  In this paper we investigate the structure of flip graphs on non-crossing
perfect matchings in the plane. Specifically, consider all non-crossing
straight-line perfect matchings on a set of $2n$ points that are placed
equidistantly on the unit circle. A flip operation on such a matching replaces
two matching edges that span an empty quadrilateral with the other two edges of
the quadrilateral, and the flip is called centered if the quadrilateral
contains the center of the unit circle. The graph $\mathcal{G}_n$ has those
matchings as vertices, and an edge between any two matchings that differ in a
flip, and it is known to have many interesting properties. In this paper we
focus on the spanning subgraph $\mathcal{H}_n$ of $\mathcal{G}_n$ obtained by
taking all edges that correspond to centered flips, omitting edges that
correspond to non-centered flips. We show that the graph $\mathcal{H}_n$ is
connected for odd $n$, but has exponentially many small connected components
for even $n$, which we characterize and count via Catalan and generalized
Narayana numbers. For odd $n$, we also prove that the diameter of
$\mathcal{H}_n$ is linear in $n$. Furthermore, we determine the minimum and
maximum degree of $\mathcal{H}_n$ for all $n$, and characterize and count the
corresponding vertices. Our results imply the non-existence of certain rainbow
cycles in $\mathcal{G}_n$, and they resolve several open questions and
conjectures raised in a recent paper by Felsner, Kleist, M\"utze, and Sering.
