### 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) |
---|---|

Pages (from-to) | 347-365 |

Number of pages | 19 |

Journal | Discrete and Computational Geometry |

Volume | 12 |

Issue number | 1 |

DOIs | |

State | Published - Dec 1994 |

### Fingerprint

### ASJC Scopus subject areas

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

### Cite this

*Discrete and Computational Geometry*,

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

**Can visibility graphs Be represented compactly?** / Agarwal, P. K.; Alon, N.; Aronov, B.; Suri, S.

Research output: Contribution to journal › Article

*Discrete and Computational Geometry*, vol. 12, no. 1, pp. 347-365. https://doi.org/10.1007/BF02574385

}

TY - JOUR

T1 - Can visibility graphs Be represented compactly?

AU - Agarwal, P. K.

AU - Alon, N.

AU - Aronov, B.

AU - Suri, S.

PY - 1994/12

Y1 - 1994/12

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={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/log2 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(nlog3 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={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/log2 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(nlog3 n), and that there are simple polygons whose visibility graphs require a clique cover of size Ω(n log n).

UR - http://www.scopus.com/inward/record.url?scp=51249162160&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=51249162160&partnerID=8YFLogxK

U2 - 10.1007/BF02574385

DO - 10.1007/BF02574385

M3 - Article

VL - 12

SP - 347

EP - 365

JO - Discrete and Computational Geometry

JF - Discrete and Computational Geometry

SN - 0179-5376

IS - 1

ER -