Distance domination of generalized de Bruijn and Kautz digraphs
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Let G = (V,A) be a digraph and k ≥ 1 an integer. For u, v ∈ V, we say that the vertex u distance k-dominate v if the distance from u to v at most k. A set D of vertices in G is a distance k-dominating set if each vertex of V D is distance k-dominated by some vertex of D. The distance k-domination number of G, denoted by γ k (G), is the minimum cardinality of a distance k-dominating set of G. Generalized de Bruijn digraphs G B (n, d) and generalized Kautz digraphs G K (n, d) are good candidates for interconnection networks. Denote Δ k := (∑ j=0 k d j )−1. F. Tian and J. Xu showed that ⌈nΔ k ⌉ γ k (G B (n, d)) ≤⌈n/d k⌉ and ⌈nΔ k ⌉ ≤ γ k (G K (n, d)) ≤ ⌈n/d k ⌉. In this paper, we prove that every generalized de Bruijn digraph G B (n, d) has the distance k-domination number ⌈nΔ k ⌉ or ⌈nΔ k ⌉+1, and the distance k-domination number of every generalized Kautz digraph G K (n, d) bounded above by ⌈n/(d k−1+d k )⌉. Additionally, we present various sufficient conditions for γ k (G B (n, d)) = ⌈nΔ k ⌉ and γ k (G K (n, d)) = ⌈nΔ k ⌉.
KeywordsCombinatorial problems dominating set distance dominating set generalized de Bruijn digraph generalized Kautz digraph
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This work was supported in part by the National Natural Science Foundation of China (Grant Nos. 11471210, 11571222, 11601262).
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