Algorithmic aspects of secure domination in unit disk graphs.

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 K 1 , t -free for some integer t ≥ 3. Finally, we propose a PTAS for the MSDS problem in unit disk graphs. [ABSTRACT FROM AUTHOR]