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 \(n \geq 3\), the locating-total domination numbers of the Cartesian product of cycles \(C_3\) and \(C_n\) is equal to \(n\) for \(n \equiv 0 \pmod{6}\) or \(n + 1\) otherwise.