A polynomial Turing-kernel for weighted independent set in bull-free graphs

The maximum stable set problem is NP-hard, even when restricted to triangle-free graphs. In particular, one cannot expect a polynomial time algorithm deciding if a bull-free graph has a stable set of size k, when k is part of the instance. Our main result in this paper is to show the existence of an FPT algorithm when we parameterize the problem by the solution size k. A polynomial kernel is unlikely to exist for this problem. We show however that our problem has a polynomial size Turingkernel. More precisely, the hard cases are instances of size O(k5). As a byproduct, if we forbid odd holes in addition to the bull, we show the existence of a polynomial time algorithm for the stable set problem. We also prove that the chromatic number of a bull-free graph is bounded by a function of its clique number and the maximum chromatic number of its triangle-free induced subgraphs. All our results rely on a decomposition theorem for bull-free graphs due to Chudnovsky which is modified here, allowing us to provide extreme decompositions, adapted to our computational purpose

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 odd-hole- and dart-free graphs as well as for odd-hole- and 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