• DOI:10.48550/arXiv.2206.02700
• Corpus ID: 249394893

Forbidding Edges between Points in the Plane to Disconnect the Triangulation Flip Graph

@article{Bigdeli2022ForbiddingEB,
  title={Forbidding Edges between Points in the Plane to Disconnect the Triangulation Flip Graph},
  author={Reza Bigdeli and Anna Lubiw},
  journal={ArXiv},
  year={2022},
  volume={abs/2206.02700}
}

• Reza Bigdeli, A. Lubiw
• Published 6 June 2022
• Mathematics
• ArXiv

The flip graph for a set P of points in the plane has a vertex for every triangulation of P , and an edge when two triangulations differ by one flip that replaces one triangulation edge by another. The flip graph is known to have some connectivity properties: (1) the flip graph is connected; (2) connectivity still holds when restricted to triangulations containing some constrained edges between the points; (3) for P in general position of size n , the flip graph is (cid:100) n 2 − 2 (cid:101) -connected, a recent result of Wagner and Welzl (SODA 2020). We introduce the study of connectivity properties of the flip graph when some edges between points are forbidden . An edge e between two points is a flip cut edge if eliminating triangulations containing e results in a disconnected flip graph. More generally, a set X of edges between points of P is a flip cut set if eliminating all triangulations that contain edges of X results in a disconnected flip graph. The flip cut number of P is the minimum size of a flip cut set. We give a characterization of flip cut edges that leads to an O ( n log n ) time algorithm to test if an edge is a flip cut edge and, with that as preprocessing, an O ( n ) time algorithm to test if two triangulations are in the same connected component of the flip graph. For a set of n points in convex position (whose flip graph is the 1-skeleton of the associahedron) we prove that the flip cut number is n − 3.