Disjoint compatibility graph 1 of non - crossing matchings 2 of points in convex position

6 Let X2k be a set of 2k labeled points in convex position in the plane. We consider geometric 7 non-intersecting straight-line perfect matchings of X2k. Two such matchings, M and M , are 8 disjoint compatible if they do not have common edges, and no edge of M crosses an edge of M . 9 Denote by DCMk the graph whose vertices correspond to such matchings, and two vertices 10 are adjacent if and only if the corresponding matchings are disjoint compatible. We show that 11 for each k ≥ 9, the connected components of DCMk form exactly three isomorphism classes 12 – namely, there is a certain number of isomorphic small components, a certain number of 13 isomorphic medium components, and one big component. The number and the structure of 14 small and medium components is determined precisely. 15