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The flip‐graph of a convex polygon   π   is the graph whose vertices are the triangulations of   π   and whose edges correspond to flips between them. The eccentricity of a triangulation   T   of   π   is the largest possible distance in this graph from  T  to any triangulation of   π  . It is well known that when all   n  −  3   interior edges of   T   are incident to the same vertex, the eccentricity of   T   in the flip‐graph of   π   is exactly   n  −  3  , where   n   denotes the number of vertices of   π  . Here, this statement is generalized to arbitrary triangulations. Denoting by   n  −  3  −  k   the largest number of interior edges of   T   incident to a vertex, it is shown that the eccentricity of   T   in the flip‐graph of   π   is exactly   n  −  3  +  k  , provided   k  ≤  n  ∕  2  −  2  . Inversely, the eccentricity of a triangulation, when small enough, allows to recover the value of   k  . More precisely, if   k  ≤  n  ∕  8  −  5  ∕  2  , it is also shown that   T   has eccentricity   n  −  3  +  k   if and only if exactly   n  −  3  −  k   of its interior edges are incident to a given vertex. When   k  >  n  ∕  2  −  2  , bounds on the eccentricity of   T   are also given and discussed.

Eccentricities in the flip‐graphs of convex polygons