On the Maximum Weight Independent Set Problem in graphs without induced cycles of length at least five
Abstract
A hole in a graph is an induced cycle of length at least , and an antihole is the complement of an induced cycle of length at least . A hole or antihole is long if its length is at least . For an integer , the -prism is the graph consisting of two cliques of size joined by a matching. The complexity of Maximum (Weight) Independent Set (MWIS) in long-hole-free graphs remains an important open problem. In this paper we give a polynomial time algorithm to solve MWIS in long-hole-free graphs with no -prism (for any fixed integer ), and a subexponential algorithm for MWIS in long-hole-free graphs in general. As a special case this gives a polynomial time algorithm to find a maximum weight clique in perfect graphs with no long antihole, and no hole of length . The algorithms use the framework of minimal chordal completions and potential maximal cliques.
Keywords
Cite
@article{arxiv.1903.04761,
title = {On the Maximum Weight Independent Set Problem in graphs without induced cycles of length at least five},
author = {Maria Chudnovsky and Marcin Pilipczuk and Michał Pilipczuk and Stéphan Thomassé},
journal= {arXiv preprint arXiv:1903.04761},
year = {2020}
}