Visibility Graphs and Oriented Matroids

Abstract. We describe a set of necessary conditions for a given graph to be the visibility graph of a simple polygon. For every graph satisfying these conditions we show that a uniform rank 3 oriented matroid can be constructed in polynomial time, which if affinely coordinatizable yields a simple polygon whose visibility graph is isomorphic to the given graph.     

Visibility graphs of staircase polygons and the weak Bruhat order,I: From visibility graphs to maximal chains

The recognition problem for visibility graphs of simple polygons is not known to be in NP, nor is it known to be NP-hard. It is, however, known to be inPSPACE. Further, every such visibility graph can be dismantled as a sequence of visibility graphs of convex fans. Any nondegenerated configuration ofn points can be associated with amaximal chain in the weak Bruhat order of the symmetric groupS _n . The visibility graph ofany simple polygon defined on this configuration is completely determined by this maximal chain via a one-to-one correspondence between maximal chains andbalanced tableaux of a certain shape. In the case of staircase polygons (special convex fans), we define a class of graphs calledpersistent graphs and show that the visibility graph of a staircase polygon is persistent. We then describe a polynomial-time algorithm that recovers a representative maximal chain in the weak Bruhat order from a given persistent graph, thus characterizing the class of persistent graphs. The question of recovering a staircase polygon from a given persistent graph, via a maximal chain, is studied in the companion paper [4]. The overall goal of both papers is to offer a characterization of visibility graphs, of convex fans. The research of J. Abello was supported by NSF Grants Nos. DCR 8603722 and DCR 8896281. This research was done while K. Kumar was at the Department of Computer Science, Texas A & M University.     

Visibility Queries and Maintenance in Simple Polygons

Abstract. In this paper we explore some novel aspects of visibility for stationary and moving points inside a simple polygon P . We provide a mechanism for expressing the visibility polygon from a point as the disjoint union of logarithmically many canonical pieces using a quadratic-space data structure. This allows us to report visibility polygons in time proportional to their size, but without the cubic space overhead of earlier methods. The same canonical decomposition can be used to determine visibility within a frustum, or to compute various attributes of the visibility polygon efficiently. By exploring the connection between visibility polygons and shortest-path trees, we obtain a kinetic algorithm that can track the visibility polygon as the viewpoint moves along polygonal paths inside P , at a polylogarithmic cost per combinatorial change in the visibility or in the flight plan of the point. The combination of the static and kinetic algorithms leads to a new static algorithm in which we can trade off space for increased overhead in the query time. As another application, we obtain an algorithm which computes the weak visibility polygon from a query segment inside P in output-sensitive time.     

Simple agents learn to find their way: An introduction on mapping polygons

This paper gives an introduction to the problem of mapping simple polygons with autonomous agents. We focus on minimalistic agents that move from vertex to vertex along straight lines inside a polygon, using their sensors to gather local data at each vertex. Our attention revolves around the question whether a given configuration of sensors and movement capabilities of the agents allows them to capture enough data in order to draw conclusions regarding the global layout of the polygon.In particular, we study the problem of reconstructing the visibility graph of a simple polygon by an agent moving either inside or on the boundary of the polygon. Our aim is to provide insight about the algorithmic challenges faced by an agent trying to map a polygon. We present an overview of techniques for solving this problem with agents that are equipped with simple sensorial capabilities. We illustrate these techniques on examples with sensors that measure angles between lines of sight or identify the previous location. We also give an overview over related problems in combinatorial geometry as well as graph exploration.     

Flipturning Polygons

   Abstract. A flipturn transforms a nonconvex simple polygon into another simple polygon by rotating a concavity 180° around the midpoint of its bounding convex hull edge. Joss and Shannon proved in 1973 that a sequence of flipturns eventually transforms any simple polygon into a convex polygon. This paper describes several new results about such flipturn sequences. We show that any orthogonal polygon is convexified after at most n-5 arbitrary flipturns, or at most

A new necessary condition for the vertex visibility graphs of simple polygons

A new necessary condition conjectured by Everett [2], which is essentially a stronger version of a necessary condition by Ghosh [3], for a graph to be the vertex visibility graph of a simple polygon is established. This work was carried out while G. Srinivasaraghavan was at the Indian Institute of Technology, Kanpur, India.     

Visibility with One Reflection

We extend the concept of the polygon visible from a source point S in a simple polygon by considering visibility with two types of reflection, specular and diffuse. In specular reflection a light ray reflects from an edge of the polygon according to the rule: the angle of incidence equals the angle of reflection. In diffuse reflection a light ray reflects from an edge of the polygon in all inward directions. Several geometric and combinatorial properties of visibility polygons under these two types of reflection are described, when at most one reflection is permitted. We show that the visibility polygon Vs(S) under specular reflection may be nonsimple, while the visibility polygon Vd(S) under diffuse reflection is always simple. We present a Θ(n ² ) worst-case bound on the combinatorial complexity of both Vs(S) and Vd(S) and describe simple O(n ² log ² n) time algorithms for constructing the sets. Received September 27, 1995, and in revised form October 24, 1997.     

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

In this paper, we present an algorithm for computing the maximum clique in the visibility graph G of a simple polygon P in O(n²e) 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.     

Isomorphism of spiral polygons

We call two polygonsisomorphic if there is a one-to-one mapping between theirpoints (not vertices) that preserves visibility. In this paper we establish necessary and sufficient conditions for two spiral polygons to be isomorphic, and give anO(n ² ) algorithm to detect such isomorphism. We also show that the continuous graph of visibility on the points of a spiral polygon is an (uncountably infinite) interval graph, and that no other polygons have this property. This research was supported by the Natural Sciences and Engineering Research Council of Canada under Research Grant Number OGP0046218 and a Post-Graduate Scholarship.     

Visibility graph approach to the analysis of ocean tidal records

By using the recent method of the visibility graph, three time series of oceanic tide level in central Argentina were investigated. The degree distributions show a rich structure; in particular the maximum is due to the main periodic oscillations at 24 hours and 12 hours and higher harmonics. The degree distributions of the residuals (obtained removing from the original signals the cyclic components) suggest that the local effects, linked with the particular coastal conditions of the sites, are discernible for the degree k < 20, while the global effects, linked with linked with the more general and common atmospheric forcing and ocean current conditions, are visible for k > 100. Although a relationship between the spectral exponent α and the exponent of the degree distribution γ of tidal signals can be recognized, this cannot be simply stated due to the very rich and complex structure of time dynamics of tides. The present study, even if still preliminary, show the importance of the visibility graph method in investigating the complex time dynamics of observational and experimental signals.     

Covering and guarding polygons using L k -sets

The Art Gallery Problem is the problem of finding a minimum number of points (called guards) in a given polygon such that every point in the polygon is visible to at least one of the guards. Chvátal [5] was the first to show that, in the worst case, [n/3] such points will suffice for any polygon of n sides. O'Rourke [15] later showed that only [n/4] guards were needed if line segments, rather than points, were allowed as guards. In this paper, we unify these results, and extend them to many other classes of guards, while using a generalization of visibility known as link-visibility. In particular, we present the following theorems:

Untangling a Polygon

   Abstract. The following problem was raised by M. Watanabe. Let P be a self-intersecting closed polygon with n vertices in general position. How manys steps does it take to untangle P , i.e., to turn it into a simple polygon, if in each step we can arbitrarily relocate one of its vertices. It is shown that in some cases one has to move all but at most O((n log n) ^2/3 ) vertices. On the other hand, every polygon P can be untangled in at most

An algorithm for constructing star-shaped drawings of plane graphs

A straight-line planar drawing of a plane graph is called a convex drawing if every facial cycle is drawn as a convex polygon. Convex drawings of graphs is a well-established aesthetic in graph drawing, however not all planar graphs admit a convex drawing. Tutte [W.T. Tutte, Convex representations of graphs, Proc. of London Math. Soc. 10 (3) (1960) 304–320] showed that every triconnected plane graph admits a convex drawing for any given boundary drawn as a convex polygon. Thomassen [C. Thomassen, Plane representations of graphs, in: Progress in Graph Theory, Academic Press, 1984, pp. 43–69] gave a necessary and sufficient condition for a biconnected plane graph with a prescribed convex boundary to have a convex drawing.In this paper, we initiate a new notion of star-shaped drawing of a plane graph as a straight-line planar drawing such that each inner facial cycle is drawn as a star-shaped polygon, and the outer facial cycle is drawn as a convex polygon. A star-shaped drawing is a natural extension of a convex drawing, and a new aesthetic criteria for drawing planar graphs in a convex way as much as possible. We give a sufficient condition for a given set A of corners of a plane graph to admit a star-shaped drawing whose concave corners are given by the corners in A, and present a linear time algorithm for constructing such a star-shaped drawing.     

Enumeration of ladder graphs

Consider N points labelled cyclically forming the vertex set of a planar convex polygon, pairs of which may be connected by bonds. Define a ladder graph to be a graph with no crossing bonds. New simple proofs are given for two results of Kirkman relating to the total number of ladder graphs with A bonds. Relations are derived for calculating the number of connected ladder graphs with A bonds, and asymptotic expressions are obtained.     

On （g, f）-Uniform Graphs

A graph G is called a (g, f)-uniform graph if for each edge of G, there is a (g, f)-factor containing it and another (g, f)-factor excluding it. In this paper a necessary and sufficient condition for a graph to be a (g, f)-uniform graph is given and some applications of this condition are discussed. In particular, some simple sufficient conditions for a graph to be an [a, b]-uniform graph are obtained for a≤b.     

Efficient visibility queries in simple polygons

We present a method of decomposing a simple polygon that allows the preprocessing of the polygon to efficiently answer visibility queries of various forms in an output sensitive manner. Using O(n³logn) preprocessing time and O(n³) space, we can, given a query point q inside or outside an n vertex polygon, recover the visibility polygon of q in O(logn+k) time, where k is the size of the visibility polygon, and recover the number of vertices visible from q in O(logn) time.

The key notion behind the decomposition is the succinct representation of visibility regions, and tight bounds on the number of such regions. These techniques are extended to handle other types of queries, such as visibility of fixed points other than the polygon vertices, and for visibility from a line segment rather than a point. Some of these results have been obtained independently by Guibas, Motwani and Raghavan [18] .     

Recognizing String Graphs Is Decidable

   Abstract. A graph is called a string graph if its vertices can be represented by continuous curves (``strings') in the plane so that two of them cross each other if and only if the corresponding vertices are adjacent. It is shown that there exists a recursive function f(n) with the property that every string graph of n vertices has a representation in which any two curves cross at most f(n) times. We obtain as a corollary that there is an algorithm for deciding whether a given graph is a string graph. This solves an old problem of Benzer (1959), Sinden (1966), and Graham (1971).     

A General Notion of Visibility Graphs

   Abstract. We define a natural class of graphs by generalizing prior notions of visibility, allowing the representing regions and sightlines to be arbitrary. We consider mainly the case of compact connected representing regions, proving two results giving necessary properties of visibility graphs, and giving some examples of classes of graphs that can be so represented. Finally, we give some applications of the concept, and we provide potential avenues for future research in the area.     

A Normal Law for Matchings

Dedicated to the memory of Paul Erdős, both for his pioneering discovery of normality in unexpected places, and for his questions, some of which led (eventually) to the present work.   For a simple graph G, let be the size of a matching drawn uniformly at random from the set of all matchings of G. Motivated by work of C. Godsil [11], we give, for a sequence and , several necessary and sufficient conditions for asymptotic normality of the distribution of , for instance