Algorithmic study on liar's vertex-edge domination problem

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 $O(\ln Δ(G))$-approximation algorithm for \textsc{MinLVEDP} in general graphs, where $Δ(G)$ 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 $\frac{1}{2}(\frac{1}{8}-ε)\ln|V|$ for any $ε>0$, 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 $p\geq 6$.