Edge-Vertex Dominating Set in Unit Disk Graphs

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 ev-dominating set) of $G$ if every vertex of $G$ is ev-dominated by at least one edge of $S^{ev}$. The minimum cardinality of an ev-dominating set is the ev-domination number. The edge-vertex dominating set problem is to find a minimum ev-domination number. In this paper we prove that the 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

Vertex-edge perfect Roman domination number

A vertex-edge perfect Roman dominating function on a graph $G = (V, E)$ (denoted by ve-PRDF) is a function $f:V\left(G\right)\longrightarrow\{0, 1, 2\}$ such that for every edge $uv\in E$, $\max\{f(u), f(v)\}\neq0$, or $u$ is adjacent to exactly one neighbor $w$ such that $f(w) = 2$, or $v$ is adjacent to exactly one neighbor $w$ such that $f(w) = 2$. The weight of a ve-PRDF on $G$ is the sum $w(f) = \sum_{v\in V}f(v)$. The vertex-edge perfect Roman domination number of $G$ (denoted by $\gamma_{veR}^{p}(G)$) is the minimum weight of a ve-PRDF on $G$. In this paper, we first show that vertex-edge perfect Roman dominating is NP-complete for bipartite graphs. Also, for a tree $T$, we give upper and lower bounds for $\gamma_{veR}^{p}(T)$ in terms of the order $n$, $l$ leaves and $s$ support vertices. Lastly, we determine $\gamma_{veR}^{p}(G)$ for Petersen, cycle and Flower snark graphs