Disjoint compatible geometric matchings
Mashhood Ishaque, Diane L. Souvaine, Csaba D. Tóth · 2011
We prove that for every even set of $n$ pairwise disjoint line segments in the plane in general position, there is another set of n segments such that the 2n segments form pairwise disjoint simple polygons in the plane. This settles in the affirmative the Disjoint Compatible Matching Conjecture by Aichholzer et al. [ABD08]. The key tool in our proof is a novel subdivision of the free space around n disjoint line segments into at most n+1 convex cells such that the dual graph of the subdivision contains two edge-disjoint spanning trees.