Efficient Generation of Triconnected Plane Triangulations

Abstract

A “rooted” plane triangulation is a plane triangulation with one designated vertex on the outer face. In this paper we give a simple algorithm to generate all triconnected rooted plane triangulations with at most n vertices. The algorithm uses O(n) space and generates such triangulations in O(1) time per triangulation without duplications. The algorithm does not output entire triangulations but the difference from the previous triangulation. By modifying the algorithm we can generate all triconnected rooted plane triangulation having exactly n vertices including exactly r vertices on the outer face in O(r) time per triangulation without duplications, while the previous best algorithm generates such triangulations in O(n ²) time per triangulation. Also we can generate without duplications all triconnected (non-rooted) plane triangulations having exactly n vertices including exactly r vertices on the outer face in O(r ² n) time per triangulation, and all maximal planar graphs in O(n ³) time per graph.