Generating all triangulations of plane graphs : Extended abstract

Abstract

In this paper, we deal with the problem of generating all triangulations of plane graphs. We give an algorithm for generating all triangulations of a triconnected plane graph G of n vertices. Our algorithm establishes a tree structure among the triangulations of G, called the "tree of triangulations," and generates each triangulation of G in O(1) time. The algorithm uses O(n) space and generates all triangulations of G without duplications. To the best of our knowledge, our algorithm is the first algorithm for generating all triangulations of a triconnected plane graph; although there exist algorithms for generating triangulated graphs with certain properties. Our algorithm for generating all triangulations of a triconnected plane graph needs to find all triangulations of a convex polygon. We give an algorithm to generate all triangulations of a convex polygon P of n vertices in time O(1) per triangulation, where the vertices of P are numbered. Our algorithm for generating all triangulations of a convex polygon also improves previous results; existing algorithms need to generate all triangulations of convex polygons of less than n vertices before generating the triangulations of a convex polygon of n vertices. Finally, we give an algorithm for generating all triangulations of a convex polygon P of n vertices in time O(n2) per triangulation, where vertices of P are not numbered. © Springer-Verlag Berlin Heidelberg 2009.