Eccentricities in the flip‐graphs of convex polygons

Abstract

AbstractThe flip‐graph of a convex polygon is the graph whose vertices are the triangulations of and whose edges correspond to flips between them. The eccentricity of a triangulation of is the largest possible distance in this graph from T to any triangulation of . It is well known that when all interior edges of are incident to the same vertex, the eccentricity of in the flip‐graph of is exactly , where denotes the number of vertices of . Here, this statement is generalized to arbitrary triangulations. Denoting by the largest number of interior edges of incident to a vertex, it is shown that the eccentricity of in the flip‐graph of is exactly , provided . Inversely, the eccentricity of a triangulation, when small enough, allows to recover the value of . More precisely, if , it is also shown that has eccentricity if and only if exactly of its interior edges are incident to a given vertex. When , bounds on the eccentricity of are also given and discussed.