On the Complexity of Hop Domination and 2-Step Domination in Graph Classes
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arXiv
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| Natura: | Preprint |
| Pubblicazione: |
2026
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| author | Das, Sandip Das, Sweta Islam, Sk Samim |
| author_facet | Das, Sandip Das, Sweta Islam, Sk Samim |
| contents | 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2605_20970 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | On the Complexity of Hop Domination and 2-Step Domination in Graph Classes Das, Sandip Das, Sweta Islam, Sk Samim Discrete Mathematics Combinatorics 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. |
| title | On the Complexity of Hop Domination and 2-Step Domination in Graph Classes |
| topic | Discrete Mathematics Combinatorics |
| url | https://arxiv.org/abs/2605.20970 |