Combinatorial structure of visibility is probably one of the most fascinating and interesting areas of engineering and computer science. The usefulness of visibility graphs in computational geometry and robotic navigation problems like motion planning, unknown-terrain learning, shortest-path planning, etc., cannot be overstressed. The visibility graph, apart from being an important data structu...

The visibility graph V(P ) of a point set P ⊆ R 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 2and 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 r...

Consider a simple polygon P and a point s on the frontier ∂P of P . For any real δ > 0 there exists a shortest path ρ inside of P such that s is on the path ρ, and for each point p in ∂P , there exists a point q in ρ at Euclidean distance less than or equal δ to p such that the line segment pq is in P . Such an optimum path ρ is called a shortest route for ∂P visibility under δ-visibility that ...

We examine several types of visibility graphs: bar and semi-bar k-visibility graphs, rectangle k-visibility graphs, arc and circle k-visibility graphs, and compact visibility graphs. We improve the upper bound on the thickness of bar k-visibility graphs from 2k(9k − 1) to 6k, and prove that the upper bound must be at least k + 1. We also show that the upper bound on the thickness of semi-bar k-...

The subject of multi-robot systems has been thoroughly investigated in the past decade (e.g., [9, 12, 11, 7]). Current research in multi-robot systems can be divided, roughly, according to two aspects: theoretical robotics (e.g. [12]) and practical robotics (e.g. [10]). Theoretical robotics research, in many cases, make unrealistic assumptions on the robots’ capabilities. For example, robots ar...

Recently, the visibility graph has been introduced as a novel view for analyzing time series, which maps it to a complex network. In this paper, we introduce a new algorithm of visibility, ”cross-visibility”, which reveals the conjugation of two coupled time series. The correspondence between the two time series is mapped to a network, ”the cross-visibility graph”, to demonstrate the correlatio...

A point visibility graph is a graph induced by a set of points in the plane where the vertices of the graph represent the points in the point set and two vertices are adjacent if and only if no other point from the point set lies on the line segment between the two corresponding points. The set of all point visibility graphs form a graph class which is examined from a computational complexity p...

A visibility representation of a graph G is to represent the nodes of G with non-overlapping horizontal line segments such that the line segments representing any two distinct adjacent nodes are vertically visible to each other. If G is a plane graph, i.e., a planar graph equipped with a planar embedding, a visibility representation of G has the additional requirement of reflecting the given pl...

A topographic surface can be modeled as a graph, a visibility graph in terms of how each point location is visible to every other. This paper demonstrates various structural properties of visibility patterns with a topographic surface from a graph perspective, which could be important for landscape planning. This paper illustrates the fact that the visibility graph with a topographic surface is...