{'fraction_non_alphanumeric': 0.03894927536231884, 'fraction_numerical': 0.01177536231884058, 'mean_word_length': 4.261904761904762, 'pattern_counts': {'":': 0, '<': 0, '<?xml version=': 0, '>': 1, 'https://': 0, 'lorem ipsum': 0, 'www.': 0, 'xml': 0}, 'pii_count': 0, 'substrings_counts': 0, 'word_list_counts': {'cursed_substrings.json': 0, 'profanity_word_list.json': 0, 'sexual_word_list.json': 0, 'zh_pornsignals.json': 0}}
Eccentricities in the flip‐graphs of convex polygons

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.