Bounds on Locating-Total Domination Number of the Cartesian Product of Cycles

Abstract

A total dominating set $S$ of a graph $G$ with no isolated vertex is a locating-total dominating set of $G$ if for every pair of distinct vertices $u$ and $v$ in $V–S$ are totally dominated by distinct subsets of the total dominating set. The minimum cardinality of a locating-total dominating set is the locating-total domination number. In this paper, we obtain new upper bounds for locating-total domination numbers of the Cartesian product of cycles $C_m$ and $C_n$, and prove that for any positive integer $n\ge 3$, the locating-total domination numbers of the Cartesian product of cycles $C_3$ and $C_n$ is equal to $n$ for $n\equiv 0(\mathrm{mod}6)$ or $n+1$ otherwise.