Bounds on the Locating-Domination Number and Differentiating-Total Domination Number in Trees

• Main Authors: Rad Nader Jafari, Rahbani Hadi
• Format: Article
• Language: English
• Published: Sciendo 2018-05-01
• Series: Discussiones Mathematicae Graph Theory
• Subjects: locating-dominating set; differentiating-total dominating set; tree; 05c69
• Online Access: https://doi.org/10.7151/dmgt.2012

A subset S of vertices in a graph G = (V,E) is a dominating set of G if every vertex in V − S has a neighbor in S, and is a total dominating set if every vertex in V has a neighbor in S. A dominating set S is a locating-dominating set of G if every two vertices x, y ∈ V − S satisfy N(x) ∩ S ≠ N(y) ∩ S. The locating-domination number γL(G) is the minimum cardinality of a locating-dominating set of G. A total dominating set S is called a differentiating-total dominating set if for every pair of distinct vertices u and v of G, N[u] ∩ S ≠ N[v] ∩ S. The minimum cardinality of a differentiating-total dominating set of G is the differentiating-total domination number of G, denoted by γtD(G)$\gamma _t^D (G)$. We obtain new upper bounds for the locating-domination number, and the differentiating-total domination number in trees. Moreover, we characterize all trees achieving equality for the new bounds.