![]() Noga Alon, Raphael Yuster, and Uri Zwick.Geometric separation and exact solutions for the parameterized independent set problem on disk graphs. In Surveys in Discrete and Computational Geometry: Twenty Years Later. State of the union (of geometric objects). Independent set of intersection graphs of convex objects in 2D. Approximation and inapproximability results for maximum clique of disc graphs in high dimensions. In Proceedings of the 17th Canadian Conference on Computational Geometry (CCCG’05). Approximation algorithms for maximum cliques in 3D unit-disk graphs. Indeed, we show that, for all those problems, there is a constant ratio of approximation that cannot be attained even in time 2 n 1−ɛ, unless the Exponential Time Hypothesis fails. In stark contrast, M AXIMUM C LIQUE on ball graphs and unit 4-dimensional ball graphs, as well as intersection graphs of filled ellipses (even close to unit disks) or filled triangles is unlikely to have such algorithms. M AX C LIQUE on unit ball graphs is equivalent to finding, given a collection of points in R 3, a maximum subset of points with diameter at most some fixed value. ![]() This, in combination with our structural results, yields a randomized EPTAS for M AX C LIQUE on disk and unit ball graphs. We then obtain a randomized EPTAS for computing the independence number on graphs having no disjoint union of two odd cycles as an induced subgraph, bounded VC-dimension, and linear independence number. From that fact and existing results, we derive a simple QPTAS and a subexponential algorithm running in time 2 Õ( n 2/3) for M AXIMUM C LIQUE on disk and unit ball graphs. We show that the disjoint union of two odd cycles is never the complement of a disk graph nor of a unit (3-dimensional) ball graph. Since then, it has been an intriguing open question whether or not tractability can be extended to general disk graphs. Almost three decades ago, an elegant polynomial-time algorithm was found for M AXIMUM C LIQUE on unit disk graphs. A (unit) disk graph is the intersection graph of closed (unit) disks in the plane. Matsui, Tomomi (2000), "Approximation Algorithms for Maximum Independent Set Problems and Fractional Coloring Problems on Unit Disk Graphs", Lecture Notes in Computer Science, Lecture Notes in Computer Science 1763: 194–200, doi: 10.1007/978-5-7_16, ISBN 978-1-7.(1994), Geometry based heuristics for unit disk graphs. Müller, Tobias (2011), "Sphere and dot product representations of graphs", Proceedings of the Twenty-Seventh Annual Symposium on Computational Geometry (SoCG'11), June 13–15, 2011, Paris, France, pp. 308–314. An example of a graph that is not a unit disk graph is the star \displaystyle-free graphs", Discrete Applied Mathematics 284: 53–60, doi: 10.1016/j.dam.2020.03.024 Unit disk graphs may be formed in a different way from a collection of equal-radius circles, by connecting two circles with an edge whenever one circle contains the center of the other circle.Įvery induced subgraph of a unit disk graph is also a unit disk graph.These graphs have a vertex for each circle or disk, and an edge connecting each pair of circles or disks that have a nonempty intersection. ![]()
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