The domination problem is a well-studied problem in graph theory. In this paper, we study two natural variants: the hop domination problem and the $2$-step domination problem. Let $G$ be a graph with vertex set $V$ and edge set $E$. For a graph $G$, a subset $S \subseteq V(G)$ is called an \emph{hop dominating set} if every vertex not in $S$ lies at distance of exactly $2$ from at least one vertex in $S$. For $v\in V(G)$, let $N(v,2)$ denote the set of vertices in $V(G)$ that are at distance exactly $2$ from $v$. For a graph $G$, a subset $S \subseteq V(G)$ is called an \emph{$2$-step dominating set} if every vertex $v\in V(G)$ lies at a distance of exactly $2$ from at least one vertex in $S$. The \textsc{Hop Domination} (HD) problem and the \textsc{$2$-Step Domination} ($2$SD) problems ask whether a graph contains a hop domination set or a $2$-step domination set of size at most $k$, respectively. We study the computational complexity of these problems, and show that both are NP-complete, even when restricted to $d$-regular graphs for every $d\geq 3$, claw-free graphs and also unit disk graphs.

翻译：支配问题是图论中一个被广泛研究的问题。本文研究其两个自然变体：跳跃支配问题和二步支配问题。设$G$是一个具有顶点集$V$和边集$E$的图。对于图$G$，若每个不在$S$中的顶点与$S$中至少一个顶点的距离都恰好为$2$，则子集$S \subseteq V(G)$称为一个\emph{跳跃支配集}。对于$v\in V(G)$，令$N(v,2)$表示$V(G)$中与$v$距离恰好为$2$的顶点集。对于图$G$，若每个顶点$v\in V(G)$与$S$中至少一个顶点的距离都恰好为$2$，则子集$S \subseteq V(G)$称为一个\emph{二步支配集}。\textsc{Hop Domination} (HD)问题和\textsc{$2$-Step Domination} ($2$SD)问题分别询问图是否包含大小至多为$k$的跳跃支配集或二步支配集。我们研究了这些问题的计算复杂性，并证明即使在限制为每个$d\geq 3$的$d$-正则图、无爪图以及单位圆盘图时，这两个问题都是NP完全的。