Computing Simple Circuits from a Set of Line Segments is NP-Complete

Abstract

Given a collection of line segments in the plane, the segments are connected by their endpoints to construct a simple circuit. (A simple circuit is the boundary of a simple polygon.) However, there are collections of line segments where this cannot be done. In this note it is proved that deciding whether a set of line segments admits a simple circuit is NP-complete. The NP-completeness proof relies on the fact that line segments may intersect at their endpoints. Deciding whether a set of horizontal line segments can be connected with horizontal and vertical line segments to construct an orthogonal simple circuit is also shown to be NP-complete.