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 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 ⌈nΔk⌉ γk(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⌉.
Combinatorial problems dominating set distance dominating set generalized de Bruijn digraph generalized Kautz digraph
This is a preview of subscription content, log in to check access
Bermond J C, Peyrat C. De Bruijn and Kautz networks: a competitor for the hypercube? In: André F, Verjus J P, eds. Hypercube and Distributed Computers. North-Holland: Elsevier Science Publishers, 1989, 279–293Google Scholar
Ghoshal J, Laskar R, Pillone D. Topics on domination in directed graphs, In: Haynes T W, Hedetniemi S T, Slater P J, eds. Domination in Graphs: Advanced Topics. New York: Marcel Dekker, 1998, 401–437Google Scholar
Haynes T W, Hedetniemi S T, Slater P J. Fundamentals of Domination in Graphs. New York: Marcel Dekker, 1998MATHGoogle Scholar
Hosseinabady M, Kakoee M R, Mathew J, Pradhan D K. Low latency and energy efficient scalable architecture for massive NoCs using generalized de Bruijn graph. IEEE Trans on Very Large Scale Integration Systems, 2011, 19: 1469–1480CrossRefGoogle Scholar