Gerard Jennhwa Chang (1) (2) (3) (4) (5) (6), Paul Dorbec (7), Hye Kyung Kim (8), André Raspaud (9), Haichao Wang (10), Weiliang Zhao (11)
(1) Department of Mathematics [Taiwan]
(2) Taida Institute for Mathematical Sciences [Taipei]
(3) National Center for Theoretical Sciences [Taiwan]
(4) Department of Mathematics [National Taiwan University]
(5) Taida Institute for Mathematical Sciences [National Taiwan University]
(6) National Center for Theoretical Sciences [National Taiwan University]
(7) Combinatoire et Algorithmique
(8) Department of Mathematics Education [Daegu]
(9) Laboratoire Bordelais de Recherche en Informatique
(10) Department of Mathematics [Shanghai]
(11) Zhejiang Industry Polytechnic College [Shaoxing]
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.
