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 $N(u)\cap S\ne N(v)\cap S$, and $S$ is a differentiating-total dominating set if for every pair of distinct vertices $u$ and $v$ in $V$, we have $N(u)\cap S\ne N(v)\cap S$. 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.