Perfect Domination Number of the Degree Splitting Graph of Some Graphs.

Abstract

Let G be a simple connected graph. A set S\subseteq V(G) is called a perfect dominating set if every vertex v\in V(G)\backslash S is adjacent to exactly one vertex in S. The perfect domination number of G, denoted by \gamma_p(G), is the minimum cardinality of such a set. In this paper, we investigate the perfect domination number of the degree splitting graph DS(G) of the corona, join and cartesian product of regular graphs G and H. In addition, we determine the perfect domination number of the degree splitting graph of a complete k-partite graph K_{n_1,n_2,\ldots,n_k}.