On the Convexity of Independent Set Games

TL;DR

This paper characterizes when independent set games on graphs are convex, providing a polynomial-time recognition algorithm and introducing a new class of such games with an efficient convexity criterion.

Contribution

It offers a necessary and sufficient graph-theoretic characterization for convexity in independent set games, including a new class with an efficient recognition method.

Findings

Convex independent set games are characterized by every non-pendant edge being incident to a pendant edge.

A polynomial-time algorithm is provided for recognizing convex instances.

A new class of independent set games with an efficient convexity characterization is introduced.

Abstract

Independent set games are cooperative games defined on graphs, where players are edges and the value of a coalition is the maximum cardinality of independent sets in the subgraph defined by the coalition. In this paper, we investigate the convexity of independent set games, as convex games possess many nice properties both economically and computationally. For independent set games introduced by Deng et al. (Math. Oper. Res., 24:751-766, 1999), we provide a necessary and sufficient characterization for the convexity, i.e., every non-pendant edge is incident to a pendant edge in the underlying graph. Our characterization immediately yields a polynomial time algorithm for recognizing convex instances of independent set games. Besides, we introduce a new class of independent set games and provide an efficient characterization for the convexity.