Visibility Graphs Of Simple Polygons Research Articles

On colourability of polygon visibility graphs

We study the problem of colouring visibility graphs of polygons. In particular, for visibility graphs of simple polygons, we provide a polynomial algorithm for 4-colouring, and prove that the 5-colourability question is already NP-complete for them. For visibility graphs of polygons with holes, we prove that the 4-colourability question is NP-complete.

On the number of visibility graphs of simple polygons

We present upper and lower bounds for the number of labelled visibility graphs of simple polygons of n vertices. The lower bound is roughly in the order of O(n 2n) and the upper bound is O(n 3n) .

A P-Completeness Result for Visibility Graphs of Simple Polygons

For each vertex of a simple polygon P an integer valued weight is given. We consider the path p1, p2, ..., pk in P which is created according to the following strategy: p1 is a designated start vertex s and pi+1 is obtained by choosing the vertex with smallest weight among all vertices visible from pi and different from p1, p2, ..., pi. If there is no such vertex the path is finished. This path is called geometric lexicographic dead end path. We shall prove the problem of determining whether a distinguished vertex t of P is on the geometric lexicographic dead end path or not to be P-complete.

On recognizing and characterizing visibility graphs of simple polygons

In this paper we establish four necessary conditions for recognizing visibility graphs of simple polygons and conjecture that these conditions are sufficient. We present an 0(n2)-time algorithm for testing the first and second necessary conditions and leave it open whether the third and fourth necessary conditions can be tested in polynomial time. We also show that visibility graphs of simple polygons do not possess the characteristics of a few special classes of graphs

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.

Negative results on characterizing visibility graphs

There is no known combinatorial characterization of the visibility graphs of simple polygons. In this paper we show negative results on two different approaches to finding such a characterization. We show that Ghosh's three necessary conditions for a graph to be a visibility graph are not sufficient thus disproving his conjecture. We also show that there is no finite set of minimal forbidden induced subgraphs that characterize visibility graphs.

COMPLEXITY ASPECTS OF VISIBILITY GRAPHS

In this paper, we consider two distinct problems related to complexity aspects of the visibility graphs of simple polygons. Recognizing visibility graphs is a long-standing open problem. It is not even known whether visibility graph recognition is in NP. That visibility graph recognition is in NP would be established if we could demonstrate that any n vertex visibility graph is realized by a polygon which can be drawn on an exponentially-sized grid. This motivates a study of the area requirements for realizing visibility graphs. In this paper, we prove: • Θ(n3) area is necessary and sufficient to realize the complete visibility graph Kn. • There exist visibility graphs which require exponential area to realize. • Any maximal outerplanar graph of diameter d can be realized in O(d2 · 2d) area, which can be as small as O(n log2 n) for a balanced mop. Linear maximal outer-planar graphs can be realized in O(n8) area. The second part of this paper considers the complexity of specific optimization problems on visibility graphs. Given a polygon P, we show that finding a maximum independent set, minimum vertex cover, or maximum dominating set in the visibility graph of P are all NP-complete. Further we show that for polygons P1 and P2, the problem of testing if they have isomorphic visibility graphs is isomorphism-complete. These problems remain hard when given the visibility graphs as input.

A note on the combinatorial structure of the visibility graph in simple polygons

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 structure for storing and updating geometric information, is a valuable mathematical tool in probing and understanding the nature of shapes of polygonal and polyhedral objects. In this research we wish to initially focus our attention on a fundamental class of geometric objects. These geometric objects may be looked upon as building blocks for more complex geometric objects, and which offer an ideal balance between complexity and simplicity, namely simple polygons. A major theme of the proposed paper is the investigation of the combinatorial structure of the visibility graph. More importantly, the goals of this paper are: 1. (i) To characterize the visibility graphs of simple polygons by obtaining necessary and sufficient conditions a graph must satisfy to qualify for the visibility graph of a simple polygon 2. (ii) To obtain hierarchical relationships between visibility graphs of simple polygons of a given number of vertices by treating them as representing simple polygons that are deformations of one another. 3. (iii) To exploit the potential of complete graphs to be natural coordinate systems for addressing the problem of reconstructing a simple polygon from visibility graph. We intend to achieve this by defining appropriate “betweenness” relationships on points with respect to the edges of the complete graphs.

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.