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The complexity of dissociation set problems in graphs

Y. Orlovich Alexandre Dolgui 1 Gerd Finke 2 Valery Gordon Frank Werner 3 
1 Laboratoire en Sciences et Technologies de l'Information
Division for Indl Engineering and Computer Sciences, DEMO-ENSMSE - Département Décision en Entreprise : Modélisation, Optimisation
2 G-SCOP_ROSP - Recherche Opérationnelle pour les Systèmes de Production
G-SCOP - Laboratoire des sciences pour la conception, l'optimisation et la production
Abstract : A subset of vertices in a graph is called a dissociation set if it induces a subgraph with a vertex degree of at most 1. The maximum dissociation set problem, i.e., the problem of finding a dissociation set of maximum size in a given graph is known to be NP-hard for bipartite graphs. We show that the maximum dissociation set problem is NP-hard for planar line graphs of planar bipartite graphs. In addition, we describe several polynomially solvable cases for the problem under consideration. One of them deals with the subclass of the so-called chair-free graphs. Furthermore, the related problem of finding a maximal (by inclusion) dissociation set of minimum size in a given graph is studied, and NP-hardness results for this problem, namely for weakly chordal and bipartite graphs, are derived. Finally, we provide inapproximability results for the dissociation set problems mentioned above.
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Submitted on : Friday, August 26, 2011 - 10:30:07 AM
Last modification on : Wednesday, September 22, 2021 - 3:05:14 AM

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Y. Orlovich, Alexandre Dolgui, Gerd Finke, Valery Gordon, Frank Werner. The complexity of dissociation set problems in graphs. Discrete Applied Mathematics, Elsevier, 2011, 159 (13), pp.1352-1366. ⟨10.1016/j.dam.2011.04.023⟩. ⟨hal-00617109⟩



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