Average distances and distance domination numbers

Authors:
Fang Tian;Jun-Ming Xu
Affiliations:
Department of Applied Mathematics, Shanghai University of Finance and Economics, Shanghai, 200433, China;Department of Mathematics, University of Science and Technology of China, Hefei, 230026, China
Venue:
Discrete Applied Mathematics
Year:
2009

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Abstract

Let k be a positive integer and G be a simple connected graph with order n. The average distance @m(G) of G is defined to be the average value of distances over all pairs of vertices of G. A subset D of vertices in G is said to be a k-dominating set of G if every vertex of V(G)-D is within distance k from some vertex of D. The minimum cardinality among all k-dominating sets of G is called the k-domination number @c"k(G) of G. In this paper tight upper bounds are established for @m(G), as functions of n, k and @c"k(G), which generalizes the earlier results of Dankelmann [P. Dankelmann, Average distance and domination number, Discrete Appl. Math. 80 (1997) 21-35] for k=1.