Frontiers of Mathematics in China

, Volume 12, Issue 2, pp 339–357 | Cite as

Distance domination of generalized de Bruijn and Kautz digraphs

  • Yanxia Dong
  • Erfang Shan
  • Xiao Min
Research Article


Let G = (V,A) be a digraph and k ≥ 1 an integer. For u, vV, 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 ⌉.


Combinatorial problems dominating set distance dominating set generalized de Bruijn digraph generalized Kautz digraph 


05C69 05C20 


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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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© Higher Education Press and Springer-Verlag Berlin Heidelberg 2017

Authors and Affiliations

  1. 1.Department of MathematicsShanghai UniversityShanghaiChina
  2. 2.School of ManagementShanghai UniversityShanghaiChina
  3. 3.School of Statistics and InformationShanghai University of International Business and EconomicsShanghaiChina
  4. 4.College of Mathematics, Physics and Information EngineeringJiaxing UniversityJiaxingChina

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