Upper k-tuple domination in graphs
2012 ◽
Vol Vol. 14 no. 2
(Graph Theory)
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Keyword(s):
Graph Theory International audience 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.
2013 ◽
Vol 05
(04)
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pp. 1350024
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Keyword(s):
2021 ◽
Vol vol. 23 no. 1
(Discrete Algorithms)
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Keyword(s):
2019 ◽
Vol 11
(06)
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pp. 1950063
Keyword(s):
2010 ◽
Vol 4
(2)
◽
pp. 241-252
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2015 ◽
Vol 23
(2)
◽
pp. 187-199