title={MAXIMUM CLIQUE and MINIMUM CLIQUE PARTITION in Visibility Graphs},
  author={Stephan Johannes Eidenbenz and Christoph Stamm},
  booktitle={IFIP TCS},
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… 

Unsolved Problems in Visibility Graph Theory

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

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

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

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

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

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

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

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

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



An Approximation Algorithm for Minimum Convex Cover with Logarithmic Performance Guarantee

The problem Minimum Convex Cover is APX-hard, i.e., there exists a constant δ > 0 such that no polynomial-time algorithm can achieve an approximation ratio of 1 + δ, and this result is obtained by analyzing and slightly modifying an already existing reduction.

Inapproximability of some art gallery problems

We prove that the three art gallery problems Vertex Guard Edge Guard and Point Guard for simple polygons with holes cannot be approximated by any polynomial time algorithm with a ratio of  lnn for

How Many People Can Hide in a Terrain?

The problem of placing a maximum number of hiding people is almost as hard to approximate as the Maximum Clique problem, i.e., it cannot be approximated by any polynomial-time algorithm with an approximation ratio of nƐ for some Ɛ > 0, unless P = NP.

Negative Results on Characterizing Visibility Graphs

On recognizing and characterizing visibility graphs of simple polygons

Three necessary conditions for recognizing visibility graphs of simple polygons are established and conjecture that these conditions are sufficient and it is shown that visibility graph do not posses the characteristics of several special classes of graphs.

Topologically sweeping an arrangement

The advantages of sweeping with a topological line that is not necessarily straight are demonstrated and an arrangement of n lines in the plane can be swept over in O ( n 2 ) time and O(n) space by a such a line.

Open Problems in the Combinatorics of Visibility and Illumination

1991 Primary 52C99. Visibility, illumination, visibility graphs, computational geometry. Supported by NSF grant CCR-9421670. URL: . The \art gallery theorem," that 3 guards su ce and are sometimes

Searching for empty convex polygons

It is shown that finding empty triangles is related to the problem of determining pairs of vertices that see each other in a star-shaped polygon, and a linear time algorithm for this problem which is of independent interest yields an optimal algorithm for finding all empty triangles.