the vertex tuples will not be updated if the matroid is modified:
sage: G = graphs.DiamondGraph()
sage: M1 = Matroid(G)
sage: N1 = M1.contract((0,1))
sage: N1.graph().edges_incident(0, sort=True)
[(0, 2, (0, 2)), (0, 2, (1, 2)), (0, 3, (1, 3))]
sage: M2 = Matroid(range(G.num_edges()), G)
sage: N2 = M2.contract(0)
sage: N1.is_isomorphic(N2)
True
AUTHORS:
Zachary Gershkoff (2017-07-07): initial version
Methods#
- class sage.matroids.graphic_matroid.GraphicMatroid(G, groundset=None)#
Bases:
Matroid
The graphic matroid class.
INPUT:
G
– a Graphgroundset
– (optional) a list in 1-1 correspondence withG.edge_iterator()
OUTPUT:
A
GraphicMatroid
instance where the ground set elements are the edges ofG
.Note
If a disconnected graph is given as input, the instance of
GraphicMatroid
will connect the graph components and store this as its graph.EXAMPLES:
sage: from sage.matroids.advanced import * sage: M = GraphicMatroid(graphs.BullGraph()); M Graphic matroid of rank 4 on 5 elements sage: N = GraphicMatroid(graphs.CompleteBipartiteGraph(3,3)); N Graphic matroid of rank 5 on 9 elements
A disconnected input will get converted to a connected graph internally:
sage: G1 = graphs.CycleGraph(3); G2 = graphs.DiamondGraph() sage: G = G1.disjoint_union(G2) sage: len(G) 7 sage: G.is_connected() False sage: M = GraphicMatroid(G) sage: M Graphic matroid of rank 5 on 8 elements sage: H = M.graph() sage: H Looped multi-graph on 6 vertices sage: H.is_connected() True sage: M.is_connected() False
You can still locate an edge using the vertices of the input graph:
sage: G1 = graphs.CycleGraph(3); G2 = graphs.DiamondGraph() sage: G = G1.disjoint_union(G2) sage: M = Matroid(G) sage: H = M.graph() sage: vm = M.vertex_map() sage: (u, v, l) = G.random_edge() sage: H.has_edge(vm[u], vm[v]) True
- graph()#
Return the graph that represents the matroid.
The graph will always have loops and multiedges enabled.
OUTPUT:
A Graph.
EXAMPLES:
sage: M = Matroid(Graph([(0, 1, 'a'), (0, 2, 'b'), (0, 3, 'c')])) sage: M.graph().edges(sort=True) [(0, 1, 'a'), (0, 2, 'b'), (0, 3, 'c')] sage: M = Matroid(graphs.CompleteGraph(5)) sage: M.graph() Looped multi-graph on 5 vertices
- graphic_coextension(u, v=None, X=None, element=None)#
Return a matroid coextended by a new element.
A coextension in a graphic matroid is the opposite of contracting an edge; that is, a vertex is split, and a new edge is added between the resulting vertices. This method will create a new vertex \(v\) adjacent to \(u\), and move the edges indicated by \(X\) from \(u\) to \(v\).
INPUT:
u
– the vertex to be splitv
– (optional) the name of the new vertex after splittingX
– (optional) a list of the matroid elements corresponding to edges incident tou
that move to the new vertex after splittingelement
– (optional) The name of the newly added element
OUTPUT:
An instance of GraphicMatroid coextended by the new element. If
X
is not specified, the new element will be a coloop.Note
A loop on
u
will stay a loop unless it is inX
.EXAMPLES:
sage: G = Graph([(0, 1, 0), (0, 2, 1), (0, 3, 2), (0, 4, 3), (1, 2, 4), (1, 4, 5), (2, 3, 6), (3, 4, 7)]) sage: M = Matroid(G) sage: M1 = M.graphic_coextension(0, X=[1,2], element='a') sage: M1.graph().edges(sort=True) [(0, 1, 0), (0, 4, 3), (0, 5, 'a'), (1, 2, 4), (1, 4, 5), (2, 3, 6), (2, 5, 1), (3, 4, 7), (3, 5, 2)]
- graphic_coextensions(vertices=None, v=None, element=None, cosimple=False)#
Return an iterator of graphic coextensions.
This method iterates over the vertices in the input. If
cosimple == False
, it first coextends by a coloop and series edge for every edge incident with the vertices. For vertices of degree four or higher, it will consider the ways to partition the vertex into two sets of cardinality at least two, and these will be the edges incident with the vertices after splitting.At most one series coextension will be taken for each series class.
INPUT:
vertices
– (optional) the vertices to be splitv
– (optional) the name of the new vertexelement
– (optional) the name of the new elementcosimple
– (default:False
) if true, coextensions by a coloop or series elements will not be taken
OUTPUT:
An iterable containing instances of GraphicMatroid. If
vertices
is not specified, the method iterates over all vertices.EXAMPLES:
sage: G = Graph([(0, 1), (0, 2), (0, 3), (0, 4), (1, 2), (1, 4), (2, 3), (3, 4)]) sage: M = Matroid(range(8), G) sage: I = M.graphic_coextensions(vertices=[0], element='a') sage: sorted([N.graph().edges_incident(0, sort=True) for N in I],key=str) [[(0, 1, 0), (0, 2, 1), (0, 3, 2), (0, 4, 3), (0, 5, 'a')], [(0, 1, 0), (0, 2, 1), (0, 3, 2), (0, 5, 'a')], [(0, 1, 0), (0, 2, 1), (0, 4, 3), (0, 5, 'a')], [(0, 1, 0), (0, 2, 1), (0, 5, 'a')], [(0, 1, 0), (0, 3, 2), (0, 4, 3), (0, 5, 'a')], [(0, 1, 0), (0, 3, 2), (0, 5, 'a')], [(0, 2, 1), (0, 3, 2), (0, 4, 3), (0, 5, 'a')], [(0, 2, 1), (0, 3, 2), (0, 5, 'a')]]
sage: N = Matroid(range(4), graphs.CycleGraph(4)) sage: I = N.graphic_coextensions(element='a') sage: for N1 in I: # random ....: N1.graph().edges(sort=True) [(0, 1, 0), (0, 3, 1), (0, 4, 'a'), (1, 2, 2), (2, 3, 3)] [(0, 1, 0), (0, 3, 1), (1, 4, 2), (2, 3, 3), (2, 4, 'a')] sage: sum(1 for n in N.graphic_coextensions(cosimple=True)) 0
- graphic_extension(u, v=None, element=None)#
Return a graphic matroid extended by a new element.
A new edge will be added between
u
andv
. Ifv
is not specified, then a loop is added onu
.INPUT:
u
– a vertex in the matroid’s graphv
– (optional) another vertexelement
– (optional) the label of the new element
OUTPUT:
A GraphicMatroid with the specified element added. Note that if
v
is not specifies or ifv
isu
, then the new element will be a loop. If the new element’s label is not specified, it will be generated automatically.EXAMPLES:
sage: M = matroids.CompleteGraphic(4) sage: M1 = M.graphic_extension(0,1,'a'); M1 Graphic matroid of rank 3 on 7 elements sage: list(M1.graph().edge_iterator()) [(0, 1, 'a'), (0, 1, 0), (0, 2, 1), (0, 3, 2), (1, 2, 3), (1, 3, 4), (2, 3, 5)] sage: M2 = M1.graphic_extension(3); M2 Graphic matroid of rank 3 on 8 elements
sage: M = Matroid(range(10), graphs.PetersenGraph()) sage: sorted(M.graphic_extension(0, 'b', 'c').graph().vertex_iterator(), key=str) [0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 'b'] sage: M.graphic_extension('a', 'b', 'c').graph().vertices(sort=False) Traceback (most recent call last): ... ValueError: u must be an existing vertex
- graphic_extensions(element=None, vertices=None, simple=False)#
Return an iterable containing the graphic extensions.
This method iterates over the vertices in the input. If
simple == False
, it first extends by a loop. It will then add an edge between every pair of vertices in the input, skipping pairs of vertices with an edge already between them ifsimple == True
.This method only considers the current graph presentation, and does not take 2-isomorphism into account. Use
twist
orone_sum
if you wish to change the graph presentation.INPUT:
element
– (optional) the name of the newly added element in each extensionvertices
– (optional) a set of vertices over which the extension may be takensimple
– (default:False
) if true, extensions by loops and parallel elements are not taken
OUTPUT:
An iterable containing instances of GraphicMatroid. If
vertices
is not specified, every vertex is used.Note
The extension by a loop will always occur unless
simple == True
. The extension by a coloop will never occur.EXAMPLES:
sage: M = Matroid(range(5), graphs.DiamondGraph()) sage: I = M.graphic_extensions('a') sage: for N in I: ....: list(N.graph().edge_iterator()) [(0, 0, 'a'), (0, 1, 0), (0, 2, 1), (1, 2, 2), (1, 3, 3), (2, 3, 4)] [(0, 1, 'a'), (0, 1, 0), (0, 2, 1), (1, 2, 2), (1, 3, 3), (2, 3, 4)] [(0, 1, 0), (0, 2, 'a'), (0, 2, 1), (1, 2, 2), (1, 3, 3), (2, 3, 4)] [(0, 1, 0), (0, 2, 1), (0, 3, 'a'), (1, 2, 2), (1, 3, 3), (2, 3, 4)] [(0, 1, 0), (0, 2, 1), (1, 2, 'a'), (1, 2, 2), (1, 3, 3), (2, 3, 4)] [(0, 1, 0), (0, 2, 1), (1, 2, 2), (1, 3, 'a'), (1, 3, 3), (2, 3, 4)] [(0, 1, 0), (0, 2, 1), (1, 2, 2), (1, 3, 3), (2, 3, 'a'), (2, 3, 4)]
sage: M = Matroid(graphs.CompleteBipartiteGraph(3,3)) sage: I = M.graphic_extensions(simple=True) sage: sum (1 for i in I) 6 sage: I = M.graphic_extensions(vertices=[0,1,2]) sage: sum (1 for i in I) 4
- groundset()#
Return the ground set of the matroid as a frozenset.
EXAMPLES:
sage: M = Matroid(graphs.DiamondGraph()) sage: sorted(M.groundset()) [(0, 1), (0, 2), (1, 2), (1, 3), (2, 3)] sage: G = graphs.CompleteGraph(3).disjoint_union(graphs.CompleteGraph(4)) sage: M = Matroid(range(G.num_edges()), G); sorted(M.groundset()) [0, 1, 2, 3, 4, 5, 6, 7, 8] sage: M = Matroid(Graph([(0, 1, 'a'), (0, 2, 'b'), (0, 3, 'c')])) sage: sorted(M.groundset()) ['a', 'b', 'c']
- groundset_to_edges(X)#
Return a list of edges corresponding to a set of ground set elements.
INPUT:
X
– a subset of the ground set
OUTPUT:
A list of graph edges.
EXAMPLES:
sage: M = Matroid(range(5), graphs.DiamondGraph()) sage: M.groundset_to_edges([2,3,4]) [(1, 2, 2), (1, 3, 3), (2, 3, 4)] sage: M.groundset_to_edges([2,3,4,5]) Traceback (most recent call last): ... ValueError: input must be a subset of the ground set
- is_valid()#
Test if the data obey the matroid axioms.
Since a graph is used for the data, this is always the case.
OUTPUT:
True
.EXAMPLES:
sage: M = matroids.CompleteGraphic(4); M M(K4): Graphic matroid of rank 3 on 6 elements sage: M.is_valid() True
- one_sum(X, u, v)#
Arrange matroid components in the graph.
The matroid’s graph must be connected even if the matroid is not connected, but if there are multiple matroid components, the user may choose how they are arranged in the graph. This method will take the block of the graph that represents \(X\) and attach it by vertex \(u\) to another vertex \(v\) in the graph.
INPUT:
X
– a subset of the ground setu
– a vertex spanned by the edges of the elements inX
v
– a vertex spanned by the edges of the elements not inX
OUTPUT:
An instance of
GraphicMatroid
isomorphic to this matroid but with a graph that is not necessarily isomorphic.EXAMPLES:
sage: edgedict = {0:[1, 2], 1:[2, 3], 2:[3], 3:[4, 5], 6:[4, 5]} sage: M = Matroid(range(9), Graph(edgedict)) sage: M.graph().edges(sort=True) [(0, 1, 0), (0, 2, 1), (1, 2, 2), (1, 3, 3), (2, 3, 4), (3, 4, 5), (3, 5, 6), (4, 6, 7), (5, 6, 8)] sage: M1 = M.one_sum(u=3, v=1, X=[5, 6, 7, 8]) sage: M1.graph().edges(sort=True) [(0, 1, 0), (0, 2, 1), (1, 2, 2), (1, 3, 3), (1, 4, 5), (1, 5, 6), (2, 3, 4), (4, 6, 7), (5, 6, 8)] sage: M2 = M.one_sum(u=4, v=3, X=[5, 6, 7, 8]) sage: M2.graph().edges(sort=True) [(0, 1, 0), (0, 2, 1), (1, 2, 2), (1, 3, 3), (2, 3, 4), (3, 6, 7), (3, 7, 5), (5, 6, 8), (5, 7, 6)]
- regular_matroid()#
Return an instance of RegularMatroid isomorphic to this GraphicMatroid.
EXAMPLES:
sage: M = matroids.CompleteGraphic(5); M M(K5): Graphic matroid of rank 4 on 10 elements sage: N = M.regular_matroid(); N Regular matroid of rank 4 on 10 elements with 125 bases sage: M.equals(N) True sage: M == N False
- subgraph_from_set(X)#
Return the subgraph corresponding to the matroid restricted to \(X\).
INPUT:
X
– a subset of the ground set
OUTPUT:
A Graph.
EXAMPLES:
sage: M = Matroid(range(5), graphs.DiamondGraph()) sage: M.subgraph_from_set([0,1,2]) Looped multi-graph on 3 vertices sage: M.subgraph_from_set([3,4,5]) Traceback (most recent call last): ... ValueError: input must be a subset of the ground set
- twist(X)#
Perform a Whitney twist on the graph.
\(X\) must be part of a 2-separation. The connectivity of \(X\) must be 1, and the subgraph induced by \(X\) must intersect the subgraph induced by the rest of the elements on exactly two vertices.
INPUT:
X
– the set of elements to be twisted with respect to the rest of the matroid
OUTPUT:
An instance of
GraphicMatroid
isomorphic to this matroid but with a graph that is not necessarily isomorphic.EXAMPLES:
sage: edgelist = [(0,1,0), (1,2,1), (1,2,2), (2,3,3), (2,3,4), (2,3,5), (3,0,6)] sage: M = Matroid(Graph(edgelist, multiedges=True)) sage: M1 = M.twist([0,1,2]); M1.graph().edges(sort=True) [(0, 1, 1), (0, 1, 2), (0, 3, 6), (1, 2, 0), (2, 3, 3), (2, 3, 4), (2, 3, 5)] sage: M2 = M.twist([0,1,3]) Traceback (most recent call last): ... ValueError: the input must display a 2-separation that is not a 1-separation
- vertex_map()#
Return a dictionary mapping the input vertices to the current vertices.
The graph for the matroid is always connected. If the constructor is given a graph with multiple components, it will connect them. The Python dictionary given by this method has the vertices from the input graph as keys, and the corresponding vertex label after any merging as values.
OUTPUT:
A dictionary.
EXAMPLES:
sage: G = Graph([(0, 1), (0, 2), (1, 2), (3, 4), (3, 5), (4, 5), ....: (6, 7), (6, 8), (7, 8), (8, 8), (7, 8)], multiedges=True, loops=True) sage: M = Matroid(range(G.num_edges()), G) sage: M.graph().edges(sort=True) [(0, 1, 0), (0, 2, 1), (1, 2, 2), (1, 4, 3), (1, 5, 4), (4, 5, 5), (4, 7, 6), (4, 8, 7), (7, 8, 8), (7, 8, 9), (8, 8, 10)] sage: M.vertex_map() {0: 0, 1: 1, 2: 2, 3: 1, 4: 4, 5: 5, 6: 4, 7: 7, 8: 8}