Disjunctive domination in trees

In this paper, we study a parameter that is a relaxation of arguably the most important domination parameter, namely the domination number. Given the sheer scale of modern networks, many existing domination type structures are expensive to implement. Variations on the theme of dominating and total dominating sets studied to date tend to focus on adding restrictions which in turn raises their implementation costs. As an alternative route a relaxation of the domination number, called disjunctive domination, was proposed and studied by Goddard et al. A set $D$ of vertices in $G$ is a disjunctive dominating set in $G$ if every vertex not in $D$ is adjacent to a vertex of $D$ or has at least two vertices in $D$ at distance $2$ from it in $G$. The disjunctive domination number, ${\gamma }_2^d(G)$, of $G$ is the minimum cardinality of a disjunctive dominating set in $G$. We show that if $T$ is a tree of order $n$ with $l$ leaves and $s$ support vertices, then $\frac{n-l+3}{4}\leq \gamma^{d}_2(T)\leq \frac{n+l+s}{4}$. Moreover, we characterize the families of trees which attain these bounds.