Total Resolving Number of Power Graphs

Abstract

Let $G = (V, E)$ be a simple connected graph. An ordered subset $W$ of $V$ is said to be a resolving set of $G$ if every vertex is uniquely determined by its vector of distances to the vertices in $W.$ The minimum cardinality of a resolving set is called the \textit{resolving number} of $G$ and is denoted by $r(G).$ As an extension, the total resolving number was introduced in [5] as the minimum cardinality taken over all resolving sets in which $\left\langle W \right\rangle$ has no isolates and it is denoted by $tr(G).$ In this paper, we obtain the bounds on the total resolving number of power graphs. Also, we characterize the extremal graphs.