Research output per year
Research output per year
Martin Balko, Steven Chaplick, Robert Ganian, Siddharth Gupta, Michael (M.) Hoffmann, Pavel Valtr, Alexander Wolff
Research output: Chapter in Book/Report/Conference proceeding › Chapter › Academic
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) [Dujmovic 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 Dujmovic 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.
Original language | English |
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Title of host publication | 30th Annual European Symposium on Algorithms, {ESA} 2022, |
Subtitle of host publication | September 5-9, 2022, Berlin/Potsdam, Germany |
Publisher | Schloss Dagstuhl - Leibniz-Zentrum fuer Informatik |
Pages | 11:1-11:13 |
Volume | 244 |
DOIs | |
Publication status | Published - 2022 |
Event | 30th Annual European Symposium on Algorithms - Berlin/Potsdam, Germany Duration: 5 Sept 2022 → 7 Sept 2022 Conference number: 30 http://esa-symposium.org/ |
Series | Leibniz International Proceedings in Informatics (LIPIcs) |
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Volume | 244 |
ISSN | 1868-8969 |
Symposium | 30th Annual European Symposium on Algorithms |
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Abbreviated title | ESA 2022 |
Country/Territory | Germany |
City | Berlin/Potsdam |
Period | 5/09/22 → 7/09/22 |
Internet address |
Research output: Contribution to journal › Article › Academic › peer-review
Research output: Working paper / Preprint › Preprint