Can visibility graphs Be represented compactly? Authors P. K. Agarwal Department of Computer Science Duke University N. Alon Department of Mathematics, Raymond and Beverly Sackler Faculty of Exact Sciences Tel Aviv University Bell Communications Research B. Aronov Computer Science Department Polytechnic University, Six Metro Tech Center S. Suri Bell Communications Research 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.

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