Superpatterns and Universal Point Sets
Vol. 18, no. 2, pp. 177-209, 2014. Regular paper.
Abstract An old open problem in graph drawing asks for the size of a universal point set, a set of points that can be used as vertices for straight-line drawings of all n-vertex planar graphs. We connect this problem to the theory of permutation patterns, where another open problem concerns the size of superpatterns, permutations that contain all patterns of a given size. We generalize superpatterns to classes of permutations determined by forbidden patterns, and we construct superpatterns of size n2/4+Θ(n) for the 213-avoiding permutations, half the size of known superpatterns for unconstrained permutations. We use our superpatterns to construct universal point sets of size n2/4−Θ(n), smaller than the previous bound by a 9/16 factor. We prove that every proper subclass of the 213-avoiding permutations has superpatterns of size O(nlogO(1)n), which we use to prove that the planar graphs of bounded pathwidth have near-linear universal point sets.
Submitted: December 2013.
Reviewed: February 2014.
Revised: March 2014.
Accepted: March 2014.
Final: April 2014.
Published: May 2014.
Communicated by Stephen K. Wismath and Alexander Wolff
article (PDF)