# Can visibility graphs be represented compactly?

Published

Journal Article

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).

### Full Text

### Duke Authors

### Cited Authors

- Agarwal, PK; Alon, N; Aronov, B; Suri, S

### Published Date

- January 1, 1993

### Published In

- Proceedings of the 9th Annual Symposium on Computational Geometry

### Start / End Page

- 338 - 347

### Digital Object Identifier (DOI)

- 10.1145/160985.161160

### Citation Source

- Scopus