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

TL;DR

This paper presents polynomial-time algorithms for solving the Maximum Weight Independent Set problem in specific classes of graphs characterized by the absence of certain induced subgraphs, expanding the understanding of its complexity.

Contribution

The paper introduces polynomial algorithms for MWIS in odd-hole- and dart-free, as well as odd-hole- and bull-free graphs, using clique separator and modular decomposition techniques.

Findings

Polynomial-time solutions for MWIS in odd-hole- and dart-free graphs

Polynomial-time solutions for MWIS in odd-hole- and bull-free graphs

Structural results improve for hole-free graphs

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 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.