The complexity of dissociation set problems in graphs

• Published in: Discrete Applied Mathematics 2011-08, Vol.159 (13), p.1352-1366
• Main Authors: Orlovich, Yury, Dolgui, Alexandre, Finke, Gerd, Gordon, Valery, Werner, Frank
• Format: Article
• Language: English
• Subjects: Algorithmics. Computability. Computer arithmetics; Applied sciences; Approximability; Combinatorics; Combinatorics. Ordered structures; Complexity; Computational complexity; Computer Science; Computer science; control theory; systems; Dissociation set; Exact sciences and technology; Forbidden induced subgraph; Graphs; Inclusions; Information retrieval. Graph; Mathematical analysis; Mathematics; Operations Research; Sciences and techniques of general use; Theoretical computing
• ISSN: 0166-218X; 1872-6771
• DOI: 10.1016/j.dam.2011.04.023

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.