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A good characterization of cograph contractions. (English) Zbl 0922.05045
All 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.

##### MSC:
 05C75 Structural characterization of families of graphs 05C60 Isomorphism problems in graph theory (reconstruction conjecture, etc.) and homomorphisms (subgraph embedding, etc.)
##### Keywords:
cograph contraction; characterization problem
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##### References:
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