Sage Reference Manual: Graph Theory - Mirrors

Sage Reference Manual: Graph Theory, Release 6.1.1
sage: 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)]]
sage: ss = (graphs.WheelGraph(5)).line_graph(labels=False)
sage: ss.coarsest_equitable_refinement(prt)
[[(0, 1)], [(1, 2), (1, 4)], [(0, 3)], [(0, 2), (0, 4)], [(2, 3), (3, 4)]]
ALGORITHM: Brendan D. McKay’s Master’s Thesis, University of Melbourne, 1976.
complement()
Returns 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
sage: PC = P.complement()
sage: PC.plot() # long time
sage: graphs.TetrahedralGraph().complement().size()
0
sage: graphs.CycleGraph(4).complement().edges()
[(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):
...
TypeError: complement not well defined for (di)graphs with multiple edges
connected_component_containing_vertex(vertex)
Returns a list of the vertices connected to vertex.
EXAMPLES:
sage: G = Graph( { 0 : [1, 3], 1 : [2], 2 : [3], 4 : [5, 6], 5 : [6] } )
sage: G.connected_component_containing_vertex(0)
[0, 1, 2, 3]
sage: D = DiGraph( { 0 : [1, 3], 1 : [2], 2 : [3], 4 : [5, 6], 5 : [6] } )
sage: D.connected_component_containing_vertex(0)
[0, 1, 2, 3]
1.1. Generic graphs 29