Embedding Planar Graphs at Fixed Vertex Locations

  • János Pach
  • Rephael Wenger
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 1547)

Abstract

Let G be a planar graph of n vertices, v1,..., vn, and let p1,... ,pn be a set of n points in the plane. We present an algorithm for constructing in O(n2) time a planar embedding of G, where vertex vi is represented by point pi and each edge is represented by a polygonal curve with O(n) bends (internal vertices.) This bound is asymptotically optimal in the worst case. In fact, if G is a planar graph containing at least m pairwise independent edges and the vertices of G are randomly assigned to points in convex position, then, almost surely, every planar embedding of G mapping vertices to their assigned points and edges to polygonal curves has at least m/20 edges represented by curves with at least m/403 bends.

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Copyright information

© Springer-Verlag Berlin Heidelberg 1998

Authors and Affiliations

  • János Pach
    • 1
  • Rephael Wenger
    • 2
  1. 1.City CollegeNew York and the Hungarian Academy of SciencesBudapest
  2. 2.The Ohio State UniversityColumbus

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