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Smallest Universal Set of Points for Planar Graphs: An Open Problem

This problem, presented by Andy Giese, asks how many points are needed in the plane to support the planar drawing of all graphs with n vertices. Specifically, it asks whether this can be achieved with O(n) points. A universal point set allows mapping for all planar graphs, with previous work suggesting values between Θ(n) and Θ(n²). While some results indicate at least Θ(1.098n), the quest for a formal proof remains elusive. Investigation into the smallest n for which such a point set does not exist is crucial to advancing our understanding of planar graph representations.

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Smallest Universal Set of Points for Planar Graphs: An Open Problem

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  1. Problem 45: Smallest Universal Set of Points for Planar Graphs Presented by Andy Giese Problem #45 from the Open Problems Project http://maven.smith.edu/~orourke/TOPP/

  2. Statement of Problem • How many points must be placed in the plane to support planar drawing of all planar graphs on n vertices? • Is it O(n)? Kittell Graph Image from Wolfram MathWorld

  3. Some Definitions • Planar Graph • No intersecting segments • Universal Point Set • Set of vertices that allows mapping of all planar graphs of size n • E.G. NxN grid • Maximal Planar Graph • 3n-6 edges • Universal set Maximal Planar Graph Image from MathWorks.de

  4. Progress • Between Θ(n) andΘ(n2) • At most O(n2) • H. de Fraysseix, J. Pach, and R. Pollack, 1990 • W. Schnyder, 1990 • At least Θ(n), more accurately, Θ (1.098n) • M. Chrobak and H.Karloff. 1989 • Allowing 1 “bend” per edge = O(n) • Hazel Everett, Sylvain Lazard, Giuseppe Liotta, and Stephen Wismath, 2010

  5. Closing Thoughts • Intuition says O(n), but no formal proof exists • Exhaustive proof: What is the smallest value of n for which a universal point set of size n does not exist? • Stephen Kobourov, 2002, proved exhaustively for n<=14 • Could we prove by contradiction? • Why is this problem important?

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