Liar's domination in unit disk graphs

Abstract

In this article, we study a variant of the minimum dominating set problem known as the minimum liar's dominating set (MLDS) problem. We prove that the MLDS problem is NP-hard in unit disk graphs. Next, we show that the recent sub-quadratic time 11 2 -factor approximation algorithm [2] for the MLDS problem is erroneous and propose a simple O(n+m) time 7.31-factor approximation algorithm, where n and m are the number of vertices and edges in the input unit disk graph, respectively. Finally, we prove that the MLDS problem admits a polynomial-time approximation scheme.

Edge-Vertex Dominating Set in Unit Disk Graphs

“…It is based on the concept of -separated collection of subsets, which was introduced by Nieberg and Hurink [19]. This concept was used by many other authors to develop PTAS (for e.g., the Roman dominating set [20], minimum Liar's dominating set [21], vertex-edge dominating set [5]). However, the way we use it here is quite different from these as we have to select edges to dominate vertices in the EVDS problem.…”