The history of degenerate (bipartite) extremal graph problems

This paper is a survey on Extremal Graph Theory, primarily focusing on the case when one of the excluded graphs is bipartite. On one hand we give an introduction to this field and also describe many important results, methods, problems, and constructions.Comment: 97 pages, 11 figures, many problems. This is the preliminary version of our survey presented in Erdos 100. In this version 2 only a citation was complete

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