Let $G=(V,E)$ be a graph. For an edge $e=xy\in E$, the closed neighbourhood of $e$, denoted by $N_G[e]$ or $N_G[xy]$, is the set $N_G[x]\cup N_G[y]$. 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 $e_j$, $|(N_G[e_i]\cup N_G[e_j])\cap L|\geq 3$. This paper introduces the notion of liar's vertex-edge domination which arises naturally from some applications in communication networks. Given a graph $G$, the \textsc{Minimum Liar's Vertex-Edge Domination Problem} (\textsc{MinLVEDP}) asks to find a liar's vertex-edge dominating set of $G$ of minimum cardinality. In this paper, we study 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 chordal graphs, bipartite graphs, and $p$-claw free graphs for $p\geq 4$. We further study approximation algorithms for this problem. We propose two approximation algorithms for \textsc{MinLVEDP} in general graphs and $p$-claw free graphs. %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. Also, we design a constant factor approximation algorithm for $p$-claw free graphs. 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 $NP\subseteq DTIME(|V|^{O(\log(\log|V|)})$. Finally, we prove that the \textsc{MinLVEDP} is APX-complete for bounded degree graphs and $p$-claw free graphs for $p\geq 6$.

翻译：令 $G=(V,E)$ 为一图。对于边 $e=xy\in E$，其闭邻域记为 $N_G[e]$ 或 $N_G[xy]$，定义为集合 $N_G[x]\cup N_G[y]$。若对于每条边 $e_i\in E$，有 $|N_G[e_i]\cap L|\geq 2$，且对于任意两条不同边 $e_i$ 与 $e_j$，有 $|(N_G[e_i]\cup N_G[e_j])\cap L|\geq 3$，则顶点集 $L\subseteq V$ 称为图 $G=(V,E)$ 的说谎者顶点-边支配集。本文引入了说谎者顶点-边支配的概念，该概念自然产生于通信网络中的某些应用。给定图 $G$，最小说谎者顶点-边支配问题 (MinLVEDP) 要求找出 $G$ 的基数最小的说谎者顶点-边支配集。本文从算法角度研究该问题。我们证明 MinLVEDP 可在线性时间内对树求解，而该问题的判定版本对于弦图、二分图以及 $p\geq 4$ 的 $p$-无爪图是 NP-完全的。我们进一步研究该问题的近似算法。针对一般图和 $p$-无爪图中的 MinLVEDP，我们提出了两个近似算法。在否定方面，我们证明除非 $NP\subseteq DTIME(|V|^{O(\log(\log|V|)})$，否则对于任意 $\epsilon >0$，MinLVEDP 无法在 $\frac{1}{2}(\frac{1}{8}-\epsilon)\ln|V|$ 因子内近似。最后，我们证明 MinLVEDP 对于有界度图和 $p\geq 6$ 的 $p$-无爪图是 APX-完全的。