Algorithmic aspects of secure domination in unit disk graphs

Abstract

Given a graph G with vertex set V, a set S⊆V is a secure dominating set of G if S is a dominating set of G and if for every vertex u∈V∖S, there exists a vertex v∈S adjacent to u such that (S∪{u})∖{v} is a dominating set of G.The minimum secure dominating set (or, for short, MSDS) problem asks to find an MSDS in a given graph. In this paper, we first show that the decision version of the MSDS problem is NP-complete in unit disk graphs, even in grid graphs. Secondly, we give an O(n+m) time t-approximation algorithm for the MSDS problem in several geometric intersection graphs which are K1,t-free for some integer t≥3. Finally, we propose a PTAS for the MSDS problem in unit disk graphs.