On the semitotal dominating sets of graphs

TL;DR

This paper investigates the properties and enumeration of semitotal dominating sets in graphs, including calculations of their minimum size and counts in specific and arbitrary graphs.

Contribution

It introduces the concept of semitotal domination number, computes it for specific graphs, and counts the number of such sets of arbitrary size in some graphs.

Findings

Computed semitotal domination number for specific graphs

Counted the number of semitotal dominating sets of arbitrary size in some graphs

Provided new insights into the structure of semitotal dominating sets

Abstract

A set $D$ of vertices in an isolate-free graph $G$ is a semitotal dominating set of $G$ if $D$ is a dominating set of $G$ and every vertex in $D$ is within distance $2$ from another vertex of $D$.The semitotal domination number of $G$ is the minimum cardinality of a semitotal dominating set of $G$ and is denoted by $\gamma_{t2}(G)$. In this paper after computation of semitotal domination number of specific graphs, we count the number of this kind of dominating sets of arbitrary size in some graphs.