A Vizing-type result for semi-total domination

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 $H$, ${\gamma }_{t2}(G\square H)\ge \frac{1}{3}{\gamma }_{t2}(G){\gamma }_{t2}(H)$.

9 pages