Total (restrained) domination in unit disk graphs

Abstract

Let G=(V,E) be a graph. A set D⊆V is considered a total dominating set of G if every vertex in V is adjacent to a vertex in D. If every vertex in V is adjacent to a vertex in D, and every vertex in V∖D is adjacent to a vertex in V∖D, then D is a total restrained dominating set. A unit disk is a disk with a diameter of 1. A unit disk representation of a graph G is a set S={Su}u∈V of unit disks, such that uv∈E if and only if Su∩Sv≠∅. A graph is a unit disk graph (or, for short, UDG) if it has a unit disk representation.In this paper, we first show that the total domination decision problem and the total restrained domination decision problem are NP-complete for grid graphs (a subclass of UDGs), strengthening the result on the total domination decision problem in UDGs by Jena et al. [11]. Then, we present a linear time 6.387-approximation algorithm for the minimum total domination problem in UDGs. This improves on a ratio 8 algorithm with linear time by Jena et al. [11]. Finally, we initially propose a linear time 10.387-approximation algorithm and a PTAS for the minimum total restrained domination problem in UDGs.