Frontiers of Mathematics in China

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

Distance domination of generalized de Bruijn and Kautz digraphs

Research Article

Abstract

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 VD is distance k-dominated by some vertex of D. The distance k-domination number of G, denoted by γk(G), is the minimum cardinality of a distance k-dominating set of G. Generalized de Bruijn digraphs GB(n, d) and generalized Kautz digraphs GK(n, d) are good candidates for interconnection networks. Denote Δk:= (∑j=0kdj)−1. F. Tian and J. Xu showed that ⌈nΔkγk(GB(n, d)) ≤⌈n/dk⌉ and ⌈nΔk⌉ ≤ γk(GK(n, d)) ≤ ⌈n/dk⌉. In this paper, we prove that every generalized de Bruijn digraph GB(n, d) has the distance k-domination number ⌈nΔk⌉ or ⌈nΔk⌉+1, and the distance k-domination number of every generalized Kautz digraph GK(n, d) bounded above by ⌈n/(dk−1+dk)⌉. Additionally, we present various sufficient conditions for γk(GB(n, d)) = ⌈nΔk⌉ and γk(GK(n, d)) = ⌈nΔk⌉.

Keywords

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

MSC

05C69 05C20 

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Copyright information

© 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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