ISOLATE SEMITOTAL DOMINATION IN GRAPHS

Abstract

Let $G=(V, E)$ be a nontrivial connected graph. A set $S$ of vertices of $G$ is a semitotal dominating set if every vertex outside of $S$ is adjacent to a vertex inside of $S$ and every vertex inside of $S$ is of distance at most 2 units from another vertex in $S$. A semitotal dominating set $S$ of $G$ is an isolate semitotal dominating set if the induced subgraph $\langle S\rangle$ contains at least one isolated vertex. The smallest cardinality $\gamma_{0 t 2}(G)$ of an isolate semitotal dominating set is the isolate semitotal domination number of $G$.

This paper initiates the study of isolate semitotal domination in graphs. It determines the specific values of $\gamma_{0 t 2}(G)$ for some special graphs and characterizes graphs $G$ with small values of $\gamma_{0 t 2}(G)$. Furthermore, this paper investigates the isolate semitotal dominating sets in the join and corona of graphs and, as a consequence, determines their corresponding isolate semitotal domination numbers.

Received: May 30, 2022
Revised: June 21, 2022
Accepted: June 30, 2022