Augmenting a Geometric Matching is $NP$-complete

Abstract

Given $2n$ points in the plane, it is well-known that there always exists a perfect straight-line non-crossing matching. We show that it is $NP$-complete to decide if a partial matching can be augmented to a perfect one, via a reduction from 1-in-3-SAT. This result also holds for bichromatic matchings.