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# Weakly chordal graphs

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/C-Papers/C-2004-SODA.pdf

## Methods

`sage.graphs.weakly_chordal.``is_long_antihole_free`(g, certificate=False)

  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 \(O(m^2)\) [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

  TESTS:

  Further tests:

      sage: g = Graph({0:[6,7],1:[7,8],2:[8,9],3:[9,10],4:[10,11],5:[11,6],6:[0,5,7],7:[0,1,6],8:[1,2,9],9:[2,3,8],10:[3,4,11],11:[4,5,10]}).complement()
      sage: r,a = g.is_long_antihole_free(certificate=True)
      sage: r
      False
      sage: a.complement().is_isomorphic( graphs.CycleGraph(9) )
      True

`sage.graphs.weakly_chordal.``is_long_hole_free`(g, certificate=False)

  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 \(O(m^2)\) [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

  TESTS:

  Another graph with vertices 2, ..., 8, 10:

      sage: g = Graph({2:[3,8],3:[2,4],4:[3,8,10],5:[6,10],6:[5,7],7:[6,8],8:[2,4,7,10],10:[4,5,8]})
      sage: r,hole = g.is_long_hole_free(certificate=True)
      sage: r
      False
      sage: hole
      Subgraph of (): Graph on 5 vertices
      sage: hole.is_isomorphic(graphs.CycleGraph(hole.order()))
      True

`sage.graphs.weakly_chordal.``is_weakly_chordal`(g, certificate=False)

  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 \(O(m^2)\) (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

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