Disjoint compatible geometric matchings

Authors:
Mashhood Ishaque;Diane L. Souvaine;Csaba D. Toth
Affiliations:
Tufts University, Medford, MA, USA;Tufts University, Medford, MA, USA;University of Calgary, Calgary, AB, Canada
Venue:
Proceedings of the twenty-seventh annual symposium on Computational geometry
Year:
2011

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Abstract

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.