A good characterization of cograph contractions.

*(English)*Zbl 0922.05045All graphs considered in the present paper are finite, undirected and simple ones. Such graphs \(H\) with no induced path on four vertices are called complement-reducible graphs or cographs. A graph \(G\) is defined to be a cograph contraction if it is obtained from a cograph \(H\) by contracting some pairwise disjoint independent sets and then making the contracted vertices pairwise adjacent. For this new graph \(H^*\) we have \(G= H^*\). M. Hujter and Zs. Tuza proved that cograph contractions are perfect, and they posed the characterization problem of cograph contractions which is solved by the present article.

In Section 2 two necessary conditions for cograph contractions are proved, namely: (1) each induced \(P_4\) in \(G\) has at least one midpoint in a clique \(Q\) in \(H^*\) (\(P_4\)-condition); (2) each induced \(\overline P_5\) in \(G\) has both midpoints in a clique \(Q\) in \(H^*\) (\(\overline P_5\)-condition). These conditions imply that cograph contractions are weakly triangulated graphs, that means, graphs without induced \(C_\ell\) and \(\overline C_\ell\) \((\ell\geq 5)\).

In Section 3 the main result of this paper is given by Theorem 3.1: A graph \(G\) is a cograph contraction iff it has a clique satisfying the \(P_4\)-condition and the \(\overline P_5\)-condition. This characterization yields a polynomial recognition algorithm for cograph contractions, by which the author gets a “good” clique in \(G\). Therefore the proof of Theorem 3.1 is a constructive one. This is described in detail and Sections 4 and 5. The construction shows that, in most cases, cograph contractions are obtained from a disconnected cograph \(H\). Therefore, the author also investigates the case of a connected cograph \(H\).

Section 6 contains the following result: A graph \(G\) is a connected-cograph contraction iff it is the join of two cograph contractions (Theorem 6.1). Moreover, an open problem is formulated.

In Section 2 two necessary conditions for cograph contractions are proved, namely: (1) each induced \(P_4\) in \(G\) has at least one midpoint in a clique \(Q\) in \(H^*\) (\(P_4\)-condition); (2) each induced \(\overline P_5\) in \(G\) has both midpoints in a clique \(Q\) in \(H^*\) (\(\overline P_5\)-condition). These conditions imply that cograph contractions are weakly triangulated graphs, that means, graphs without induced \(C_\ell\) and \(\overline C_\ell\) \((\ell\geq 5)\).

In Section 3 the main result of this paper is given by Theorem 3.1: A graph \(G\) is a cograph contraction iff it has a clique satisfying the \(P_4\)-condition and the \(\overline P_5\)-condition. This characterization yields a polynomial recognition algorithm for cograph contractions, by which the author gets a “good” clique in \(G\). Therefore the proof of Theorem 3.1 is a constructive one. This is described in detail and Sections 4 and 5. The construction shows that, in most cases, cograph contractions are obtained from a disconnected cograph \(H\). Therefore, the author also investigates the case of a connected cograph \(H\).

Section 6 contains the following result: A graph \(G\) is a connected-cograph contraction iff it is the join of two cograph contractions (Theorem 6.1). Moreover, an open problem is formulated.

Reviewer: H.-J.Presia (Ilmenau)

##### MSC:

05C75 | Structural characterization of families of graphs |

05C60 | Isomorphism problems in graph theory (reconstruction conjecture, etc.) and homomorphisms (subgraph embedding, etc.) |

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\textit{Van Bang Le}, J. Graph Theory 30, No. 4, 309--318 (1999; Zbl 0922.05045)

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