2026-07-22T11:08:04Z https://drops.dagstuhl.de/oai

oai:drops-oai.dagstuhl.de:16949 2024-03-06T10:58:47Z ddc:004 open_access

Bounding and Computing Obstacle Numbers of Graphs Balko, Martin Chaplick, Steven Ganian, Robert Gupta, Siddharth Hoffmann, Michael Valtr, Pavel Wolff, Alexander Obstacle representation Obstacle number Visibility NP-hardness FPT 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. Schloss Dagstuhl – Leibniz-Zentrum für Informatik Martin Balko and Steven Chaplick and Robert Ganian and Siddharth Gupta and Michael Hoffmann and Pavel Valtr and Alexander Wolff 2022 Is Part Of LIPIcs, Volume 244, 30th Annual European Symposium on Algorithms (ESA 2022) InProceedings Text doc-type:ResearchArticle publishedVersion application/pdf doi:10.4230/LIPIcs.ESA.2022.11 urn:nbn:de:0030-drops-169495 https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.ESA.2022.11 eng https://creativecommons.org/licenses/by/4.0/legalcode