Decomposing Berge graphs and detecting balanced skew partitions

TL;DR

This paper proves that detecting balanced skew partitions in general graphs is NP-hard, but provides an $O(n^9)$-time algorithm for Berge graphs using a new decomposition theorem that refines previous results.

Contribution

It introduces a more precise decomposition theorem for Berge graphs and an efficient algorithm for detecting balanced skew partitions within this class.

Findings

Deciding balanced skew partitions is NP-hard in general graphs.

An $O(n^9)$-time algorithm for Berge graphs is developed.

Every Berge graph can be decomposed using only balanced skew partitions and 2-joins.

Abstract

A hole in a graph is an induced cycle on at least four vertices. A graph is Berge if it has no odd hole and if its complement has no odd hole. In 2002, Chudnovsky, Robertson, Seymour and Thomas proved a decomposition theorem for Berge graphs saying that every Berge graph either is in a well understood basic class, or has some kind of decomposition. Then, Chudnovsky proved stronger theorems. One of them restricts the allowed decompositions to 2-joins and balanced skew partitions. We prove that the problem of deciding whether a graph has a balanced skew partition is NP-hard. We give an $O(n^9)$-time algorithm for the same problem restricted to Berge graphs. Our algorithm is not constructive: it only certifies whether a graph has a balanced skew partition or not. It relies on a new decomposition theorem for Berge graphs that is more precise than the previously known theorems. Our theorem also implies that every Berge graph can be decomposed in a first step by using only balanced skew partitions, and in a second step by using only 2-joins. Our proof of this new theorem uses at an essential step one of the theorems of Chudnovsky.