Abstract
The class of even-hole-free graphs is structurally quite similar to the class of perfect graphs, which was the key initial motivation for their study. The techniques developed in the study of even-hole-free graphs were generalized to other complex hereditary graph classes, and in the case of perfect graphs led to the famous resolution of the Strong Perfect Graph Conjecture and their polynomial time recognition. The class of even-holefree graphs is also of independent interest due to its relationship to ?-perfect graphs. In this survey we describe all the different structural characterizations of even-hole-free graphs, focusing on their algorithmic consequences.
Highlights
All graphs in this paper are finite, simple and undirected
Theorem 3.1 (Conforti, Cornuejols, Kapoor and Vuskovic [15]) A connected 4hole-free odd-signable graph is either a clique, a hole, a long 3P C(∆, ·) or a nontrivial basic graph, or it has a 2-join or k-star cutset, for k ≤ 3. This theorem was strong enough to be used in construction of the first known polynomial time recognition algorithm for even-hole-free graphs [16], as we shall see in Section 4, but even at that time it was suspected that a stronger decomposition theorem was possible
We describe the ideas behind a decomposition based recognition algorithm
Summary
All graphs in this paper are finite, simple and undirected. We say that a graph G contains a graph F if F is isomorphic to an induced subgraph of G. A hole of length n is called an n-hole In this survey we focus on the class of even-hole-free graphs, i.e. graphs that are F-free where F denotes the family of all even holes. Note that by excluding the 4-hole, one excludes all antiholes of length at least 6, so the similarity between even-hole-free graphs and Berge graphs is higher than with the class odd-hole-free graphs. Another motivation for the study of even-hole-free graphs is their connection to β-perfect graphs introduced by Markossian, Gasparian and Reed [34].
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