The logic engine and the realization problem for nearest neighbor graphs

Abstract

Roughly speaking, a “nearest neighbor graph” is formed from a set of points in the plane by joining two points if one is the nearest neighbor of the other. There are several ways in which this intuitive concept can be made precise. This paper investigates the complexity of determining whether, for a given graph G, there is a set of points P in the plane such that G is isomorphic to a nearest neighbor graph on P. We show that this problem is NP-hard for several definitions of nearest neighbor graph. Our proof technique uses an interesting simulation of a mechanical device called a “logic engine”.