Flipping Matchings is Hard

Given a point set $\mathcal{P}$ and a plane perfect matching $\mathcal{M}$ on $\mathcal{P}$, a flip is an operation that replaces two edges of $\mathcal{M}$ such that another plane perfect matching on $\mathcal{P}$ is obtained. Given two plane perfect matchings on $\mathcal{P}$, we show that it is NP-hard to minimize the number of flips that are needed to transform one matching into the other.