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

Towards an Isomorphism Dichotomy for Hereditary Graph Classes

In this paper we resolve the complexity of the isomorphism problem on all but finitely many of the graph classes characterized by two forbidden induced subgraphs. To this end we develop new techniques applicable for the structural and algorithmic analysis of graphs. First, we develop a methodology to show isomorphism completeness of the isomorphism problem on graph classes by providing a general framework unifying various reduction techniques. Second, we generalize the concept of the modular decomposition to colored graphs, allowing for non-standard decompositions. We show that, given a suitable decomposition functor, the graph isomorphism problem reduces to checking isomorphism of colored prime graphs. Third, we extend the techniques of bounded color valence and hypergraph isomorphism on hypergraphs of bounded color size as follows. We say a colored graph has generalized color valence at most k if, after removing all vertices in color classes of size at most k, for each color class C every vertex has at most k neighbors in C or at most k non-neighbors in C. We show that isomorphism of graphs of bounded generalized color valence can be solved in polynomial time.Comment: 37 pages, 4 figure