Sage Reference Manual: Graph Theory - Mirrors

Sage Reference Manual: Graph Theory, Release 6.1.1
The empty graph is not considered to be a tree:
sage: graphs.EmptyGraph().is_tree()
False
With certificates:
sage: g = graphs.RandomTree(30)
sage: g.is_tree(certificate=True)
(True, None)
sage: g.add_edge(10,-1)
sage: g.add_edge(11,-1)
sage: isit, cycle = g.is_tree(certificate=True)
sage: isit
False
sage: -1 in cycle
True
One can also ask for the certificate as a list of edges:
sage: g = graphs.CycleGraph(4)
sage: g.is_tree(certificate=True, output=’edge’)
(False, [(3, 2, None), (2, 1, None), (1, 0, None), (0, 3, None)])
This is useful for graphs with multiple edges:
sage: G = Graph([(1, 2, ’a’), (1, 2, ’b’)])
sage: G.is_tree(certificate=True)
(False, [1, 2])
sage: G.is_tree(certificate=True, output=’edge’)
(False, [(1, 2, ’a’), (2, 1, ’b’)])
TESTS:
trac ticket #14434 is fixed:
sage: g = Graph({0:[1,4,5],3:[4,8,9],4:[9],5:[7,8],7:[9]})
sage: _,cycle = g.is_tree(certificate=True)
sage: g.size()
10
sage: g.add_cycle(cycle)
sage: g.size()
10
is_triangle_free(algorithm=’bitset’)
Returns whether self is triangle-free
INPUT:
•algorithm – (default: ’bitset’) specifies the algorithm to use among:
EXAMPLE:
–’matrix’ – tests if the trace of the adjacency matrix is positive.
–’bitset’ – encodes adjacencies into bitsets and uses fast bitset operations to test if the input
graph contains a triangle. This method is generaly faster than stantard matrix multiplication.
The Petersen Graph is triangle-free:
sage: g = graphs.PetersenGraph()
sage: g.is_triangle_free()
True
1.2. Undirected graphs 217