Can visibility graphs Be represented compactly? P. K. Agarwal N. Alon B. Aronov S. Suri Article

First Online: 01 September 1994 Received: 16 March 1993 Revised: 07 February 1994 DOI :
10.1007/BF02574385

Cite this article as: Agarwal, P.K., Alon, N., Aronov, B. et al. Discrete Comput Geom (1994) 12: 347. doi:10.1007/BF02574385 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 ={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 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 (n log^{3} n ), and that there are simple polygons whose visibility graphs require a clique cover of size Ω(n logn ).

The work by the first author was supported by National Science Foundation Grant CCR-91-06514. The work by the second author was supported by a USA-Israeli BSF grant. The work by the third author was supported by National Science Foundation Grant CCR-92-11541.

References 1.

P. Agarwal, M. Sharir, and S. Toledo, Applications of parametric searching in geometric optimization,J. Algorithms (1993), to appear.

2.

A. Aggarwal and S. Suri, The biggest diagonal in a simple polygon,

Inform. Process. Lett.
35 (1990), 13–18.

MATH MathSciNet CrossRef 3.

N. Alon and J. H. Spencer,The Probabilistic Method , Wiley, New York, 1991.

4.

B. Chazelle, A polygon cutting theorem,Proc. 23rd IEEE Symp. on Foundations of Computer Science , 1982, pp. 339–349.

5.

B. Chazelle, Lower bounds on the complexity of polytope range searching,

J. Amer. Math. Soc. ,

2 (1989), 637–666.

MATH MathSciNet CrossRef 6.

B. Chazelle, Lower bounds for the orthogonal range searching: II. The arithmetic model,

J Assoc. Comput. Mach.
37 (1990), 439–463.

MATH MathSciNet CrossRef 7.

B. Chazelle, H. Edelsbrunner, L. Guibas, and M. Sharir, Algorithms for bichromatic line segment problems and polyhedral terrains.

Algorithmica
11 (1994), 116–132.

MATH MathSciNet CrossRef 8.

B. Chazelle and L. Guibas, Visibility and intersection problems in plane geometry,

Discrete Comput. Geom.
4 (1989), 551–589.

MATH MathSciNet CrossRef 9.

B. Chazelle and B. Rosenberg, The complexity of computing partial sums off-line,

Internat. J. Comput. Geom. Appl.
1 (1991), 33–46.

MATH MathSciNet CrossRef 10.

P. Erdős and P. Turán, On a problem of Sidon in additive number theory, and on some related problems,

Proc. London Math. Soc.
16 (1941), 212–215.

CrossRef 11.

M. L. Fredman, A lower bound on the complexity of orthogonal range queries,

J. Assoc. Comput. Mach.
28 (1981), 696–705.

MATH MathSciNet CrossRef 12.

S. Ghosh and D. Mount, An output-sensitive algorithm for computing visibility graphs,

SIAM J. Comput.
20 (1991), 888–910.

MATH MathSciNet CrossRef 13.

G. Hardy and E. Wright,An Introduction to the Theory of Numbers , Oxford University Press, London 1959.

14.

G. Katona and E. Szemerédi, On a problem in a graph theory,

Studia Sci. Math. Hungar.
2 (1967), 23–28.

MATH MathSciNet 15.

J. Singer, A theorem in finite projective geometry and some applications to number theory,

Trans. Amer. Math. Soc.
43 (1938), 377–385.

MathSciNet CrossRef 16.

Z. Tuzua, Covering of graphs by complete bipartite subgraphs: complexity of 0–1 matrices,

Combinatorica
4 (1984), 111–116.

MathSciNet CrossRef 17.

E. Welzl, Constructing the visibility graph for

n line segments in

O(n
^{2} ) time,

Inform. Process. Lett.
20 (1985), 167–171.

MATH MathSciNet CrossRef © Springer-Verlag New York Inc. 1994

Authors and Affiliations P. K. Agarwal N. Alon B. Aronov S. Suri 1. Department of Computer Science Duke University Durham USA 2. Department of Mathematics, Raymond and Beverly Sackler Faculty of Exact Sciences Tel Aviv University Tel Aviv Israel 3. Computer Science Department Polytechnic University, Six Metro Tech Center Brooklyn USA 4. Bell Communications Research Morristown USA