### 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 ∑_{ i=1}^{ k} n_{ i}, where n_{ i} is the number of vertices in G_{ i} . Our main result is that there are 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 of size O(nlog^{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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Pages (from-to) | 347-365 |

Number of pages | 19 |

Journal | Discrete & Computational Geometry |

Volume | 12 |

Issue number | 1 |

DOIs | |

State | Published - Dec 1 1994 |

Externally published | Yes |

### All Science Journal Classification (ASJC) codes

- Theoretical Computer Science
- Geometry and Topology
- Discrete Mathematics and Combinatorics
- Computational Theory and Mathematics

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## Cite this

*Discrete & Computational Geometry*,

*12*(1), 347-365. https://doi.org/10.1007/BF02574385