### Abstract

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 = {G_{1}, G_{2},..., G_{k}} is called a clique cover of G if (i) each G_{i} is a clique or a bipartite clique, and (ii) the union of G_{i} is G. The size of the clique cover G is defined as Σ_{i}^{k}=1 n_{i}, where n_{i} is the number of vertices in G_{i}. Our main result is that there exist visibility graphs of n nonintersecting line segments in the plane whose smallest clique cover has size Ω(n^{2}/log^{2} n). An upper bound of O(n^{2}/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 log^{3} n), and that there are simple polygons whose visibility graphs require a clique cover of size Ω(n log n).

Original language | English (US) |
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Title of host publication | Proceedings of the 9th Annual Symposium on Computational Geometry |

Publisher | Publ by ACM |

Pages | 338-347 |

Number of pages | 10 |

ISBN (Print) | 0897915828, 9780897915823 |

DOIs | |

State | Published - 1993 |

Externally published | Yes |

Event | Proceedings of the 9th Annual Symposium on Computational Geometry - San Diego, CA, USA Duration: May 19 1993 → May 21 1993 |

### Publication series

Name | Proceedings of the 9th Annual Symposium on Computational Geometry |
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### Other

Other | Proceedings of the 9th Annual Symposium on Computational Geometry |
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City | San Diego, CA, USA |

Period | 5/19/93 → 5/21/93 |

### All Science Journal Classification (ASJC) codes

- Engineering(all)

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## Cite this

*Proceedings of the 9th Annual Symposium on Computational Geometry*(pp. 338-347). (Proceedings of the 9th Annual Symposium on Computational Geometry). Publ by ACM. https://doi.org/10.1145/160985.161160