Gerard Jennhwa Chang ; Paul Dorbec ; Hye Kyung Kim ; André Raspaud ; Haichao Wang et al. - Upper k-tuple domination in graphs

dmtcs:593 - Discrete Mathematics & Theoretical Computer Science, December 12, 2012, Vol. 14 no. 2 - https://doi.org/10.46298/dmtcs.593
Upper k-tuple domination in graphsArticle

Authors: Gerard Jennhwa Chang 1,2,3; Paul Dorbec ORCID4; Hye Kyung Kim 5; André Raspaud 6; Haichao Wang 7; Weiliang Zhao 8

  • 1 Department of Mathematics [Taiwan]
  • 2 Taida Institute for Mathematical Sciences [Taipei]
  • 3 National Center for Theoretical Sciences [Taiwan]
  • 4 Combinatoire et Algorithmique
  • 5 Department of Mathematics Education [Daegu]
  • 6 Laboratoire Bordelais de Recherche en Informatique
  • 7 Department of Mathematics [Shanghai]
  • 8 Zhejiang Industry Polytechnic College [Shaoxing]

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.


Volume: Vol. 14 no. 2
Section: Graph Theory
Published on: December 12, 2012
Accepted on: June 9, 2015
Submitted on: February 22, 2012
Keywords: [INFO.INFO-DM] Computer Science [cs]/Discrete Mathematics [cs.DM]

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