Connectedness of the two-edge flip graph of plane perfect matchings in general position

Determine whether, for any finite set of an even number of points in general position in the plane, the flip graph whose vertices are plane perfect matchings and whose edges connect two matchings when one can be obtained from the other by replacing exactly two matching edges with two other non-crossing edges (yielding another plane perfect matching), is connected.

Background

In geometric matching reconfiguration, a perfect matching cannot be transformed into another perfect matching with a single edge flip. A natural flip operation replaces two matching edges with two other edges so that the result is again a perfect matching. Prior work has established that, for point sets in convex position, the corresponding flip graph is connected.

This paper proves connectedness for odd-size point sets (almost perfect matchings) under single-edge flips and bounds the diameter. However, for even-size point sets (perfect matchings) under two-edge flips, the connectedness of the flip graph on general position point sets is not resolved and remains an outstanding question.