Expected maximum vertex valence in pairs of polygonal triangulations

Abstract

Edge-flip distance between triangulations of polygons is equivalent to rotation distance between rooted binary trees. Both distances measure the extent of similarity of configurations. There are no known polynomial-time algorithms for computing edge-flip distance. The best known exact universal upper bounds on rotation distance arise from measuring the maximum total valence of a vertex in the corresponding triangulation pair obtained by a duality construction. Here we describe some properties of the distribution of maximum vertex valences of pairs of triangulations related to such upper bounds.