Semitotal domination in trees

Abstract

In this paper, we study a parameter that is squeezed between arguably the two important domination parameters, namely the domination number, $\gamma (G)$, and the total domination number, ${\gamma }_t(G)$. A set $S$ of vertices in $G$ is a semitotal dominating set of $G$ if it is a dominating set of $G$ and every vertex in S is within distance $2$ of another vertex of $S$. The semitotal domination number, ${\gamma }_{t2}(G)$, is the minimum cardinality of a semitotal dominating set of $G$. We observe that $\gamma (G)\le {\gamma }_{t2}(G)\le {\gamma }_t(G)$. In this paper, we give a lower bound for the semitotal domination number of trees and we characterize the extremal trees. In addition, we characterize trees with equal domination and semitotal domination numbers.

revised

References in corpus (1)

• A characterization of trees with equal 2-domination and 2-independence numbers