On the Complexity of Hop Domination and 2-Step Domination in Graph Classes

2026-05-20

AI Summary

This study investigates the computational complexity of the Hop Domination and 2-Step Domination problems across various graph classes. By employing graph-theoretic modeling and polynomial-time reductions, it establishes for the first time that both problems are NP-complete on d-regular graphs (for d ≥ 3), claw-free graphs, and unit disk graphs. These results rigorously characterize the intrinsic intractability of Hop Domination and 2-Step Domination within several well-studied restricted graph families, thereby delineating fundamental theoretical limits for the design of approximation or exact algorithms targeting these problems.

Abstract

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.

Problem

• hop domination
• 2-step domination
• computational complexity
• NP-completeness
• graph classes

Innovation

• hop domination
• 2-step domination
• NP-completeness
• graph classes
• computational complexity