Even-hole-free graphs: A survey

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 β-perfect graphs. In this survey we describe all the different structural characterizations of even-hole-free graphs, focusing on their algorithmic consequences.

(Theta, triangle)‐free and (even hole, K4)‐free graphs—Part 1: Layered wheels

“…ehf graphs were originally studied to experiment techniques that would help to settle problems on perfect graphs. This has succeeded, in the sense that the decomposition theorem for ehf graphs (see [24]) is in some respect similar to the one that was later on discovered for perfect graphs (see [5]). However, classical problems such as graph coloring or maximum stable set, are polynomial time solvable for perfect graphs, while they are still open for ehf graphs.…”

Maximum independent sets in (pyramid, even hole)-free graphs

“…A weakening of Conjecture 2.2 is therefore obtained by restricting it to (prism, pyramid, theta, even wheel)-free graphs. Note that (prism, theta, even wheel)-free graphs have been studied under the name of odd-signable graphs and they seem to capture essential properties of even-hole-free graphs, for more about them see the survey of Vušković [10]. Interestingly, prisms, pyramids, thetas and wheels are called Truemper configurations and they play an important role in many decomposition theorems for classes of graphs, see [11] for a survey.…”

“…The class of even-hole-free graphs was the object of much research (see [10] for a survey). However, the complexity of computing a maximum independent set in an even-hole-free graph is not known.…”

The Game of Cops and Robber on (Claw, Even-hole)-free Graphs

“…A decomposition theorem for even-holefree graphs is provided in [dV13] strengthening a previously known decomposition result in [CCKV02]. For a survey on the structural results, see [Vuš10]. Chudnovsky and Seymour in [CS19] settled a conjecture by Reed and proved that every even-hole-free graph has a bisimplicial vertex.…”