Distance domination in graphs with given minimum and maximum degree

Michael A. Henning () and Nicolas Lichiardopol ()
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Michael A. Henning: University of Johannesburg
Nicolas Lichiardopol: Lycée A. de Craponne

Journal of Combinatorial Optimization, 2017, vol. 34, issue 2, No 17, 545-553

Abstract: Abstract For an integer $$k \ge 1$$ k ≥ 1 , a distance k-dominating set of a connected graph G is a set S of vertices of G such that every vertex of V(G) is at distance at most k from some vertex of S. The distance k-domination number $$\gamma _k(G)$$ γ k ( G ) of G is the minimum cardinality of a distance k-dominating set of G. In this paper, we establish an upper bound on the distance k-domination number of a graph in terms of its order, minimum degree and maximum degree. We prove that for $$k \ge 2$$ k ≥ 2 , if G is a connected graph with minimum degree $$\delta \ge 2$$ δ ≥ 2 and maximum degree $$\Delta $$ Δ and of order $$n \ge \Delta + k - 1$$ n ≥ Δ + k - 1 , then $$\gamma _k(G) \le \frac{n + \delta - \Delta }{\delta + k - 1}$$ γ k ( G ) ≤ n + δ - Δ δ + k - 1 . This result improves existing known results.

Keywords: Distance domination; Minimum degree; Maximum degree; 05C69 (search for similar items in EconPapers)
Date: 2017
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Citations: View citations in EconPapers (1)

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DOI: 10.1007/s10878-016-0091-z

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