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 ) implicit in their work, 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), 2ν s (G)} ≤ diss(G) ≤ α(G) + ν s (G) ≤ 2α(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 lower and upper bounding diss(G) is NP-hard.

Relating the independence number and the dissociation number

“…Section 2. For cubic graphs though, our first main result is the following best possible improvement of (1). Note that the connected cubic graph K 4 satisfies…”

“…We proceed to the proof of the characterization of the extremal subcubic graphs for (1). For the converse, let G be a connected subcubic graph with 2α(G) = diss(G) that is distinct from K 4 .…”

“…While ( 1) is trivial, it is NP-hard [1] to recognize the extremal graphs for (1). The dissociation number is algorithmically hard even when restricted, for instance, to subcubic bipartite graphs [3,11,15].…”

“…The starting point for the present paper were possible improvements of (1). As it turns out, (sub)cubic graphs play a special role in this context, and it is not difficult to construct arbitrarily large connected r-regular extremal graphs for (1) as soon as r ≥ 4, cf.…”