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 ∑ ki=1 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/log2 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(nlog3 n), and that there are simple polygons whose visibility graphs require a clique cover of size Ω(n logn).
Article PDF
Similar content being viewed by others
References
P. Agarwal, M. Sharir, and S. Toledo, Applications of parametric searching in geometric optimization,J. Algorithms (1993), to appear.
A. Aggarwal and S. Suri, The biggest diagonal in a simple polygon,Inform. Process. Lett. 35 (1990), 13–18.
N. Alon and J. H. Spencer,The Probabilistic Method, Wiley, New York, 1991.
B. Chazelle, A polygon cutting theorem,Proc. 23rd IEEE Symp. on Foundations of Computer Science, 1982, pp. 339–349.
B. Chazelle, Lower bounds on the complexity of polytope range searching,J. Amer. Math. Soc.,2 (1989), 637–666.
B. Chazelle, Lower bounds for the orthogonal range searching: II. The arithmetic model,J Assoc. Comput. Mach. 37 (1990), 439–463.
B. Chazelle, H. Edelsbrunner, L. Guibas, and M. Sharir, Algorithms for bichromatic line segment problems and polyhedral terrains.Algorithmica 11 (1994), 116–132.
B. Chazelle and L. Guibas, Visibility and intersection problems in plane geometry,Discrete Comput. Geom. 4 (1989), 551–589.
B. Chazelle and B. Rosenberg, The complexity of computing partial sums off-line,Internat. J. Comput. Geom. Appl. 1 (1991), 33–46.
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.
M. L. Fredman, A lower bound on the complexity of orthogonal range queries,J. Assoc. Comput. Mach. 28 (1981), 696–705.
S. Ghosh and D. Mount, An output-sensitive algorithm for computing visibility graphs,SIAM J. Comput. 20 (1991), 888–910.
G. Hardy and E. Wright,An Introduction to the Theory of Numbers, Oxford University Press, London 1959.
G. Katona and E. Szemerédi, On a problem in a graph theory,Studia Sci. Math. Hungar. 2 (1967), 23–28.
J. Singer, A theorem in finite projective geometry and some applications to number theory,Trans. Amer. Math. Soc. 43 (1938), 377–385.
Z. Tuzua, Covering of graphs by complete bipartite subgraphs: complexity of 0–1 matrices,Combinatorica 4 (1984), 111–116.
E. Welzl, Constructing the visibility graph forn line segments inO(n 2) time,Inform. Process. Lett. 20 (1985), 167–171.
Author information
Authors and Affiliations
Additional information
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.
Rights and permissions
About this article
Cite this article
Agarwal, P.K., Alon, N., Aronov, B. et al. Can visibility graphs Be represented compactly?. Discrete Comput Geom 12, 347–365 (1994). https://doi.org/10.1007/BF02574385
Received:
Revised:
Published:
Issue Date:
DOI: https://doi.org/10.1007/BF02574385