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

摘要

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 - 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 C-m square P-n of cycles C-m and paths P-n. Exact values for the locating-total domination number of the Cartesian product C-3 square P-n are found, and it is shown that for the locating-total domination number of the Cartesian product C-4 square P-n this number is between [3n/2] and [3n/2] + 1 with two sharp bounds. (C) 2015 Elsevier B.V. All rights reserved.