Witness Gabriel Graphs

We consider a generalization of the Gabriel graph, the witness Gabriel graph. Given a set of vertices P and a set of witnesses W in the plane, there is an edge ab between two points of P in the witness Gabriel graph GG-(P,W) if and only if the closed disk with diameter ab does not contain any witness point (besides possibly a and/or b). We study several properties of the witness Gabriel graph, both as a proximity graph and as a new tool in graph drawing.Comment: 23 pages. EuroCG 200

Witness (Delaunay) Graphs

Proximity graphs are used in several areas in which a neighborliness relationship for input data sets is a useful tool in their analysis, and have also received substantial attention from the graph drawing community, as they are a natural way of implicitly representing graphs. However, as a tool for graph representation, proximity graphs have some limitations that may be overcome with suitable generalizations. We introduce a generalization, witness graphs, that encompasses both the goal of more power and flexibility for graph drawing issues and a wider spectrum for neighborhood analysis. We study in detail two concrete examples, both related to Delaunay graphs, and consider as well some problems on stabbing geometric objects and point set discrimination, that can be naturally described in terms of witness graphs.Comment: 27 pages. JCCGG 200

How to Cover a Point Set with a V-Shape of Minimum Width

A balanced V-shape is a polygonal region in the plane contained in the union of two crossing equal-width strips. It is delimited by two pairs of parallel rays that emanate from two points x, y, are contained in the strip boundaries, and are mirror-symmetric with respect to the line xy. The width of a balanced V-shape is the width of the strips. We first present an O(n^2 log n) time algorithm to compute, given a set of n points P, a minimum-width balanced V-shape covering P. We then describe a PTAS for computing a (1+epsilon)-approximation of this V-shape in time O((n/epsilon)log n+(n/epsilon^(3/2))log^2(1/epsilon)). A much simpler constant-factor approximation algorithm is also described.Comment: In Proceedings of the 12th International Symposium on Algorithms and Data Structures (WADS), p.61-72, August 2011, New York, NY, US

Witness gabriel graphs

We consider a generalization of the Gabriel graph, the witness Gabriel graph. Given a set of vertices P and a set of witness points W in the plane, there is an edge ab between two points of P in the witness Gabriel graph $GG^-$(P,W) if and only if the closed disk with diameter ab does not contain any witness point (besides possibly a and/or b). We study several properties of the witness Gabriel graph, both as a proximity graph and as a new tool in graph drawing.Postprint (published version

Witness proximity graphs and other geometric problems

We introduce a generalization of the proximity graphs, the witness proximity graphs. We study in detail many concrete examples, and consider as well some problems on stabbing geometric objects and point set discrimination, that can be naturally described in terms of witness (proximity) graphs. The witness graph of a point set P of vertices in the plane, with respect to a point set W of witnesses, is the graph with vertex set P in which two points x,y in P are adjacent if and only if they define a region, the region of influence, that does (positive version) or does not (negative version) contain a witness w in W. In a positive witness graph, witnesses, which are said to be positive, are necessary to have an edge between a pair of points, and in a negative witness graph, witnesses, which are said to be negative, remove edges between pairs of points. As examples of witness graphs, we introduce two generalizations of the Delaunay graph, the witness Delaunay graph and the square graph for which the regions of influence are respectively the Delaunay disks (disks with two vertices on their boundary), and squares with two vertices on their boundary. We introduce also generalizations of the Gabriel graph and the rectangular influence graph, the witness Gabriel graph and the witness rectangle graph, for which the regions of influence are respectively the Gabriel disks (disks with diameters defined by a pair of vertices), and axis-parallel rectangles with diagonals defined by a pair of vertices. Finally we introduce a generalization of the relative neighborhood graph, the witness relative neighborhood graph with negative witnesses. In this witness graph, the region of influence is a lens, which is the intersection of two disks with a pair of vertices as centers and radii equal to the distance between these two vertices. For all these witness proximity graphs, we present some construction algorithms, worst-case and/or output-sensitive, as well as partial or complete characterization of the graphs. Subsequently, we introduce some variations on the witness graphs. In one variation we show the interrelation between two witness proximity graphs defined by two sets of points A and B such that in the first witness graph, A is the set of vertices and B is the set of witnesses, and in the second witness graph, B is the set of vertices and A is the set of witnesses. The three witness proximity graphs studied in this particular case are the witness Delaunay graph, the witness Gabriel graph, and the witness rectangle graph. In another variation, we study a special case of witness rectangle graph with positive and negative witnesses at the same time. Finally, we conclude with some related results on the number of points required to stab all the Delaunay disks, squares, Gabriel disks, rectangles, or lenses defined by pairs of points in a set of n points

Witness gabriel graphs

We consider a generalization of the Gabriel graph, the witness Gabriel graph. Given a set of vertices P and a set of witness points W in the plane, there is an edge ab between two points of P in the witness Gabriel graph $GG^-$(P,W) if and only if the closed disk with diameter ab does not contain any witness point (besides possibly a and/or b). We study several properties of the witness Gabriel graph, both as a proximity graph and as a new tool in graph drawing

Witness rectangle graphs

In a witness rectangle graph (WRG) on vertex point set P with respect to witness point set W in the plane, two points x, y in P are adjacent whenever the open isothetic rectangle with x and y as opposite corners contains at least one point in W. WRGs are representative of a larger family of witness proximity graphs introduced in two previous papers. We study graph-theoretic properties of WRGs. We prove that any WRG has at most two non-trivial connected components. We bound the diameter of the non-trivial connected components of a WRG in both the one-component and two-component cases. In the latter case, we prove that a graph is representable as a WRG if and only if each component is a connected co-interval graph, thereby providing a complete characterization of WRGs of this type. We also completely characterize trees drawable as WRGs. In addition, we prove that a WRG with no isolated vertices has domination number at most four. Moreover, we show that any combinatorial graph can be drawn as a WRG using a combination of positive and negative witnesses. Finally, we conclude with some related results on the number of points required to stab all the rectangles defined by a set of n points.Peer ReviewedPostprint (published version

Packing 2 × 2 unit squares into grid polygons is NP-complete

In a packing problem, the goal is to put some small objects disjointly into a large container, while optimizing some objective function. The packing problem is very general, and a rich variety of objects and containers ar