Drawing Graphs

2017 ◽  
Author(s):  
Arthur Benjamin ◽  
Gary Chartrand ◽  
Ping Zhang
Keyword(s):  
Planar Graph ◽  
Crossing Number ◽  
Art Gallery ◽  
Art Gallery Problem ◽  
Underlying Theory ◽  
Euler Identity

This chapter considers the concept of planar graph and its underlying theory. It begins with a discussion of the Three Houses and Three Utilities Problem and how it can be represented by a graph. It shows that solving the Three Houses and Three Utilities Problem is equivalent to the problem of determining whether the graph that represents it can be drawn in the plane without any of its edges crossing. It then describes the Euler Identity and the Euler Polyhedron Formula, along with the proposition that every planar graph contains a vertex of degree 5 or less. It also examines Kuratowski's Theorem, introduced by the Polish topologist Kazimierz Kuratowski, and the problem of crossing number. Finally, it provides an overview of the Art Gallery Problem, Wagner's Conjecture, and the Brick-Factory Problem.

Tight bounds for the rectangular art gallery problem

1992 ◽  
pp. 105-112 ◽  
Author(s):  
J. Czyzowicz ◽  
E. Rivera-Campo ◽  
N. Santoro ◽  
J. Urrutia ◽  
J. Zaks
Keyword(s):  
Art Gallery ◽  
Art Gallery Problem

POLYGON DECOMPOSITION AND THE ORTHOGONAL ART GALLERY PROBLEM

2007 ◽  
Vol 17 (02) ◽  
pp. 105-138 ◽  
Author(s):  
CHRIS WORMAN ◽  
J. MARK KEIL
Keyword(s):  
Polynomial Time ◽  
Art Gallery ◽  
Minimum Cardinality ◽  
Decomposition Problem ◽  
Art Gallery Problem ◽  
Orthogonal Polygon ◽  
Polygon Decomposition

A decomposition of a polygon P is a set of polygons whose geometric union is exactly P. We study a polygon decomposition problem that is equivalent to the Orthogonal Art Gallery problem. Two points are r-visible if the orthogonal bounding rectangle for p and q lies within P. A polygon P is an r-star if there exists a point k ∈ P such that for each point q ∈ P, q is r-visible from k. In this problem we seek a minimum cardinality decomposition of a polygon into r-stars. We show how to compute the minimum r-star cover of an orthogonal polygon in polynomial time.

The Art Gallery Problem is ∃ℝ-complete

2022 ◽  
Vol 69 (1) ◽  
pp. 1-70
Author(s):  
Mikkel Abrahamsen ◽  
Anna Adamaszek ◽  
Tillmann Miltzow
Keyword(s):  
Simple Polygon ◽  
Art Gallery ◽  
Arbitrary Degree ◽  
Minimum Cardinality ◽  
Art Gallery Problem ◽  
Algebraic Set ◽  
Classic Problem ◽  
Finite Set ◽  
Roots Of Polynomials ◽  
System Of Polynomial Equations

The Art Gallery Problem (AGP) is a classic problem in computational geometry, introduced in 1973 by Victor Klee. Given a simple polygon 풫 and an integer k , the goal is to decide if there exists a set G of k guards within 풫 such that every point p ∈ 풫 is seen by at least one guard g ∈ G . Each guard corresponds to a point in the polygon 풫, and we say that a guard g sees a point p if the line segment pg is contained in 풫. We prove that the AGP is ∃ ℝ-complete, implying that (1) any system of polynomial equations over the real numbers can be encoded as an instance of the AGP, and (2) the AGP is not in the complexity class NP unless NP = ∃ ℝ. As a corollary of our construction, we prove that for any real algebraic number α, there is an instance of the AGP where one of the coordinates of the guards equals α in any guard set of minimum cardinality. That rules out many natural geometric approaches to the problem, as it shows that any approach based on constructing a finite set of candidate points for placing guards has to include points with coordinates being roots of polynomials with arbitrary degree. As an illustration of our techniques, we show that for every compact semi-algebraic set S ⊆ [0, 1] 2 , there exists a polygon with corners at rational coordinates such that for every p ∈ [0, 1] 2 , there is a set of guards of minimum cardinality containing p if and only if p ∈ S . In the ∃ ℝ-hardness proof for the AGP, we introduce a new ∃ ℝ-complete problem ETR-INV. We believe that this problem is of independent interest, as it has already been used to obtain ∃ ℝ-hardness proofs for other problems.

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