Generating Rooted Triangulations with Minimum Degree Four

A graph is a triangulation if it is planar and every face is a triangle. A triangulation is rooted if the external triangular face is labelled. Two rooted triangulations with the same external face labels are isomorphic if their internal vertices can be labelled so that both triangulations have identical edge lists. In this article, we show that in the set of rooted triangulations on n points with minimum degree four, there exists a target triangulation E n such that any other triangulation E n 6 = E n in the set can be transformed to E n via a nite sequence of single and double diagonal transformations. Using this result with the reverse search technique, we present an algorithm for generating all non-isomorphic rooted triangulations on n points with minimum degree four. The triangulations are produced without repetitions in O(n 2) time per triangulation. The algorithm uses O(n) space. A simple planar graph is maximal planar if and only if it is triangulated. Hence a triangulation of a set of points in the plane is a maximal planar graph. In a maximal planar graph(MPG) G = (V; E), jEj = 3jV j ? 6. An edge in a triangulation is external if it is contained in exactly one triangle (not the external face) of the triangulation. The vertices of the external edges are external vertices. Vertices that are not external are internal vertices. An edge in a triangulation is internal if it is contained in exactly two triangles (neither is the external face) of the triangulation. Such an edge bounds the two triangles. The objects we want to generate are rooted triangulations with minimum degree four on n points (given n), which, hereafter, will be referred to as rooted maximal planar graphs with minimum degree four (rooted MPG4). In a maximal planar graph G, let 4abc and 4abd be two adjoining triangular faces. The edges of these faces form a quadrangle acbd with the diagonal (a; b). When this diagonal is replaced by (c; d), one obtains a new maximal planar graph G 0 with the same vertices and the same number of edges and faces. We say that G 0 has been obtained from G by a diagonal transformation or a diagonal ip. This is possible only if (c; d) is not an edge in G already. Additional details may be found in 4, 6]. The reverse search …