Maximum Weight Independent Sets in Odd-Hole-Free Graphs Without Dart or Without Bull

The Maximum Weight Independent Set (MWIS) Problem on graphs with vertex weights asks for a set of pairwise nonadjacent vertices of maximum total weight. Being one of the most investigated and most important problems on graphs, it is well known to be NP-complete and hard to approximate. The complexity of MWIS is open for hole-free graphs (i.e., graphs without induced subgraphs isomorphic to a chordless cycle of length at least five). By applying clique separator decomposition as well as modular decomposition, we obtain polynomial time solutions of MWIS for oddholeand dart-free graphs as well as for odd-holeand bull-free graphs (dart and bull have five vertices, say a, b, c, d, e, and dart has edges ab, ac, ad, bd, cd, de, while bull has edges ab, bc, cd, be, ce). If the graphs are hole-free instead of odd-hole-free then stronger structural results and better time bounds are obtained.