article

Can visibility graphs Be represented compactly?

Authors:

S. SuriAuthors Info & Claims

Discrete & Computational Geometry, Volume 12, Issue 3

Pages 347 - 365

https://doi.org/10.1007/BF02574385

Published: 01 December 1994 Publication History

Abstract

We consider the problem of representing the visibility graph of line segments as a union of cliques and bipartite cliques. Given a graphG, a familyG={G1,G2,...,Gk} is called aclique cover ofG if (i) eachGi is a clique or a bipartite clique, and (ii) the union ofGi isG. The size of the clique coverG is defined as i=1kni, whereni is the number of vertices inGi. Our main result is that there are visibility graphs ofn nonintersecting line segments in the plane whose smallest clique cover has size Ω(n2/log2n). An upper bound ofO(n2/logn) on the clique cover follows from a well-known result in extremal graph theory. On the other hand, we show that the visibility graph of a simple polygon always admits a clique cover of sizeO(nlog3n), and that there are simple polygons whose visibility graphs require a clique cover of size Ω(n logn).

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Cited By

Mrazović R(2016)Extractors in Paley graphsEuropean Journal of Combinatorics10.1016/j.ejc.2015.12.00954:C(154-162)Online publication date: 1-May-2016
https://dl.acm.org/doi/10.1016/j.ejc.2015.12.009
Friedrich THercher C(2015)On the kernel size of clique cover reductions for random intersection graphsJournal of Discrete Algorithms10.1016/j.jda.2015.05.01434:C(128-136)Online publication date: 1-Sep-2015
https://dl.acm.org/doi/10.1016/j.jda.2015.05.014
Cygan MKratsch SPilipczuk MPilipczuk MWahlström M(2014)Clique Cover and Graph SeparationACM Transactions on Computation Theory10.1145/25944396:2(1-19)Online publication date: 1-May-2014
https://dl.acm.org/doi/10.1145/2594439
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Can visibility graphs Be represented compactly?
1. Mathematics of computing
  1. Discrete mathematics
    1. Graph theory
2. Theory of computation
  1. Randomness, geometry and discrete structures

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cover image Discrete & Computational Geometry

Discrete & Computational Geometry Volume 12, Issue 3

September 1994

142 pages

ISSN:0179-5376

Issue’s Table of Contents

Copyright © Copyright © 1994 Springer-Verlag New York Inc.

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Springer-Verlag

Berlin, Heidelberg

Publication History

Published: 01 December 1994

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Cited By

Mrazović R(2016)Extractors in Paley graphsEuropean Journal of Combinatorics10.1016/j.ejc.2015.12.00954:C(154-162)Online publication date: 1-May-2016
https://dl.acm.org/doi/10.1016/j.ejc.2015.12.009
Friedrich THercher C(2015)On the kernel size of clique cover reductions for random intersection graphsJournal of Discrete Algorithms10.1016/j.jda.2015.05.01434:C(128-136)Online publication date: 1-Sep-2015
https://dl.acm.org/doi/10.1016/j.jda.2015.05.014
Cygan MKratsch SPilipczuk MPilipczuk MWahlström M(2014)Clique Cover and Graph SeparationACM Transactions on Computation Theory10.1145/25944396:2(1-19)Online publication date: 1-May-2014
https://dl.acm.org/doi/10.1145/2594439
Ghosh SGoswami P(2013)Unsolved problems in visibility graphs of points, segments, and polygonsACM Computing Surveys10.1145/2543581.254358946:2(1-29)Online publication date: 27-Dec-2013
https://dl.acm.org/doi/10.1145/2543581.2543589
Ennis JFayle CEnnis D(2012)Assignment-minimum clique coveringsACM Journal of Experimental Algorithmics10.1145/2133803.227559617(1.1-1.17)Online publication date: 19-Jul-2012
https://dl.acm.org/doi/10.1145/2133803.2275596
Christofides DMarkstrom K(2012)Random Latin square graphsRandom Structures & Algorithms10.1002/rsa.2039041:1(47-65)Online publication date: 1-Aug-2012
https://dl.acm.org/doi/10.1002/rsa.20390
Gramm JGuo JHüffner FNiedermeier R(2009)Data reduction and exact algorithms for clique coverACM Journal of Experimental Algorithmics10.1145/1412228.141223613(2.2-2.15)Online publication date: 23-Feb-2009
https://dl.acm.org/doi/10.1145/1412228.1412236
Kumar RTomkins AVee ENajork MBroder AChakrabarti S(2008)Connectivity structure of bipartite graphs via the KNC-plotProceedings of the 2008 International Conference on Web Search and Data Mining10.1145/1341531.1341550(129-138)Online publication date: 11-Feb-2008
https://dl.acm.org/doi/10.1145/1341531.1341550
Franzblau DXenakis G(2008)An algorithm for the difference between set coversDiscrete Applied Mathematics10.1016/j.dam.2007.10.031156:10(1623-1632)Online publication date: 1-May-2008
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Agarwal PBrady DMatoušek J(2006)Segmenting object space by geometric reference structuresACM Transactions on Sensor Networks10.1145/1218556.12185572:4(455-465)Online publication date: 1-Nov-2006
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