Given an undirected graph $G=(V,E)$, a vertex $v\in V$ is edge-vertex (ev) dominated by an edge $e\in E$ if $v$ is either incident to $e$ or incident to an adjacent edge of $e$. A set $S^{ev}\subseteq E$ is an edge-vertex dominating set (referred to as \textit{ev}-dominating set and in short as \textit{EVDS}) of $G$ if every vertex of $G$ is \textit{ev}-dominated by at least one edge of $S^{ev}$. The minimum cardinality of an \textit{ev}-dominating set is the \textit{ev}-domination number. The edge-vertex dominating set problem is to find a minimum \textit{ev}-domination number. In this paper, we prove that the \textit{ev}-dominating set problem is {\tt NP-hard} on unit disk graphs. We also prove that this problem admits a polynomial-time approximation scheme on unit disk graphs. Finally, we give a simple 5-factor linear-time approximation algorithm.

翻译：给定一个无向图 $G=(V,E)$，若顶点 $v\in V$ 与边 $e\in E$ 关联，或与 $e$ 的邻接边关联，则称 $v$ 被边 $e$ 边-顶点支配（edge-vertex dominated，简称 ev 支配）。集合 $S^{ev}\subseteq E$ 是图 $G$ 的一个边-顶点支配集（称为 \textit{ev}-支配集，简记为 \textit{EVDS}），当且仅当 $G$ 中每个顶点至少被 $S^{ev}$ 中的一条边 \textit{ev}-支配。最小 \textit{ev}-支配集的基数称为 \textit{ev}-支配数。边-顶点支配集问题旨在寻找最小的 \textit{ev}-支配数。本文证明了在单位圆盘图上，\textit{ev}-支配集问题是 {\tt NP-困难}的。同时，我们证明了该问题在单位圆盘图上存在多项式时间近似方案。最后，我们给出一个简单的5因子线性时间近似算法。