Liar's vertex-edge domination in unit disk graph

Abstract

Let $G=(V,E)$ be a simple undirected graph. A closed neighbourhood of an edge $e=uv$ between two vertices $u$ and $v$ of $G$, denoted by $N_G[e]$, is the set of vertices in the neighbourhood of $u$ and $v$ including $\{u,v\}$. A subset $L$ of $V$ is said to be liar's vertex-edge dominating set if $(i)$ for every edge $e\in E$, $|N_G[e]\cap L|\ge 2$ and $(ii)$ for every pair of distinct edges $e,e^{\prime }$, $|(N_G[e]\cup N_G[e^{\prime }])\cap L|\ge 3$. The minimum liar's vertex-edge domination problem is to find the liar's vertex-edge dominating set of minimum cardinality. In this article, we show that the liar's vertex-edge domination problem is NP-complete in unit disk graphs, and we design a polynomial time approximation scheme(PTAS) for the minimum liar's vertex-edge domination problem in unit disk graphs.