Abstract
Given a set P of points in general position in the plane, the graph of triangulations ** of P has a vertex for every triangulation of P, and two of them are adjacent if they differ by a single edge exchange. We prove that the subgraph ** of **, consisting of all triangulations of P that admit a perfect matching, is connected. A main tool in our proof is a result of independent interest, namely that the graph ** that has as vertices the non-crossing perfect matchings of P and two of them are adjacent if their symmetric difference is a single non-crossing cycle, is also connected.
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