TY - GEN
T1 - Can visibility graphs be represented compactly?
AU - Agarwal, Pankaj K.
AU - Alon, Noga
AU - Aronov, Boris
AU - Suri, Subhash
PY - 1993
Y1 - 1993
N2 - We consider the problem of representing the visibility graph of line segments as a union of cliques and bipartite cliques. Given a graph G, a family G = {G1, G2,..., Gk} is called a clique cover of G if (i) each Gi is a clique or a bipartite clique, and (ii) the union of Gi is G. The size of the clique cover G is defined as Σik=1 ni, where ni is the number of vertices in Gi. Our main result is that there exist visibility graphs of n nonintersecting line segments in the plane whose smallest clique cover has size Ω(n2/log2 n). An upper bound of O(n2/log n) 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 a size O(n log3 n), and that there are simple polygons whose visibility graphs require a clique cover of size Ω(n log n).
AB - We consider the problem of representing the visibility graph of line segments as a union of cliques and bipartite cliques. Given a graph G, a family G = {G1, G2,..., Gk} is called a clique cover of G if (i) each Gi is a clique or a bipartite clique, and (ii) the union of Gi is G. The size of the clique cover G is defined as Σik=1 ni, where ni is the number of vertices in Gi. Our main result is that there exist visibility graphs of n nonintersecting line segments in the plane whose smallest clique cover has size Ω(n2/log2 n). An upper bound of O(n2/log n) 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 a size O(n log3 n), and that there are simple polygons whose visibility graphs require a clique cover of size Ω(n log n).
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U2 - 10.1145/160985.161160
DO - 10.1145/160985.161160
M3 - Conference contribution
AN - SCOPUS:0027802232
SN - 0897915828
SN - 9780897915823
T3 - Proceedings of the 9th Annual Symposium on Computational Geometry
SP - 338
EP - 347
BT - Proceedings of the 9th Annual Symposium on Computational Geometry
PB - Publ by ACM
T2 - Proceedings of the 9th Annual Symposium on Computational Geometry
Y2 - 19 May 1993 through 21 May 1993
ER -