Cartesian product graphs and $k$-tuple total domination

TL;DR

This paper studies the $k$-tuple total domination number in Cartesian product graphs, establishing inequalities and bounds, and provides explicit formulas for the rook's graph case, advancing understanding of domination parameters in graph products.

Contribution

It introduces a Vizing-like inequality for $k$-tuple total domination in Cartesian products and derives bounds and explicit formulas, including for the rook's graph.

Findings

Established a Vizing-like inequality for $ ext{γ}_{\times k,t}$ in Cartesian products.

Derived bounds on $ ext{γ}_{\times k,t}$ using packing numbers.

Provided a formula for $ ext{γ}_{\times 2,t}$ in the rook's graph case.

Abstract

A $k$-tuple total dominating set ($k$TDS) of a graph $G$ is a set $S$ of vertices in which every vertex in $G$ is adjacent to at least $k$ vertices in $S$; the minimum size of a $k$TDS is denoted $\gamma_{\times k,t}(G)$. We give a Vizing-like inequality for Cartesian product graphs, namely $\gamma_{\times k,t}(G) \gamma_{\times k,t}(H) \leq 2k \gamma_{\times k,t}(G \Box H)$ provided $\gamma_{\times k,t}(G) \leq 2k\rho(G)$, where $\rho$ is the packing number. We also give bounds on $\gamma_{\times k,t}(G \Box H)$ in terms of (open) packing numbers, and consider the extremal case of $\gamma_{\times k,t}(K_n \Box K_m)$, i.e., the rook's graph, giving a constructive proof of a general formula for $\gamma_{\times 2, t}(K_n \Box K_m)$.