TOTAL RESOLVING HOP DOMINATION IN GRAPHS

Abstract

A set $S \subseteq V(G)$ is called a total resolving hop dominating set if it is both a resolving set and a hop dominating set, and every vertex of $S$ has a hop neighbor in $S$. The minimum cardinality of such a set is called the total resolving hop domination number, denoted by $\gamma_{t r h}(G)$. In this paper, we introduce this new graph parameter, establish its existence criteria and fundamental bounds, characterize total resolving hop dominating sets in graphs with universal vertices, and determine exact values of $\gamma_{\text {trh }}\left(P_n\right)$ and $\gamma_{\text {trh }}\left(C_n\right)$ for paths and cycles.