On total domination in the Cartesian product of graphs

TL;DR

This paper investigates the total domination number in Cartesian product graphs, extending known bounds, characterizing specific cases where bounds are tight, and constructing graph families that approach these bounds asymptotically.

Contribution

It characterizes pairs of graphs where the total domination number of their Cartesian product equals half the product of their individual total domination numbers, and constructs graphs approaching this bound.

Findings

Characterization of graph pairs with tight total domination bounds

Construction of graph families approaching the bound asymptotically

Extension of previous results on total domination in Cartesian products

Abstract

Ho proved in [A note on the total domination number, Util.Math. 77 (2008) 97--100] that the total domination number of the Cartesian product of any two graphs with no isolated vertices is at least one half of the product of their total domination numbers. We extend a result of Lu and Hou from [Total domination in the Cartesian product of a graph and $K_2$ or $C_n$, Util. Math. 83 (2010) 313--322] by characterizing the pairs of graphs $G$ and $H$ for which $\gamma_t(G\Box H)=\frac{1}{2}\gamma_t(G) \gamma_t(H)\,$, whenever $\gamma_t(H)=2$. In addition, we present an infinite family of graphs $G_n$ with $\gamma_t(G_n)=2n$, which asymptotically approximate the equality in $\gamma_t(G_n\Box G_n)\ge \frac{1}{2}\gamma_t(G_n)^2$.