8 research outputs found

    Drawing Planar Graphs with a Prescribed Inner Face

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    Given a plane graph GG (i.e., a planar graph with a fixed planar embedding) and a simple cycle CC in GG 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 GG. 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

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    A straight-line drawing δ\delta of a planar graph GG need not be plane, but can be made so by moving some of the vertices. Let shift(G,δ)(G,\delta) denote the minimum number of vertices that need to be moved to turn δ\delta into a plane drawing of GG. We show that shift(G,δ)(G,\delta) is NP-hard to compute and to approximate, and we give explicit bounds on shift(G,δ)(G,\delta) when GG 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

    Untangling a Planar Graph

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

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    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
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