Discrete & Computational Geometry
, Volume 12, Issue 3, pp 347365
First online:
Can visibility graphs Be represented compactly?
 P. K. AgarwalAffiliated withDepartment of Computer Science, Duke University
 , N. AlonAffiliated withDepartment of Mathematics, Raymond and Beverly Sackler Faculty of Exact Sciences, Tel Aviv UniversityBell Communications Research
 , B. AronovAffiliated withComputer Science Department, Polytechnic University, Six Metro Tech Center
 , S. SuriAffiliated withBell Communications Research
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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={G _{1},G _{2},...,G _{ k }} is called aclique cover ofG if (i) eachG _{ i } is a clique or a bipartite clique, and (ii) the union ofG _{ i } isG. The size of the clique coverG is defined as ∑ _{i=1} ^{k} n _{ i }, wheren _{ i } is the number of vertices inG _{ i }. Our main result is that there are visibility graphs ofn nonintersecting line segments in the plane whose smallest clique cover has size Ω(n ^{2}/log^{2} n). An upper bound ofO(n ^{2}/logn) on the clique cover follows from a wellknown 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(nlog^{3} n), and that there are simple polygons whose visibility graphs require a clique cover of size Ω(n logn).
 Title
 Can visibility graphs Be represented compactly?
 Journal

Discrete & Computational Geometry
Volume 12, Issue 3 , pp 347365
 Cover Date
 199409
 DOI
 10.1007/BF02574385
 Print ISSN
 01795376
 Online ISSN
 14320444
 Publisher
 Springer New York
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 Authors

 P. K. Agarwal ^{(1)}
 N. Alon ^{(2)} ^{(3)}
 B. Aronov ^{(4)}
 S. Suri ^{(3)}
 Author Affiliations

 1. Department of Computer Science, Duke University, Box 90129, 277080129, Durham, NC, USA
 2. Department of Mathematics, Raymond and Beverly Sackler Faculty of Exact Sciences, Tel Aviv University, Tel Aviv, Israel
 3. Bell Communications Research, 445 South Street, 07960, Morristown, NJ, USA
 4. Computer Science Department, Polytechnic University, Six Metro Tech Center, 11201, Brooklyn, NY, USA