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Communication Dans Un Congrès Année : 2007

Decomposing Berge graphs and detecting balanced skew partitions

Nicolas Trotignon

Résumé

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.
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Dates et versions

hal-00265245 , version 1 (18-03-2008)

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  • HAL Id : hal-00265245 , version 1

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Nicolas Trotignon. Decomposing Berge graphs and detecting balanced skew partitions. Graph Theory, organized by: Reinhard Diestel, Alexander Schrijver and Paul Seymour, Mar 2007, Oberwolfach, Germany. ⟨hal-00265245⟩
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