Bounds on locating total domination number of the Cartesian product of cycles and paths

• We show that the locating-total domination number of the Cartesian product C3□PnC3□Pn is equal to n+1n+1.
• We show that for the locating-total domination number of the Cartesian product C4□PnC4□Pn this number is between ⌈3n2⌉ and ⌈3n2⌉+1 with two sharp bounds.

The problem of placing monitoring devices in a system in such a way that every site in the safeguard system (including the monitors themselves) is adjacent to a monitor site can be modeled by total domination in graphs. Locating-total dominating sets are of interest when the intruder/fault at a vertex precludes its detection in that location. 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−SV−S are totally dominated by distinct subsets of the total dominating set. The locating-total domination number of a graph G is the minimum cardinality of a locating-total dominating set of G  . In this paper, we study the bounds on locating-total domination numbers of the Cartesian product Cm□PnCm□Pn of cycles CmCm and paths PnPn. Exact values for the locating-total domination number of the Cartesian product C3□PnC3□Pn are found, and it is shown that for the locating-total domination number of the Cartesian product C4□PnC4□Pn this number is between ⌈3n2⌉ and ⌈3n2⌉+1 with two sharp bounds.