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

Abstract

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 oddhole-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.

Induced subgraphs of bounded treewidth and the container method

“…Following the discussion in the previous paragraph, no NP-hardness result is known for MWIS in long-hole-free graphs, that is, graphs with no long holes. The question of the existence of an efficient algorithm for MWIS in this graph class remained a long-standing open problem with a number of tractability results in subclasses [3,4,9,10,12,13]. Here we answer this question in the affirmative.…”

Perfect divisibility and 2‐divisibility

“…The other referee alerted us to a serious typographical error. We would like to thank T. Karthick for pointing out to us.…”

“…One of the referees and T. Karthick (private communication) pointed out that was already proved by Brandstadt and Mosca . Actually the result also follows from Reed and Sbihi's Wheel Lemma stating that if a bull‐free graph contains a wheel, then it has a homogeneous set.…”

Independent Sets in Classes Related to Chair-Free Graphs

“…If G is odd-hole-free, then G is (odd-hole, diamond)-free. Since MWIS in (oddhole, diamond)-free graphs can be solved in polynomial time [8], MWIS can be solved in polynomial time for G. Suppose that G is prime and contains an odd-hole. Then by Theorem 3, G is claw-free.…”