Super Domination in Trees

Abstract

For $S{\subseteq } V(G)$, we define ${\overline{S}}=V(G){\setminus } S$. A set $S{\subseteq } V(G)$ is called a super dominating set if for every vertex $u\in {\overline{S}}$, there exists $v\in S$ such that $N(v)\cap {\overline{S}}=\{u\}$. The super domination number $\gamma _{sp}(G)$ of G is the minimum cardinality among all super dominating sets in G. The super domination subdivision number $sd_{\gamma _{sp}}(G)$ of a graph G is the minimum number of edges that must be subdivided in order to increase the super domination number of G. In this paper, we investigate the ratios between super domination and other domination parameters in trees. In addition, we show that for any nontrivial tree T, $1\le sd_{\gamma _{sp}}(T)\le 2$, and give constructive characterizations of trees whose super domination subdivision number are 1 and 2, respectively.