Generating All Triangulations of Plane Graphs

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 each face (a cycle) of the graph. We give an algorithm to generate all triangulations of a cycle C of n vertices in time O(1) per triangulation, where the vertices of C are numbered. Finally, we give an algorithm for generating all triangulations of a cycle C of n vertices in time O(n²) per triangulation, where vertices of C are not numbered. Key words: Triangulation; Graph; Cycle; Plane Graph; Genealogical Tree.