Point Set Isolation Using Unit Disks is NP-complete

TL;DR

This paper proves that the problem of isolating point sets with minimum unit disks and related connectivity problems are NP-complete, settling open questions and extending NP-completeness to unit disk graphs.

Contribution

It establishes NP-completeness for point set isolation and connectivity problems involving unit disks, resolving open problems in computational geometry.

Findings

Point set isolation with unit disks is NP-complete.

Connectivity of the complement of disk sets is NP-complete.

Multiterminal Cut remains NP-complete on unit disk graphs.

Abstract

We consider the situation where one is given a set S of points in the plane and a collection D of unit disks embedded in the plane. We show that finding a minimum cardinality subset of D such that any path between any two points in S is intersected by at least one disk is NP-complete. This settles an open problem raised by Matt Gibson et al[1]. Using a similar reduction, we show that finding a minimum cardinality subset D' of D such that R^2 - (D - D') consists of a single connected region is also NP-complete. Lastly, we show that the Multiterminal Cut Problem remains NP-complete when restricted to unit disk graphs.