Disjoint compatibility graph of non-crossing matchings of points in   convex position
  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. 
Oswin Aichholzer, Andrei Asinowski, Tillmann Miltzow
math.CO cs.CG cs.DM