2-Step Movability of Hop Dominating Sets in Graphs

Let $G$ be an undirected connected graph with vertex and edge sets $V(G)$ and $E(G)$, respectively. A hop dominating set $S$ in $G$ is $2$-step movable hop dominating if for each $v \in S$, $S\setminus \{v\}$ or $[S\setminus \{v\}] \cup \{w\}$ for some $w \in [V(G)\setminus S] \cap N_G^2(v)$ is a hop dominating set in $G$. The minimum cardinality of a $2$-step movable hop dominating set in $G$, denoted by $\gamma_{mh}^2(G)$, is called the $2$-step movable hop domination number of $G$. In this paper, we characterize those graphs which admit a $2$-step movable hop dominating set. We give bounds on the $2$-step movable hop domination number and give necessary and sufficient conditions for those graphs that attain these bounds. We also characterize the $2$-step movable hop dominating sets in the shadow graph and complementary prism and determine their respective $2$-step movable hop domination number.