Hardness of Liar's Domination on Unit Disk Graphs

A unit disk graph is the intersection graph of a set of unit diameter disks in the plane. In this paper we consider liar’s domination problem on unit disk graphs, a variant of dominating set problem. We call this problem as Euclidean liar’s domination problem. In the Euclidean liar’s domination problem, a set P = {p1, p2, . . . , pn} of n points (disk centers) are given in the Euclidean plane. For p ∈ P, N [p] is a subset of P such that for any q ∈ N [p], the Euclidean distance between p and q is less than or equal to 1, i.e., the corresponding unit diameter disks intersect. The objective of the Euclidean liar’s domination problem is to find a subset D (⊆ P) of minimum size having the following properties : (i) |N [pi] ∩ D| ≥ 2 for 1 ≤ i ≤ n, and (ii) |(N [pi] ∪ N [pj ]) ∩ D| ≥ 3 for i 6= j, 1 ≤ i, j ≤ n. This article aims to prove the Euclidean liar’s domination problem is NP-complete.