DOMINATING COVER OF GRAPHS

Let $G=(V,E)$ be a simple graph. Then a set $S \subseteq V(G)$ is called a dominating set if every vertex $v \in V(G) \backslash S$ is adjacent to at least one vertex $u \in S$. The domination number of $G$, denoted by $\gamma (G)$, is the minimum cardinality among all dominating sets of $G$. A dominating set $S$ is said to be minimal if no proper subset of $S$ is also a dominating set of $G$. A family $\mathscr{T}$ of minimal dominating sets in $G$ is called a dominating cover of $G$ if for every vertex $u \in V(G)$, there exists a set $S \in \mathscr{T}$ such that $u \in S$. The dominating covering number of $G,$ denoted by $d_c(G)$, is defined as $$d_c(G)= \text {min} \{|\mathscr{T}|: \mathscr{T}\, \text{is a dominating cover of} \,G\}.$$ In this paper, the authors determine the dominating covering number for several families of graphs, including the empty graph $\overline {K}_n$, the complete graph $K_n$, bipartite graphs, cycle graphs $C_n$, as well as the join and corona of two graphs $G$ and $H$. Furthermore, bounds for this parameter are established in terms of the domination number, vertex covering number, upper domination number, and independence number of a graph. Finally, the paper characterizes graphs satisfying $d_c(G)=1$ and those with $d_c(G)=n$, where $n$ is the order of a graph $G$.