Locating and Differentiating-Total Dominating Sets in Unicyclic Graphs

Abstract

Given a graph \(G = (V, E)\) with no isolated vertex, a subset \(S \subseteq V\) is a total dominating set of \(G\) if every vertex in \(V\) is adjacent to a vertex in \(S\). A total dominating set \(S\) of \(G\) is a locating-total dominating set if for every pair of distinct vertices \(u\) and \(v\) in \(V – S\), we have \(N(u) \cap S \neq N(v) \cap S\), and \(S\) is a differentiating-total dominating set if for every pair of distinct vertices \(u\) and \(v\) in \(V\), we have \(N(u) \cap S \neq N(v) \cap S\). The locating-total domination number (or the differentiating-total domination number) of \(G\), denoted by \(\gamma_t^L(G)\) (or \(\gamma_t^D(G)\)), is the minimum cardinality of a locating-total dominating set (or a differentiating-total dominating set) of \(G\). In this paper, we investigate the bounds of locating and differentiating-total domination numbers of unicyclic graphs.