# MAXIMUM CLIQUE and MINIMUM CLIQUE PARTITION in Visibility Graphs

@inproceedings{Eidenbenz2000MAXIMUMCA, title={MAXIMUM CLIQUE and MINIMUM CLIQUE PARTITION in Visibility Graphs}, author={Stephan Johannes Eidenbenz and Christoph Stamm}, booktitle={IFIP TCS}, year={2000} }

In an alternative approach to "characterizing" the graph class of visibility graphs of simple polygons, we study the problem of finding a maximum clique in the visibility graph of a simple polygon with n vertices. We show that this problem is very hard, if the input polygons are allowed to contain holes: a gap-preserving reduction from the maximum clique problem on general graphs implies that no polynomial time algorithm can achieve an approximation ratio of n1/8-Ɛ/4 for any Ɛ > 0, unless NP…

## 16 Citations

### Computing the maximum clique in the visibility graph of a simple polygon

- Mathematics, Computer ScienceJ. Discrete Algorithms
- 2007

### Unsolved Problems in Visibility Graph Theory

- Mathematics
- 2009

The visibility graph is a fundamental structure studied in the field of computational geometry, and pose some special challenges [12, 26]. Apart from theoretical interests, visibility graphs has…

### Colouring polygon visibility graphs and their generalizations

- Mathematics, Computer ScienceSoCG
- 2021

It is proved that every curve pseudo-visibility graph with clique number ω has chromatic number at most 3 ·4ω−1.

### Estimating the Maximum Hidden Vertex Set in Polygons

- Computer Science2008 International Conference on Computational Sciences and Its Applications
- 2008

It is concluded that on average the maximum number of hidden vertices in a simple polygon (arbitrary or orthogonal) with n vertices is n/4, and the best approximate algorithm is the Simulated Annealing metaheuristic.

### Applications of visibility space in polygon search problems

- Computer Science, Mathematics
- 2005

This thesis investigates the two-guard room search problem, where two guards cooperate in finding an intruder by maintaining mutual visibility, and extensively employ the visibility diagram that represents mutual visibility information for each pair of boundary points.

### On the Chromatic Number of Disjointness Graphs of Curves

- MathematicsSoCG
- 2019

The construction showing the tightness of the last result settles a 25 years old problem: it yields that there exist K_k-free disjointness graphs of x-monotone curves such that any proper coloring of them uses at least $\Omega(k^{4})$ colors.

### Minimum Hidden Guarding of Histogram Polygons

- Computer Science, MathematicsArXiv
- 2017

A linear time exact algorithm for finding minimum hidden guard set in histogram polygons under orthogonal visibility is proposed and it is allowed that guards place everywhere in the polygon.

### Guarding Path Polygons with Orthogonal Visibility

- Computer Science, MathematicsArXiv
- 2017

The problem of finding the minimum number of guards for simple polygon with general visibility (minimum guard set) becomes linear-time solvable in orthogonal path polygons and guards can be placed everywhere in the interior and boundary of polygon.

### Unsolved problems in visibility graphs of points, segments, and polygons

- MathematicsACM Comput. Surv.
- 2013

Open problems and conjectures on visibility graphs of points, segments, and polygons along with necessary backgrounds for understanding them are presented.

### (In-)Approximability of visibility problems on polygons and terrains

- Mathematics
- 2000

Visibility problems appear in a variety of applicative backgrounds. While the traditional "art gallery" problem, which consists of guarding a given floor plan of an art gallery by a minimum number of…

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