On k-tuple and k-tuple total domination numbers of regular graphs [article]

Sharareh Alipour, Amir Jafari, Morteza Saghafian
<span title="2018-01-20">2018</span> <i > arXiv </i> &nbsp; <span class="release-stage" >pre-print</span>
Let G be a connected graph of order n, whose minimum vertex degree is at least k. A subset S of vertices in G is a k-tuple total dominating set if every vertex of G is adjacent to at least k vertices in S. The minimum cardinality of a k-tuple total dominating set of G is the k-tuple total domination number of G, denoted by γ_× k,t(G). Henning and Yeo in hen proved that if G is a cubic graph different from the Heawood graph, γ_× 2, t(G) ≤5/6n, and this bound is sharp. Similarly, a k-tuple
more &raquo; ... ing set is a subset S of vertices of G, V (G) such that |N[v] ∩ S| ≥ k for every vertex v, where N[v] = {v}∪{u ∈ V(G) : uv ∈ E(G)}. The k-tuple domination number of G, denoted by γ_× k(G), is the minimum cardinality of a k-tuple dominating set of G. In this paper, we give a simple approach to compute an upper bound for (r-1)-tuple total domination number of r-regular graphs. Also, we give an upper bound for the r-tuple dominating number of r-regular graphs. In addition, our method gives algorithms to compute dominating sets with the given bounds, while the previous methods are existential.
<span class="external-identifiers"> <a target="_blank" rel="external noopener" href="https://arxiv.org/abs/1709.01245v2">arXiv:1709.01245v2</a> <a target="_blank" rel="external noopener" href="https://fatcat.wiki/release/k645vqca2bb2pg6tyoqmh3ttfe">fatcat:k645vqca2bb2pg6tyoqmh3ttfe</a> </span>
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