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 γr(G) is the minimum cardinality of a r-dominating set of G.A r-dominating 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 of G. In this paper, results involving this new parameter are derived.