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

  • Oswin Aichholzer
  • Andrei Asinowski
  • Tillmann Miltzow
Keywords: Planar straight-line graphs, disjoint compatible matchings, reconfiguration graph, non-crossing geometric drawings, non-crossing partitions, combinatorial enumeration.


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.
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