A graph is Berge if no induced subgraph of it is an odd cycle of length at least five or the complement of one. In joint work with Robertson, Seymour, and Thomas we recently proved the Strong Perfect Graph Theorem, which was a conjecture about the chromatic number of Berge graphs. The proof consisted of showing that every Berge graph either belongs to one of a few basic classes, or admits one of a few kinds of decompositions. We used three kinds of decompositions: skew-partitions, 2-joins, and proper homogeneous pairs. At that time we were not sure whether all three decompositions were necessary. In this article we show that the proper homogeneous pair decomposition is in fact unnecessary. This is a consequence of a general decomposition theorem for " Berge trigraphs." A trigraph 7 is a generalization of a graph, where the adjacency of some vertex pairs is "undecided." A trigraph is Berge if however we decide the undecided pairs, the resulting graph is Berge. We show that the decomposition result of  for Berge graphs extends (with slight modifications) to Berge trigraphs; that is for a Berge trigraph T, either T belongs to one of a few basic classes or T admits one of a few decompositions. Moreover, the decompositions are such that, however, we decide the undecided pairs of T, the resulting graph admits the same decomposition. This last property is crucial for the application. The full proof of this result is over 200 pages long and was the author's PhD thesis. In this article we present the parts that differ significantly from the proof of the decomposition theorem for Berge graphs, and only in the case needed for the application.
All Science Journal Classification (ASJC) codes
- Geometry and Topology
- Berge graphs
- Perfect graphs
- Trigraphs: homogeneous pairs