Enumeration of maximal irredundant sets for claw-free graphs

Abstract

Domination is one of the classical subjects in structural graph theory and in graph algorithms. The Minimum Dominating Set problem and many of its variants are NP-complete and have been studied from various algorithmic perspectives. One of those variants called irredundance is highly related to domination. For example, every minimal dominating set of a graph G is also a maximal irredundant set of G. In this paper we study the enumeration of the maximal irredundant sets of a claw-free graph. We show that an n-vertex claw-free graph has O(1.9341n) maximal irredundant sets and these sets can be enumerated in the same time. We complement the aforementioned upper bound with a lower bound by providing a family of graphs having 1.5848n maximal irredundant sets.