Drawing Planar Graphs with a Prescribed Inner Face

Given a plane graph $G$ (i.e., a planar graph with a fixed planar embedding) and a simple cycle $C$ in $G$ whose vertices are mapped to a convex polygon, we consider the question whether this drawing can be extended to a planar straight-line drawing of $G$. We characterize when this is possible in terms of simple necessary conditions, which we prove to be sufficient. This also leads to a linear-time testing algorithm. If a drawing extension exists, it can be computed in the same running time

Moving Vertices to Make Drawings Plane

A straight-line drawing $\delta$ of a planar graph $G$ need not be plane, but can be made so by moving some of the vertices. Let shift$(G,\delta)$ denote the minimum number of vertices that need to be moved to turn $\delta$ into a plane drawing of $G$. We show that shift$(G,\delta)$ is NP-hard to compute and to approximate, and we give explicit bounds on shift$(G,\delta)$ when $G$ is a tree or a general planar graph. Our hardness results extend to 1BendPointSetEmbeddability, a well-known graph-drawing problem.Comment: This paper has been merged with http://arxiv.org/abs/0709.017

9 1996 Springer-Verlag New York Inc. Generating Rooted Triangulations Without Repetitions 1

Abstract. We use the reverse search technique to give algorithms for generating all graphs on n points that are 2- and 3-connected planar triangulations with r points on the outer face. The triangulations are rooted, which means the outer face has a fixed labelling. The triangulations are produced without duplications in O(n 2) time per triangulation. The algorithms use O(n) space. A program for generating all 3-connected rooted triangulations based on this algorithm is available by ftp. Key Words. Rooted triangulations, 2- and 3-Connected triangulations, Reverse search technique. 1. Introduction. Let G-----(VI E) be a planar graph with vertex set V = {vl..... vn}, and let 3 < r < n be an integer. G is an r-rooted triangulation if it can be embedded in the plane such that the outer face has labels {Vl..... Vr} in clockwise order, and all interior faces are triangles. A vertex (or edge) on the external face is called external, otherwise it is internal. All r=rooted triangulations are 2-connected. It is well known that an r-roote