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 complexit...

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Bibliographic Details
Published inarXiv.org
Main Authors Brandstädt, Andreas, Mosca, Raffaele
Format Paper
LanguageEnglish
Published Ithaca Cornell University Library, arXiv.org 12.09.2012
Subjects
Apexes
Decomposition
Graph theory
Graphs
Polynomials
Separators
Weight
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Summary: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.
ISSN:2331-8422