Bounds on the locating-total domination number of a tree

Abstract

a b s t r a c tIn this paper, we continue the study of locating-total domination in graphs, introduced by Haynes et al. [T.W. Haynes, M.A. Henning, J. Howard, Locating and total dominating sets in trees, Discrete Applied Mathematics 154 (8) (2006) 1293-1300]. A total dominating set S in a graph G = (V , E) is a locating-total dominating set of G if, for every pair of distinct vertices u andThe minimum cardinality of a locating-total dominating set is the locating-total domination number γ L t (G). We show that, for a tree T of order n ≥ 3 with l leaves and s support vertices, n+l+1Moreover, we constructively characterize the extremal trees achieving these bounds.

Locating-Domination and Identification

“…• The notions of 1-LD and 1-Id codes can be modified by adding the condition that the code, instead of being dominating, must be total dominating: a total dominating code C of a graph G = (V, E) is a locating-total dominating code if for every pair of distinct vertices u and v in V \ C, one has N (u) ∩ C = N (v) ∩ C, and C is a differentiating-total dominating code, or an identifyingtotal dominating code, if for every pair of distinct vertices u and v in V , N [u] ∩ C = N [v] ∩ C. See, e.g., [103], [49], [53], [22], [105], [146], [129].…”

Locating–paired-dominating sets in square grids

“…Locating-dominating sets in infinite grids and their density were studied in [14,15,17,19]. Locating-total dominating sets in trees are studied in [4,10,13]. Only few results are known for locating-paired-dominating sets in graphs.…”

Locating-Total Domination Number of Cacti Graphs

“…e differentiating-total and locating-total domination numbers of path graphs are also determined. e improved new bound for locating-total dominating number of trees is presented by Chen and Sohn [2]. ey also gave the sharpness of these bounds and identified the trees achieving these bounds.…”