Relating dissociation, independence, and matchings

Abstract

A dissociation set in a graph is a set of vertices inducing a subgraph of maximum degree at most 1. Computing the dissociation number diss ( G ) of a given graph G , defined as the order of a maximum dissociation set in G , is algorithmically hard even when G is restricted to be bipartite. Recently, Hosseinian and Butenko proposed a simple 4 3 -approximation algorithm for the dissociation number problem in bipartite graphs. Their result relies on the inequality diss ( G ) ≤ 4 3 α ( G − M ) , where G is a bipartite graph, M is a maximum matching in G , and α ( G − M ) denotes the independence number of G − M . We show that the pairs ( G , M ) for which this inequality holds with equality can be recognized efficiently, and that a maximum dissociation set can be determined for them efficiently. The dissociation number of a graph G satisfies max { α ( G ) , 2 ν s ( G ) } ≤ diss ( G ) ≤ α ( G ) + ν s ( G ) ≤ 2 α ( G ) , where ν s ( G ) denotes the induced matching number of G . We show that deciding whether diss ( G ) equals any of the four terms α ( G ) , 2 ν s ( G ) , α ( G ) + ν s ( G ) , and 2 α ( G ) is NP-hard.