A Vizing-type result for semi-total domination

TL;DR

This paper establishes a lower bound for the semi-total domination number of Cartesian product graphs, extending Vizing-type results to this domination variant.

Contribution

It introduces a new inequality relating the semi-total domination numbers of product graphs to those of factor graphs.

Findings

Proves that  of the product's semi-total domination number is at least the product of the factors' numbers.

Extends Vizing-type bounds to semi-total domination in Cartesian products.

Provides a foundational inequality for future research in domination theory.

Abstract

A set of vertices $S$ in a simple isolate-free graph $G$ is a semi-total dominating set of $G$ if it is a dominating set of $G$ and every vertex of $S$ is within distance 2 or less with another vertex of $S$. The semi-total domination number of $G$, denoted by $\gamma_{t2}(G)$, is the minimum cardinality of a semi-total dominating set of $G$. In this paper, we study semi-total domination of Cartesian products of graphs. Our main result establishes that for any graphs $G$ and $H$, $\gamma_{t2}(G\,\square\, H)\ge \frac{1}{3}\gamma_{t2}(G)\gamma_{t2}(H)$.