### Upper k-tuple domination in graphs

*Gerard Jennhwa Chang, Paul Dorbec, Hye Kyung Kim, André Raspaud, Haichao Wang, Weiliang Zhao*

#### 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.Full Text: PDF PostScript