Upper k-tuple domination in graphs

Abstract

For a positive integer k, a k-tuple dominating set of a graph G is a subset S of V(G) such that \N[v] ∪ S\ ≥ k for every vertex v, where N[v] = {v}∩{u ∈ V(G):uv ∈ E (G)}. The upper k-tuple domination number of G, denoted by Γ × k (G), is the maximum cardinality of a minimal k-tuple dominating set of G. In this paper we present an upper bound on Γ × k(G) for r-regular graphs G with r ≥ k, and characterize extremal graphs achieving the upper bound. We also establish an upper bound on Γ × 2(G) for claw-free r-regular graphs. For the algorithmic aspect, we show that the upper k-tuple domination problem is NP-complete for bipartite graphs and for chordal graphs. © 2012 Discrete Mathematics and Theoretical Computer Science (DMTCS), Nancy, France.