# Bounding and Computing Obstacle Numbers of Graphs

Martin Balko, Steven Chaplick, Robert Ganian, Siddharth Gupta, Michael (M.) Hoffmann, Pavel Valtr, Alexander Wolff

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## Abstract

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(nlogn) [M. Balko, J. Cibulka, and P. Valtr, Discrete Comput. Geom., 59 (2018), pp. 143-164] and that there are n-vertex graphs whose obstacle number is \Omega(n/(loglogn) 2) [V. Dujmovi\'c and P. Morin, Electron. J. Combin., 22 (2015), 3.1]. We improve this lower bound to \Omega(n/loglogn) for simple polygons and to \Omega(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 Dujmovi\'c 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 \Omega(n 2) 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.

Original language English 1537-1565 29 Siam Journal on Discrete Mathematics 38 2 https://doi.org/10.1137/23M1585088 Published - 30 Jun 2024

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• ### Bounding and computing obstacle numbers of graphs

Balko, M., Chaplick, S., Ganian, R., Gupta, S., Hoffmann, M. (. )., Valtr, P. & Wolff, A., 30 Jun 2022.

Research output: Working paper / PreprintPreprint

• ### Bounding and Computing Obstacle Numbers of Graphs

Balko, M., Chaplick, S., Ganian, R., Gupta, S., Hoffmann, M. (. )., Valtr, P. & Wolff, A., 2022, 30th Annual European Symposium on Algorithms, {ESA} 2022,: September 5-9, 2022, Berlin/Potsdam, Germany. Schloss Dagstuhl - Leibniz-Zentrum fuer Informatik, Vol. 244. p. 11:1-11:13 (Leibniz International Proceedings in Informatics (LIPIcs), Vol. 244).

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