Edge-Removal and Non-Crossing Perfect Matchings [article]

Aviv Sheyn And Ran J. Tessler
2011 arXiv   pre-print
We study the following problem - How many arbitrary edges can be removed from a complete geometric graph with 2n vertices such that the resulting graph always contains a perfect non-crossing matching? We first address the case where the boundary of the convex hull of the original graph contains at most n + 1 points. In this case we show that n edges can be removed, one more than the general case. In the second part we establish a lower bound for the case where the 2n points are randomly chosen.
more » ... We prove that with probability which tends to 1, one can remove any n + Θ(n/log (n)) edges but the residual graph will still contain a non-crossing perfect matching. We also discuss the upper bound for the number of arbitrary edges one must remove in order to eliminate all the non-crossing perfect matchings.
arXiv:1107.2314v1 fatcat:tm2mfs6dizesvb2lpescx2vn4q