A vertex set $L\subseteq V$ is liar's vertex-edge dominating set of a graph $G=(V,E)$ if for every $e_i\in E$, $|N_G[e_i]\cap L|\geq 2$ and for every pair of distinct edges $e_i$ and $e_j$, $|(N_G[e_i]\cup N_G[e_j])\cap L|\geq 3$. In this paper, we introduce the notion of liar's vertex-edge domination which arise naturally from some application in communication network. Given a graph $G$, the \textsc{Minimum Liar's Vertex-Edge Domination Problem} (\textsc{MinLVEDP}) asks to find a minimum liar's vertex-edge dominating set of $G$ of minimum cardinality. We have studied this problem from algorithmic point of view. We show that \textsc{MinLVEDP} can be solved in linear time for trees, whereas the decision version of this problem is NP-complete for general graphs. We further study approximation algorithms for this problem. We propose an $O(\ln \Delta(G))$-approximation algorithm for \textsc{MinLVEDP} in general graphs, where $\Delta(G)$ is the maximum degree of the input graph. On the negative side, we show that the \textsc{MinLVEDP} cannot be approximated within $\frac{1}{2}(\frac{1}{8}-\epsilon)\ln|V|$ for any $\epsilon >0$, unless $NP\subseteq DTIME(|V|^{O(\log(\log|V|)})$.

翻译：设$L\subseteq V$是图$G=(V,E)$的一个顶点集，若对每条边$e_i\in E$，有$|N_G[e_i]\cap L|\geq 2$，且对任意两条不同边$e_i$和$e_j$，有$|(N_G[e_i]\cup N_G[e_j])\cap L|\geq 3$，则称$L$为图$G$的**说谎者顶点-边支配集**。本文提出源自通信网络应用的说谎者顶点-边支配概念。给定图$G$，**最小说谎者顶点-边支配问题**（\textsc{MinLVEDP}）要求寻找基数最小的说谎者顶点-边支配集。我们从算法角度研究该问题：证明\textsc{MinLVEDP}可在树上线性时间求解，而其判定版本在一般图中是NP完全的；进一步研究近似算法——提出一般图中\textsc{MinLVEDP}的$O(\ln \Delta(G))$-近似算法（$\Delta(G)$为输入图的最大度），并证明除非$NP\subseteq DTIME(|V|^{O(\log\log|V|)})$，否则\textsc{MinLVEDP}无法在任意$\epsilon>0$下做到$\frac{1}{2}(\frac{1}{8}-\epsilon)\ln|V|$近似比。