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 ...

Given a set of n disjoint line segments in the plane, the segment visibility graph is the graph whose 2n vertices correspond to the endpoints of the line segments and whose edges connect every pair of vertices whose corresponding endpoints can see each other. In this paper we characterize and provide a polynomial time recognition algorithm for planar segment visibility graphs. Actually, we char...

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 in PSPACE. Further, every such visibility graph can be dismantled as a sequence of visibility graphs of convex fans. Any nondegenerate configuration of n points can be associated with a maximal chain in the weak Bruhat order of the symmetric group...

A visibility relation can be viewed as a graph: the uncountable graph of a visibility relationship between points in a polygon P is called the point visibility graph (PVG) of P. In this paper we explore the use of perfect graphs to characterize tractable subprob-lems of visibility problems. Our main result is a characterization of which polygons are guaranteed to have weakly triangulated PVGs, ...

We study the sensor and movement capabilities that simple robots need in order to create a map of an unknown polygon of size n, and to meet. We consider robots that can move from vertex to vertex, can backtrack movements, and see distant vertices in counter-clockwise order but have no means of visibly identifying them. We show that such robots can always solve the weak rendezvous problem and re...

Bar k-visibility graphs are graphs admitting a representation in which the vertices correspond to horizontal line segments, called bars, and the edges correspond to vertical lines of sight which can traverse up to k bars. These graphs were introduced by Dean et al. [3] who conjectured that bar 1-visibility graphs have thickness at most 2. We construct a bar 1-visibility graph having thickness 3...

In this paper, we propose a method for maintaining the region visible from a moving point observer inside a planar scene. In this method, we check the observer position in discrete time-stamps to detect and apply changes to the visible or illuminated region of a moving point observer q, or VP(q). We efficiently maintain a list C(q) of edges in VP(q) which are subject to change during the motion...

The 0− 1 matrix A contains a 0− 1 matrix M if some submatrix of A can be transformed into M by changing some ones to zeroes. If A does not contain M , then A avoids M . Let ex(n,M) be the maximum number of ones in an n×n 0−1 matrix that avoids M , and let exk(m,M) be the maximum number of columns in a 0− 1 matrix with m rows that avoids M and has at least k ones in every column. A method for bo...

This paper presents a visibility analysis tool for 3D urban environments and its possible applications for urban design practice. Literature exists for performing visibility analysis using various methods and techniques, however, tools that result from such research are generally not suitable for use by designers in practice. Our visibility analysis tool resides in Grasshopper, Rhino. It uses a...

A unit bar-visibility graph is a graph whose vertices can be represented in the plane by disjoint horizontal unit-length bars such that two vertices are adjacent if and only if there is a unobstructed, non-degenerate, vertical band of visibility between the corresponding bars. We generalize unit bar-visibility graphs to [1, k]bar-visibility graphs by allowing the lengths of the bars to be betwe...