Super domination in trees

TL;DR

This paper explores the properties of super dominating sets in trees, analyzing their relationships with other domination parameters, and characterizes trees based on how edge subdivisions affect their super domination number.

Contribution

It introduces the concept of super domination subdivision number in trees and provides characterizations for trees with subdivision numbers 1 and 2.

Findings

Super domination number in trees is studied in relation to other parameters.

For any nontrivial tree, the subdivision number is either 1 or 2.

Characterizations of trees with subdivision numbers 1 and 2 are provided.

Abstract

For $S\subseteq V(G)$, we define $\bar{S}=V(G)\setminus S$. A set $S\subseteq V(G)$ is called a super dominating set if for every vertex $u\in \bar{S}$, there exists $v\in S$ such that $N(v)\cap \bar{S}=\{u\}$. 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 $T$, $1\leq sd_{\gamma_{sp}}(T)\leq 2$, and give constructive characterizations of trees whose super domination subdivision number are $1$ and $2$, respectively.