Drawing planar graphs with many collinear vertices

  • Giordano Da Lozzo Roma Tre University
  • Vida Dujmović University of Ottawa
  • Fabrizio Frati Roma Tre University
  • Tamara Mchedlidze Karlsruhe Institute of Technology
  • Vincenzo Roselli Roma Tre University

Abstract

Consider the following problem: Given a planar graph $G$, what is the maximum number $p$ such that $G$ has a planar straight-line drawing with $p$ collinear vertices? This problem resides at the core of several graph drawing problems, including universal point subsets, untangling, and column planarity. The following results are known for it: Every $n$-vertex planar graph has a planar straight-line drawing with $\Omega(\sqrt{n})$ collinear vertices; for every $n$, there is an $n$-vertex planar graph whose every planar straight-line drawing has $O(n^\sigma)$ collinear vertices, where $\sigma<0.986$; every $n$-vertex planar graph of treewidth at most two has a planar straight-line drawing with $\Theta(n)$ collinear vertices. We extend the linear bound to planar graphs of treewidth at most three and to triconnected cubic planar graphs. This (partially) answers two open problems posed by Ravsky and Verbitsky [WG 2011:295–306]. Similar results are not possible for all bounded-treewidth planar graphs or for all bounded-degree planar graphs. For planar graphs of treewidth at most three, our results also imply asymptotically tight bounds for all of the other above mentioned graph drawing problems.
Published
2018-05-03
How to Cite
Da Lozzo, G., Dujmović, V., Frati, F., Mchedlidze, T., & Roselli, V. (2018). Drawing planar graphs with many collinear vertices. Journal of Computational Geometry, 9(1), 94–130. https://doi.org/10.20382/jocg.v9i1a4
Section
Articles

Most read articles by the same author(s)