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Bounding and computing obstacle numbers of graphs
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Balko, Martin, Chaplick, Steven, Ganian, Robert, Gupta, Siddharth, Hoffmann, Michael, Valtr, Pavel and Wolff, Alexander (2022) Bounding and computing obstacle numbers of graphs. In: 30th Annual European Symposium on Algorithms (ESA 2022), Potsdam, Germany, 5-7 Sep 2022, 244 11:1-11:13. ISBN 9783959772471. doi:10.4230/LIPIcs.ESA.2022.11 ISSN 1868-8969.
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Official URL: https://drops.dagstuhl.de/opus/volltexte/2022/1694...
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(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.
Item Type: | Conference Item (Paper) | |||||||||||||||||||||||||||||||||
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Subjects: | Q Science > QA Mathematics | |||||||||||||||||||||||||||||||||
Divisions: | Faculty of Science, Engineering and Medicine > Science > Computer Science | |||||||||||||||||||||||||||||||||
Library of Congress Subject Headings (LCSH): | Combinatorial geometry , Geometry -- Data processing , Graph theory , Combinatorial analysis | |||||||||||||||||||||||||||||||||
Series Name: | Leibniz International Proceedings in Informatics (LIPIcs) | |||||||||||||||||||||||||||||||||
Publisher: | Schloss Dagstuhl — Leibniz-Zentrum für Informatik | |||||||||||||||||||||||||||||||||
Place of Publication: | Dagstuhl, Germany | |||||||||||||||||||||||||||||||||
ISBN: | 9783959772471 | |||||||||||||||||||||||||||||||||
ISSN: | 1868-8969 | |||||||||||||||||||||||||||||||||
Official Date: | 2022 | |||||||||||||||||||||||||||||||||
Dates: |
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Volume: | 244 | |||||||||||||||||||||||||||||||||
Page Range: | 11:1-11:13 | |||||||||||||||||||||||||||||||||
DOI: | 10.4230/LIPIcs.ESA.2022.11 | |||||||||||||||||||||||||||||||||
Status: | Peer Reviewed | |||||||||||||||||||||||||||||||||
Publication Status: | Published | |||||||||||||||||||||||||||||||||
Access rights to Published version: | Open Access (Creative Commons) | |||||||||||||||||||||||||||||||||
Date of first compliant deposit: | 10 November 2022 | |||||||||||||||||||||||||||||||||
Date of first compliant Open Access: | 10 November 2022 | |||||||||||||||||||||||||||||||||
RIOXX Funder/Project Grant: |
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Conference Paper Type: | Paper | |||||||||||||||||||||||||||||||||
Title of Event: | 30th Annual European Symposium on Algorithms (ESA 2022) | |||||||||||||||||||||||||||||||||
Type of Event: | Conference | |||||||||||||||||||||||||||||||||
Location of Event: | Potsdam, Germany | |||||||||||||||||||||||||||||||||
Date(s) of Event: | 5-7 Sep 2022 | |||||||||||||||||||||||||||||||||
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