Weakly chordal graphs#

This module deals with everything related to weakly chordal graphs. It currently contains the following functions:

`is_long_hole_free()`	Tests whether `g` contains an induced cycle of length at least 5.
`is_long_antihole_free()`	Tests whether `g` contains an induced anticycle of length at least 5.
`is_weakly_chordal()`	Tests whether `g` is weakly chordal.

Author:

Birk Eisermann (initial implementation)
Nathann Cohen (some doc and optimization)
David Coudert (remove recursion)

Methods#

sage.graphs.weakly_chordal.is_long_antihole_free(g, certificate=False)[source]#

Tests whether the given graph contains an induced subgraph that is isomorphic to the complement of a cycle of length at least 5.

INPUT:

certificate – boolean (default: False)

Whether to return a certificate. When certificate = True, then the function returns
- (False, Antihole) if g contains an induced complement of a cycle of length at least 5 returned as Antihole.
- (True, []) if g does not contain an induced complement of a cycle of length at least 5. For this case it is not known how to provide a certificate.
When certificate = False, the function returns just True or False accordingly.

ALGORITHM:

This algorithm tries to find a cycle in the graph of all induced \(\overline{P_4}\) of \(g\), where two copies \(\overline{P}\) and \(\overline{P'}\) of \(\overline{P_4}\) are adjacent if there exists a (not necessarily induced) copy of \(\overline{P_5}=u_1u_2u_3u_4u_5\) such that \(\overline{P}=u_1u_2u_3u_4\) and \(\overline{P'}=u_2u_3u_4u_5\).

This is done through a depth-first-search. For efficiency, the auxiliary graph is constructed on-the-fly and never stored in memory.

The run time of this algorithm is \(O(n+m^2)\) for SparseGraph and \(O(n^2\log{m} + m^2)\) for DenseGraph [NP2007] (where \(n\) is the number of vertices and \(m\) is the number of edges of the graph).

EXAMPLES:

The Petersen Graph contains an antihole:

Sage

sage: g = graphs.PetersenGraph()
sage: g.is_long_antihole_free()
False

Python

>>> from sage.all import *
>>> g = graphs.PetersenGraph()
>>> g.is_long_antihole_free()
False

The complement of a cycle is an antihole:

Sage

sage: g = graphs.CycleGraph(6).complement()
sage: r,a = g.is_long_antihole_free(certificate=True)
sage: r
False
sage: a.complement().is_isomorphic(graphs.CycleGraph(6))
True

Python

>>> from sage.all import *
>>> g = graphs.CycleGraph(Integer(6)).complement()
>>> r,a = g.is_long_antihole_free(certificate=True)
>>> r
False
>>> a.complement().is_isomorphic(graphs.CycleGraph(Integer(6)))
True

sage.graphs.weakly_chordal.is_long_hole_free(g, certificate=False)[source]#

Tests whether g contains an induced cycle of length at least 5.

INPUT:

certificate – boolean (default: False)

Whether to return a certificate. When certificate = True, then the function returns
- (True, []) if g does not contain such a cycle. For this case, it is not known how to provide a certificate.
- (False, Hole) if g contains an induced cycle of length at least 5. Hole returns this cycle.
If certificate = False, the function returns just True or False accordingly.

ALGORITHM:

This algorithm tries to find a cycle in the graph of all induced \(P_4\) of \(g\), where two copies \(P\) and \(P'\) of \(P_4\) are adjacent if there exists a (not necessarily induced) copy of \(P_5=u_1u_2u_3u_4u_5\) such that \(P=u_1u_2u_3u_4\) and \(P'=u_2u_3u_4u_5\).

This is done through a depth-first-search. For efficiency, the auxiliary graph is constructed on-the-fly and never stored in memory.

The run time of this algorithm is \(O(n+m^2)\) for SparseGraph and \(O(n^2 + m^2)\) for DenseGraph [NP2007] (where \(n\) is the number of vertices and \(m\) is the number of edges of the graph).

EXAMPLES:

The Petersen Graph contains a hole:

Sage

sage: g = graphs.PetersenGraph()
sage: g.is_long_hole_free()
False

Python

>>> from sage.all import *
>>> g = graphs.PetersenGraph()
>>> g.is_long_hole_free()
False

The following graph contains a hole, which we want to display:

Sage

sage: g = graphs.FlowerSnark()
sage: r,h = g.is_long_hole_free(certificate=True)
sage: r
False
sage: Graph(h).is_isomorphic(graphs.CycleGraph(h.order()))
True

Python

>>> from sage.all import *
>>> g = graphs.FlowerSnark()
>>> r,h = g.is_long_hole_free(certificate=True)
>>> r
False
>>> Graph(h).is_isomorphic(graphs.CycleGraph(h.order()))
True

sage.graphs.weakly_chordal.is_weakly_chordal(g, certificate=False)[source]#

Tests whether the given graph is weakly chordal, i.e., the graph and its complement have no induced cycle of length at least 5.

INPUT:

certificate – Boolean value (default: False) whether to return a certificate. If certificate = False, return True or False according to the graph. If certificate = True, return
- (False, forbidden_subgraph) when the graph contains a forbidden subgraph H, this graph is returned.
- (True, []) when the graph is weakly chordal.
  For this case, it is not known how to provide a certificate.

ALGORITHM:

This algorithm checks whether the graph g or its complement contain an induced cycle of length at least 5.

Using is_long_hole_free() and is_long_antihole_free() yields a run time of \(O(n+m^2)\) for SparseGraph and \(O(n^2\log{m} + m^2)\) for DenseGraph (where \(n\) is the number of vertices and \(m\) is the number of edges of the graph).

EXAMPLES:

The Petersen Graph is not weakly chordal and contains a hole:

Sage

sage: g = graphs.PetersenGraph()
sage: r,s = g.is_weakly_chordal(certificate=True)
sage: r
False
sage: l = s.order()
sage: s.is_isomorphic(graphs.CycleGraph(l))
True

Python

>>> from sage.all import *
>>> g = graphs.PetersenGraph()
>>> r,s = g.is_weakly_chordal(certificate=True)
>>> r
False
>>> l = s.order()
>>> s.is_isomorphic(graphs.CycleGraph(l))
True