  Let $X_{2k}$ be a set of $2k$ labeled points in convex position in the plane.
We consider geometric non-intersecting straight-line perfect matchings of
$X_{2k}$. Two such matchings, $M$ and $M'$, are disjoint compatible if they do
not have common edges, and no edge of $M$ crosses an edge of $M'$. Denote by
$\mathrm{DCM}_k$ the graph whose vertices correspond to such matchings, and two
vertices are adjacent if and only if the corresponding matchings are disjoint
compatible. We show that for each $k \geq 9$, the connected components of
$\mathrm{DCM}_k$ form exactly three isomorphism classes -- namely, there is a
certain number of isomorphic small components, a certain number of isomorphic
medium components, and one big component. The number and the structure of small
and medium components is determined precisely.
