Degree Equitable Regular Set Domination in Graphs

Abstract

A set D of vertices in a graph $$G =(V,E)$$ is said to be regular set dominating set if for every set $$I \subseteq V-D$$ there exists a nonempty set $$S \subseteq D$$ such that $$\langle I \cup S \rangle $$ is regular. A subset $$D \subseteq V(G)$$ is called an equitable dominating set of a graph G if every vertex $$v \in V(G) {\setminus } D$$ has a neighbor $$u \in D$$ such that $$|d_G(u)-d_G(v)| \le 1$$ . An equitable dominating set D is a DERSD-dominating set of G if D is a regular set dominating set of G. The DERSD-domination number of G, denoted by $$\gamma _{rs}^e(G)$$ , 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 $$\gamma _{rs}^e(G)$$ 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).