Local Structure When All Maximal Independent Sets Have Equal Weight

Abstract

In many combinatorial situations there is a notion of independence of a set of points. Maximal independent sets can be easily constructed by a greedy algorithm, and it is of interest to determine, for example, if they all have the same size or the same parity. Both of these questions may be formulated by weighting the points with elements of an abelian group, and asking whether all maximal independent sets have equal weight. If a set is independent precisely when its elements are pairwise independent, a graph can be used as a model. The question then becomes whether a graph, with its vertices weighted by elements of an abelian group, is well-covered, i.e., has all maximal independent sets of vertices with equal weight. This problem is known to be co-NP-complete in general. We show that whether a graph is well-covered or not depends on its local structure. Based on this, we develop an algorithm to recognize well-covered graphs. For graphs with n vertices and maximum degree $\Delta$, it runs in linear time if $\Delta$ is bounded by a constant, and in polynomial time if $\Delta = O(\root 3 \of {\log n})$. We mention various applications to areas including hypergraph matchings and radius k independent sets. We extend our results to the problem of determining whether a graph has a weighting which makes it well-covered.