Decomposing Berge graphs and detecting balanced skew partitions

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(n9)-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 partit...

• Type: Journal
• Year: 2008
• Where: JCT
• Authors: Nicolas Trotignon
• Keywords: Balanced Skew, Balanced Skew Partition, Berge Graphs, JCT 2008