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)
REFERENCES:
[NikolopoulosPalios07]  (1, 2) Nikolopoulos, S.D. and Palios, L. Detecting holes and antiholes in graphs Algorithmica, 2007 Vol. 47, number 2, pages 119–138 http://www.cs.uoi.gr/~stavros/CPapers/C2004SODA.pdf 
Methods¶

sage.graphs.weakly_chordal.
is_long_antihole_free
¶ 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)
ifg
contains an induced complement of a cycle of length at least 5 returned asAntihole
.(True, [])
ifg
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 justTrue
orFalse
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 depthfirstsearch. For efficiency, the auxiliary graph is constructed onthefly and never stored in memory.
The run time of this algorithm is \(O(m^2)\) [NikolopoulosPalios07] ( where \(m\) is the number of edges of the graph ) .
EXAMPLES:
The Petersen Graph contains an antihole:
sage: g = graphs.PetersenGraph() sage: g.is_long_antihole_free() False
The complement of a cycle is an antihole:
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

sage.graphs.weakly_chordal.
is_long_hole_free
¶ 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, [])
ifg
does not contain such a cycle. For this case, it is not known how to provide a certificate.(False, Hole)
ifg
contains an induced cycle of length at least 5.Hole
returns this cycle.
If
certificate = False
, the function returns justTrue
orFalse
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 depthfirstsearch. For efficiency, the auxiliary graph is constructed onthefly and never stored in memory.
The run time of this algorithm is \(O(m^2)\) [NikolopoulosPalios07] ( where \(m\) is the number of edges of the graph ) .
EXAMPLES:
The Petersen Graph contains a hole:
sage: g = graphs.PetersenGraph() sage: g.is_long_hole_free() False
The following graph contains a hole, which we want to display:
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

sage.graphs.weakly_chordal.
is_weakly_chordal
¶ 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. Ifcertificate = False
, returnTrue
orFalse
according to the graph. Ifcertificate = 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 \(O(m^2)\) (where \(m\) is the number of edges of the graph).
EXAMPLES:
The Petersen Graph is not weakly chordal and contains a hole:
sage: g = graphs.PetersenGraph() sage: r,s = g.is_weakly_chordal(certificate = True) sage: r False sage: l = len(s.vertices()) sage: s.is_isomorphic( graphs.CycleGraph(l) ) True