Decomposition of even-hole-free graphs with star cutsets and 2-joins

Abstract

In this paper we consider the class of simple graphs defined by excluding, as induced subgraphs, even holes (i.e. chordless cycles of even length). These graphs are known as even-hole-free graphs. We prove a decomposition theorem for even-hole-free graphs, that uses star cutsets and 2-joins. This is a significant strengthening of the only other previously known decomposition of even-hole-free graphs, by Conforti, Cornuéjols, Kapoor and Vušković, that uses 2-joins and star, double star and triple star cutsets. It is also analogous to the decomposition of Berge (i.e. perfect) graphs with skew cutsets, 2-joins and their complements, by Chudnovsky, Robertson, Seymour and Thomas. The similarity between even-hole-free graphs and Berge graphs is higher than the similarity between even-hole-free graphs and simply odd-hole-free graphs, since excluding a 4-hole, automatically excludes all antiholes of length at least 6. In a graph that does not contain a 4-hole, a skew cutset reduces to a star cutset, and a 2-join in the complement implies a star cutset, so in a way it was expected that even-hole-free graphs can be decomposed with just the star cutsets and 2-joins. A consequence of this decomposition theorem is a recognition algorithm for even-hole-free graphs that is significantly faster than the previously known ones.