The Number of 2-dominating Sets, and 2-domination Polynomial of a Graph

Abstract

Abstract: Let $$G=(V,E)$$ be a simple graph. A set $$D\subseteq V$$ is a $$2$$-dominating set of $$G$$, if every vertex of $$V\setminus D$$ has at least two neighbors in $$D$$. The $$2$$-domination number of a graph $$G$$, is denoted by $$\gamma_{2}(G)$$ and is the minimum size of the $$2$$-dominating sets of $$G$$. In this paper, we count the number of $$2$$-dominating sets of $$G$$. To do this, we consider a polynomial which is the generating function for the number of $$2$$-dominating sets of $$G$$ and call it $$2$$-domination polynomial. We study some properties of this polynomial. Furthermore, we compute the $$2$$-domination polynomial for some of the graph families.