Let X2k be a set of 2k labeled points in convex position in the plane.
We consider geometric non-intersecting straight-line perfect matchings of
X2k. 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
DCMk 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≥9, the connected components of
DCMk 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.Comment: 46 pages, 30 figure