Semitotal domination in trees

TL;DR

This paper introduces and studies the semitotal domination number in trees, providing bounds, characterizations of extremal trees, and conditions for equality with the domination number.

Contribution

It establishes a lower bound for the semitotal domination number in trees and characterizes trees where this number equals the domination number.

Findings

Lower bound for semitotal domination number in trees

Characterization of extremal trees for semitotal domination

Trees with equal domination and semitotal domination numbers

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 $\gamma(G)\leq \gamma_{t2}(G)\leq \gamma_t(G)$. 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.