Quasi-parallel segments and characterization of unique bichromatic matchings

Given n red and n blue points in general position in the plane, it is well-known that there is a perfect matching formed by non-crossing line segments. We characterize the bichromatic point sets which admit exactly one non-crossing matching. We give several geometric descriptions of such sets, and find an O ( n  log  n ) algorithm that checks whether a given bichromatic set has this property.

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