Domination cover number of graphs

A set [Formula: see text] for the graph [Formula: see text] is called a dominating set if any vertex [Formula: see text] has at least one neighbor in [Formula: see text]. Fomin et al. [Combinatorial bounds via measure and conquer: Bounding minimal dominating sets and applications, ACM Transactions on Algorithms (TALG) 5(1) (2008) 9] gave an algorithm for enumerating all minimal dominating sets with [Formula: see text] vertices in [Formula: see text] time. It is known that the number of minimal dominating sets for interval graphs and trees on [Formula: see text] vertices is at most [Formula: see text]. In this paper, we introduce the domination cover number as a new criterion for evaluating the dominating sets in graphs. The domination cover number of a dominating set [Formula: see text], denoted by [Formula: see text], is the summation of the degrees of the vertices in [Formula: see text]. Maximizing or minimizing this parameter among all minimal dominating sets has interesting applications in many real-world problems, such as the art gallery problem. Moreover, we investigate this concept for different graph classes and propose some algorithms for finding the domination cover number in trees and block graphs.