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We study noncrossing geometric graphs and their disjoint compatible geometricmatchings. Given a cycle (a polygon) P we want to draw a set of pairwisedisjoint straight-line edges with endpoints on the vertices of P such thatthese new edges neither cross nor contain any edge of the polygon. We proveNP-completeness of deciding whether there is such a perfect matching. For anyn-vertex polygon, with n > 3, we show that such a matching with less than n/7edges is not maximal, that is, it can be extended by another compatiblematching edge. We also construct polygons with maximal compatible matchingswith n/7 edges, demonstrating the tightness of this bound. Tight bounds on thesize of a minimal maximal compatible matching are also obtained for thefamilies of d-regular geometric graphs for each d in {0,1,2}. Finally weconsider a related problem. We prove that it is NP-complete to decide whether anoncrossing geometric graph G admits a set of compatible noncrossing edges suchthat G together with these edges has minimum degree five.

Augmenting Geometric Graphs with Matchings