Super domination in trees (Q2071243)

For a graph \(G\), a set \(S\subseteq V(G)\) is called a super dominating set of \(G\) if for every vertex \(u\in \overline{S}\), there exists a vertex \(v\in S\) 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, the author investigates the ratios between super domination and other domination parameters, such as domination number, total domination number, in trees. In addition, it is shown that \(1\leq sd_{\gamma_{sp}}(G)\leq 2\) for any nontrivial tree \(T\). Moreover, a constructive characterizations of trees whose super domination subdivision number are 1 and 2 are given.