Topics on flip graphs
(Jean Cardinal)

A flip in a triangulation is a simple local operation involving neighboring pairs of triangles, and consisting in exchanging the two diagonals of the quadrilateral that they form. Such operations define so-called flip graphs on sets of triangulations: every vertex of the graph is a triangulation, and two vertices are adjacent whenever the triangulations differ by a single flip. For triangulations of convex polygons, flip graphs are skeletons of associahedra, a well-known family of polytopes. Associahedra also have applications to the theory of binary search trees.

We will introduce such flip graphs and some of their generalizations, as well as a number of important open questions, both on the combinatorial and computational aspects. In particular, we plan to discuss flip graphs on rectangulations, on tubings in graphs, and on regular triangulations of point sets. These all have in common to be skeletons of polytopes.