Planar Graphs. Graphs on Higher Surfaces. Degrees. Critical Graphs. The Conjectures of Hadwiger and Hajos. Sparse Graphs. Perfect Graphs. Geometric and Combinatorial Graphs. Algorithms. Constructions. Edge Colorings. Orientations and Flows. Chromatic Polynomials. Hypergraphs. Infinite Chromatic Graphs. Miscellaneous Problems. Indexes.

/pdf/graph-coloring-problems-4v7vijos3s.pdf

Graph Coloring Problems

In 1973, Victor Klee posed the following question: How many guards are necessary, and how many are sufficient to patrol the paintings and works of art in an art gallery with n walls? This wonderfully naive question of combinatorial geometry has, since its formulation, stimulated a plethora of papers, surveys and a book, most of them written in the last fifteen years. The first result in this area, due to V. Chvatal, asserts that n 3 guards are occasionally necessary and always sufficient to guard an art gallery represented by a simple polygon with n vertices. Since ChvataFs result, numerous variations on the art gallery problem have been studied, including mobile guards, guards with limited visibility or mobility, illumination of families of convex sets on the plane, guarding of rectilinear polygons, and others. In this paper, we survey most of these results.

Art Gallery and Illumination Problems

Two points in a polygon are called visible if the straight line segment between them lies entirely inside the polygon. The art gallery problem for a polygon P is to find a minimum set of points G in P such that every point of P is visible from some point of G. This problem has been shown to be NP-hard by Lee and Lin [71]. However, Chvatal showed that the number of points of G will never exceed [n/3] for a simple polygon of n sides [21]. This latter result is referred to as the art gallery theorem

Recent Results in Art Galleries

Chapter 15 – Geometric Shortest Paths and Network Optimization

Two points in a polygon are called if the straight line between them lies entirely inside the polygon. The art gallery problem for a polygon P is to find a minimum set of points G in P such that every point in P is visible from some point of G. The author provides an introduction to art gallery theorems and surveys the recent results of the field. The emphasis is on the results rather than the techniques. Several new problems that have the same geometric flavor as art gallery problems are also examined. >

Recent results in art galleries (geometry)

A grid graph is a finite node induced subgraph of the infinite two dimensional integer grid. A solid grid graph is a grid graph without holes. For general grid graphs, the Hamiltonian cycle problem is known to be NP complete. We give a polynomial time algorithm for the Hamiltonian cycle problem in solid grid graphs, resolving a longstanding open question posed by A. Itai et al. (1982). In fact, our algorithm can identify Hamiltonian cycles in quad quad graphs, a class of graphs that properly includes solid grid graphs.

Hamiltonian cycles in solid grid graphs

The link center of a simple polygon P is the set of points x inside P at which the maximal link-distance from x to any other point in P is minimized, where the link distance between two points x, y inside P is defined as the smallest number of straight edges in a polygonal path inside P connecting x to y. We prove several geometric properties of the link center and present an algorithm that calculates this set in time O (n2), where n is the number of sides of P. We also give an O(n log n) algorithm for finding a point x in an approximate link center, namely the maximal link distance from x to any point in P is at most one more than the value attained from the link center.

/pdf/computing-the-link-center-of-a-simple-polygon-dtkghrknvi.pdf

Computing the link center of a simple polygon

Consider the problem of moving a closed chain ofn links in two or more dimensions from one given configuration to another. The links have fixed lengths and may rotate about their endpoints, possibly passing through one another. The notion of a "line-tracking motion" is defined, and it is shown that when reconfiguration is possible by any means, it can be achieved byO(n) line-tracking motions. These motions can be computed inO(n) time on real RAM. It is shown that in three or more dimensions, reconfiguration is always possible, but that in dimension two this is not the case. Reconfiguration is shown to be always possible in two dimensions if and only if the sum of the lengths of the second and third longest links add to at most the sum of the lengths of the remaining links. AnO(n) algorithm is given for determining whether it is possible to move between two given configurations of a closed chain in the plane and, if it is possible, for computing a sequence of line-tracking motions to carry out the reconfiguration.

/pdf/reconfiguring-closed-polygonal-chains-in-euclideand-space-20vwn02t8v.pdf

Reconfiguring closed polygonal chains in Euclideand-space

Increasing attention has been given recently to drawings of graphs in which edges connect vertices based on some notion of proximity. Among such drawings are Gabriel, relative neighborhood, Delaunay, sphere of influence, and minimum spanning drawings. This paper attempts to survey the work that has been done to date on proximity drawings, along with some of the problems which remain open in this area.

/pdf/proximity-drawability-a-survey-1jqtqume3o.pdf

Proximity Drawability: a Survey

Complete characterizations are given for those trees that can be drawn as either the relative neighborhood graph, relatively closest graph, Gabriel graph, or modified Gabriel graph of a set of points in the plane. The characterizations give rise to linear-time algorithms for determining whether a tree has such a drawing; if such a drawing exists one can be constructed in linear time in the real RAM model. The characterization of Gabriel graphs settles several conjectures of Matula and Sokal [17].

William Lenhart

Papers

Hamiltonian cycles in solid grid graphs

Computing the link center of a simple polygon

Reconfiguring closed polygonal chains in Euclideand-space

Proximity Drawability: a Survey

Characterizing Proximity Trees