Sage Reference Manual: Graph Theory - Mirrors

Sage Reference Manual: Graph Theory, Release 6.1.1
Giving a LollipopGraph(4,2), that is a complete graph with 4 vertices with a pending edge:
sage: G = graphs.LollipopGraph(4,2)
sage: G.is_cut_vertex(0)
False
sage: G.is_cut_vertex(3)
True
Comparing the weak and strong connectivity of a digraph:
sage: D = digraphs.Circuit(6)
sage: D.is_strongly_connected()
True
sage: D.is_cut_vertex(2)
True
sage: D.is_cut_vertex(2, weak=True)
False
Giving a vertex that is not in the graph:
sage: G = graphs.CompleteGraph(6)
sage: G.is_cut_vertex(7)
Traceback (most recent call last):
...
ValueError: The input vertex is not in the vertex set.
is_drawn_free_of_edge_crossings()
Returns True is the position dictionary for this graph is set and that position dictionary gives a planar
embedding.
This simply checks all pairs of edges that don’t share a vertex to make sure that they don’t intersect.
Note: This function require that _pos attribute is set. (Returns False otherwise.)
EXAMPLES:
sage: D = graphs.DodecahedralGraph()
sage: D.set_planar_positions()
sage: D.is_drawn_free_of_edge_crossings()
True
is_equitable(partition, quotient_matrix=False)
Checks whether the given partition is equitable with respect to self.
A partition is equitable with respect to a graph if for every pair of cells C1, C2 of the partition, the number
of edges from a vertex of C1 to C2 is the same, over all vertices in C1.
INPUT:
•partition - a list of lists
•quotient_matrix - (default False) if True, and the partition is equitable, returns a matrix over the
integers whose rows and columns represent cells of the partition, and whose i,j entry is the number of
vertices in cell j adjacent to each vertex in cell i (since the partition is equitable, this is well defined)
EXAMPLES:
sage: G = graphs.PetersenGraph()
sage: G.is_equitable([[0,4],[1,3,5,9],[2,6,8],[7]])
False
sage: G.is_equitable([[0,4],[1,3,5,9],[2,6,8,7]])
88 Chapter 1. Graph objects and methods