Strong non split r-domination number of a graph

The concept of r-domination was introduced by Meir and Moon [A. Meir and J.W. Moon: Relations between packing and covering number of a tree, Pacific J. Math. 61 (1975), 225-233]. Let G = ( V , E ) be a simple, connected and undirected graph and r be a positive integer. A subset D of V ( G ) is a r-dominating set if every vertex in V − D is within a distance r from at least one vertex of D. The r-domination number γ r ( G ) is the minimum cardinality of a r-dominating set of G. minating set D in V ( G ) is said to be a nonsplit r-dominating set if 〈 V − D 〉 is connected. The smallest cardinality of nonsplit r-dominating set is called as nonsplit r-domination number of G. A nonsplit r-dominating set is said to be a strong nonsplit r-dominating set if 〈 V − D 〉 is r-complete. The strong nonsplit r-domination number is the minimum cardinality of a strong nonsplit r-dominating set and is denoted by γ s n s r ( G ) of G. In this paper, results involving this new parameter are derived.