Visibility Graphs and Oriented Matroids

Abello,; Kumar, Anita B.

doi:10.1007/s00454-002-2881-6

Cited by 21 publications

(26 citation statements)

References 18 publications

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“…Note that visibility graphs of polygons are well studied (see [1] for example); even here, it is an open problem whether the chromatic number is bounded by the clique number. The main open problem that has been studied here is whether visibility graphs of polygons can be recognised in polynomial time (see [3] for example).…”

Section: Conjecture 1 Visibility Graphs Are χ -Bounded That Is Is mentioning

confidence: 99%

On the Chromatic Number of the Visibility Graph of a Set of Points in the Plane

Kára

Pór

Wood

2005

Discrete Comput Geom

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The visibility graph V(P) of a point set P ⊆ R 2 has vertex set P, such that two points v, w ∈ P are adjacent whenever there is no other point in P on the line segment between v and w. We study the chromatic number of V(P). We characterise the 2-and 3-chromatic visibility graphs. It is an open problem whether the chromatic number of a visibility graph is bounded by its clique number. Our main result is a super-polynomial lower bound on the chromatic number (in terms of the clique number).

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Section: Conjecture 1 Visibility Graphs Are χ -Bounded That Is Is mentioning

confidence: 99%

On the Chromatic Number of the Visibility Graph of a Set of Points in the Plane

Kára

Pór

Wood

2005

Discrete Comput Geom

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“…We feel that studying these graph-theoretic problems with additional information may lead to solving the long standing open problem in geometric graph theory of recognizing and characterizing visibility graphs of simple polygons [1,2,11,21].…”

Section: Discussionmentioning

confidence: 98%

“…The problem of actually constructing such a P is called the visibility graph reconstruction problem. The visibility graph recognition and reconstruction problems are long-standing open problems in geometric graph theory with only partial results achieved to date [1,4,5,[9][10][11]21].…”

Section: Introductionmentioning

confidence: 99%

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

Ghosh

Shermer

Bhattacharya

et al. 2007

Journal of Discrete Algorithms

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In this paper, we present an algorithm for computing the maximum clique in the visibility graph G of a simple polygon P in O(n 2 e) time, where n and e are number of vertices and edges of G respectively. We also present an O(ne) time algorithm for computing the maximum hidden vertex set in the visibility graph G of a convex fan P . We assume in both algorithms that the Hamiltonian cycle in G that corresponds to the boundary of P is given as an input along with G.

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“…For instance, spiral polygons [14], and tower polygons [7] (also called funnel polygons), can be reconstructed in linear time, and each consists of one and two reflex chains, respectively. 2-spirals can also be reconstructed in polynomial time [3], as can a more general class of visibility graphs related to 3-matroids [4].…”

Section: Special Classesmentioning

confidence: 99%

Reconstructing Generalized Staircase Polygons with Uniform Step Length

Sitchinava¹,

Strash²

2018

JGAA

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Visibility graph reconstruction, which asks us to construct a polygon that has a given visibility graph, is a fundamental problem with unknown complexity (although visibility graph recognition is known to be in PSPACE). We show that two classes of uniform step length polygons can be reconstructed efficiently by finding and removing rectangles formed between consecutive convex boundary vertices called tabs. In particular, we give an O(n 2 m)-time reconstruction algorithm for orthogonally convex polygons, where n and m are the number of vertices and edges in the visibility graph, respectively. We further show that reconstructing a monotone chain of staircases (a histogram) is fixed-parameter tractable, when parameterized on the number of tabs, and polynomially solvable in time O(n 2 m) under reasonable alignment restrictions.

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Visibility Graphs and Oriented Matroids

Cited by 21 publications

References 18 publications

On the Chromatic Number of the Visibility Graph of a Set of Points in the Plane

On the Chromatic Number of the Visibility Graph of a Set of Points in the Plane

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

Reconstructing Generalized Staircase Polygons with Uniform Step Length

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