Abstract
An obstacle representation of a graph G consists of a set of pairwise disjoint simplyconnected closed regions and a onetoone 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 nvertex graph is O(n log n) [Balko, Cibulka, and Valtr, 2018] and that there are nvertex 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 nvertex graphs with bounded obstacle number, solving a conjecture by Dujmović and Morin. We also show that if the drawing of some nvertex 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 fixedparameter tractable in the vertex cover number of G. Second, we show that, given a graph G and a simple polygon P, it is NPhard to decide whether G admits an obstacle representation using P as the only obstacle.
BibTeX  Entry
@InProceedings{balko_et_al:LIPIcs.ESA.2022.11,
author = {Balko, Martin and Chaplick, Steven and Ganian, Robert and Gupta, Siddharth and Hoffmann, Michael and Valtr, Pavel and Wolff, Alexander},
title = {{Bounding and Computing Obstacle Numbers of Graphs}},
booktitle = {30th Annual European Symposium on Algorithms (ESA 2022)},
pages = {11:111:13},
series = {Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN = {9783959772471},
ISSN = {18688969},
year = {2022},
volume = {244},
editor = {Chechik, Shiri and Navarro, Gonzalo and Rotenberg, Eva and Herman, Grzegorz},
publisher = {Schloss Dagstuhl  LeibnizZentrum f{\"u}r Informatik},
address = {Dagstuhl, Germany},
URL = {https://drops.dagstuhl.de/opus/volltexte/2022/16949},
URN = {urn:nbn:de:0030drops169495},
doi = {10.4230/LIPIcs.ESA.2022.11},
annote = {Keywords: Obstacle representation, Obstacle number, Visibility, NPhardness, FPT}
}
Keywords: 

Obstacle representation, Obstacle number, Visibility, NPhardness, FPT 
Collection: 

30th Annual European Symposium on Algorithms (ESA 2022) 
Issue Date: 

2022 
Date of publication: 

01.09.2022 