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

Let be a simple graph. A set is a -dominating set of , if every vertex of has at least two neighbors in . The -domination number of a graph , is denoted by and is the minimum size of the -dominating sets of . In this paper, we count the number of -dominating sets of . To do this, we consider a polynomial which is the generating function for the number of -dominating sets of and call it -domination polynomial. We study some properties of this polynomial. Furthermore, we compute the -domination polynomial for some of the graph families.