Graphs and Combinatorics

, Volume 17, Issue 4, pp 717–728

Embedding Planar Graphs at Fixed Vertex Locations

  • János Pach
  • Rephael Wenger

DOI: 10.1007/PL00007258

Cite this article as:
Pach, J. & Wenger, R. Graphs Comb (2001) 17: 717. doi:10.1007/PL00007258

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.

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