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Bounding and Computing Obstacle Numbers of Graphs

Authors Martin Balko , Steven Chaplick , Robert Ganian , Siddharth Gupta , Michael Hoffmann , Pavel Valtr , Alexander Wolff

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Author Details

Martin Balko
  • Faculty of Mathematics and Physics, Charles University, Prague, Czech Republic
Steven Chaplick
  • Maastricht University, The Netherlands
Robert Ganian
  • Algorithms and Complexity Group, TU Wien, Austria
Siddharth Gupta
  • Department of Computer Science, University of Warwick, Coventry, UK
Michael Hoffmann
  • Department of Computer Science, ETH Zürich, Switzerland
Pavel Valtr
  • Faculty of Mathematics and Physics, Charles University, Prague, Czech Republic
Alexander Wolff
  • Institut für Informatik, Universität Würzburg, Germany


This research was initiated at the Bertinoro Workshop on Graph Drawing 2022 (BWGD'22).

Cite AsGet BibTex

Martin Balko, Steven Chaplick, Robert Ganian, Siddharth Gupta, Michael Hoffmann, Pavel Valtr, and Alexander Wolff. Bounding and Computing Obstacle Numbers of Graphs. In 30th Annual European Symposium on Algorithms (ESA 2022). Leibniz International Proceedings in Informatics (LIPIcs), Volume 244, pp. 11:1-11:13, Schloss Dagstuhl - Leibniz-Zentrum für Informatik (2022)


An obstacle representation of a graph G consists of a set of pairwise disjoint simply-connected closed regions and a one-to-one mapping of the vertices of G to points such that two vertices are adjacent in G if and only if the line segment connecting the two corresponding points does not intersect any obstacle. The obstacle number of a graph is the smallest number of obstacles in an obstacle representation of the graph in the plane such that all obstacles are simple polygons. It is known that the obstacle number of each n-vertex graph is O(n log n) [Balko, Cibulka, and Valtr, 2018] and that there are n-vertex graphs whose obstacle number is Ω(n/(log log n)²) [Dujmović and Morin, 2015]. We improve this lower bound to Ω(n/log log n) for simple polygons and to Ω(n) for convex polygons. To obtain these stronger bounds, we improve known estimates on the number of n-vertex graphs with bounded obstacle number, solving a conjecture by Dujmović and Morin. We also show that if the drawing of some n-vertex graph is given as part of the input, then for some drawings Ω(n²) obstacles are required to turn them into an obstacle representation of the graph. Our bounds are asymptotically tight in several instances. We complement these combinatorial bounds by two complexity results. First, we show that computing the obstacle number of a graph G is fixed-parameter tractable in the vertex cover number of G. Second, we show that, given a graph G and a simple polygon P, it is NP-hard to decide whether G admits an obstacle representation using P as the only obstacle.

Subject Classification

ACM Subject Classification
  • Theory of computation → Computational geometry
  • Human-centered computing → Graph drawings
  • Obstacle representation
  • Obstacle number
  • Visibility
  • NP-hardness
  • FPT


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