Algorithmic study on liar’s vertex-edge domination problem

Let \(G=(V,E)\) $G=(V,E)$ be a graph. For an edge \(e=xy\in E\) $e=xy\in E$ , the closed neighbourhood of e, denoted by \(N_G[e]\) $N_G[e]$ or \(N_G[xy]\) $N_G[xy]$ , is the set \(N_G[x]\cup N_G[y]\) $N_G[x]\cup N_G[y]$ . A vertex set \(L\subseteq V\) $L\subseteq V$ is liar’s vertex-edge dominating set of a graph \(G=(V,E)\) $G=(V,E)$ if for every \(e_i\in E\) $e_i\in E$ , \(|N_G[e_i]\cap L|\ge 2\) $|N_G[e_i]\cap L|\ge 2$ and for every pair of distinct edges \(e_i\) $e_i$ and \(e_j\) $e_j$ , \(|(N_G[e_i]\cup N_G[e_j])\cap L|\ge 3\) $|(N_G[e_i]\cup N_G[e_j])\cap L|\ge 3$ . 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 Minimum Liar’s Vertex-Edge Domination Problem (MinLVEDP) asks to find a liar’s vertex-edge dominating set of G of minimum cardinality. In this paper, we study this problem from an algorithmic point of view. We show that MinLVEDP can be solved in linear time for trees, whereas the decision version of this problem is NP-complete for general graphs, chordal graphs, and bipartite graphs. We further study approximation algorithms for this problem. We propose two approximation algorithms for MinLVEDP in general graphs and p-claw free graphs. On the negative side, we show that the MinLVEDP cannot be approximated within \(\frac{1}{2}(\frac{1}{8}-\epsilon )\ln |V|\) $\frac{1}{2}(\frac{1}{8}-ϵ)ln|V|$ for any \(\epsilon >0\) $ϵ>0$ , unless \(NP\subseteq DTIME(|V|^{O(\log (\log |V|)})\) $NP\subseteq DTIME(|V{|}^{O(log(log|V|)})$ . Finally, we prove that the MinLVEDP is APX-complete for bounded degree graphs and p-claw-free graphs for \(p\ge 6\) $p\ge 6$ .