Graphs and Combinatorics

, Volume 17, Issue 4, pp 717–728 | Cite as

Embedding Planar Graphs at Fixed Vertex Locations

  • János Pach
  • Rephael Wenger

Abstract.

 Let G be a planar graph of n vertices, v1,…,v n , and let {p1,…,p n } 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 v i is represented by point p i 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.

Keywords

Planar Graph Mapping Vertex Internal Vertex Planar Embedding Independent Edge 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

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Copyright information

© Springer-Verlag Tokyo 2001

Authors and Affiliations

  • János Pach
    • 1
  • Rephael Wenger
    • 2
  1. 1.City College, New York and the Hungarian Academy of Sciences, Budapest e-mail: pach@cims.nyu.eduXX
  2. 2.The Ohio State University, Columbus, OH 43210, USA e-mail: wenger@cis.ohio-state.eduUS

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