Degree Equitable Regular Set Domination in Graphs

A set D of vertices in a graph G - oV; Ethorn is said to be regular set dominating set if for every set I V D there exists a nonempty set S D such that hI [Si is regular. A subset D VoGthorn is called an equitable dominating set of a graph G if every vertex v 2 VoGthornnD has a neighbor u 2 D such that jdGouthorn dGovthornj 1. An equitable dominating set D is a DERSD- dominating set of G if D is a regular set dominating set of G. The DERSDdomination number of G, denoted by ce rs oGthorn, is the minimum cardinality of a DERSD- dominating set of G. We initiate the study of DERSD- domination in graphs and obtain some sharp bounds. Finally, we show that the decision problem for determining ce rs oGthorn is NP- complete which gives a solution to the open problem posed by Sampathkumar and Pushpa Latha in their article Set domination in graphs, J Graph Theory 18(5): 489- 495, (1994).