  A set $D \subseteq V$ for the graph $G=(V, E)$ is called a dominating set if
any vertex $v\in V\setminus D$ has at least one neighbor in $D$. Fomin et
al.[9] gave an algorithm for enumerating all minimal dominating sets with $n$
vertices in $O(1.7159^n)$ time. It is known that the number of minimal
dominating sets for interval graphs and trees on $n$ vertices is at most
$3^{n/3} \approx 1.4422^n$. 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 $D$, denoted by $\mathcal{C}_D(G)$,
is the summation of the degrees of the vertices in $D$. Maximizing or
minimizing this parameter among all minimal dominating sets have 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, block graphs.
