### Disjoint Compatibility Graph of Non-Crossing Matchings of Points in Convex Position

#### Abstract

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 $\rm{DCM}_k$ 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 $\rm{DCM}_k$ 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.#### Keywords

Planar straight-line graphs; disjoint compatible matchings; reconfiguration graph; non-crossing geometric drawings; non-crossing partitions; combinatorial enumeration.