On 1-movable Strong Resolving Hop Domination in Graphs

Abstract

A set $S$ is a $1$-\emph{movable strong resolving hop dominating set} of $G$ if for every $v\in S$, either $S\setminus \{v\}$ is a strong resolving hop dominating set or there exists a vertex $u\in (V(G)\setminus S)\cap N_G(v)$ such that $(S\setminus \{v\})\cap \{u\}$ is a strong resolving hop dominating set of $G$. The minimum cardinality of a $1$-movable strong resolving hop dominating set of $G$ is denoted by ${\gamma }_{msRh}^1(G)$. In this paper, we obtained the corresponding parameter in graphs resulting from the join, corona and lexicographic product of two graphs. Specifically, we characterize the 1-movable strong resolving hop dominating sets in these types of graphs and determine the bounds or exact values of their 1-movable strong resolving hop domination numbers.