Muriel Dulieu

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

Witness (Delaunay) graphs

GG^-

Non-crossing matchings of points with geometric objects

(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

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.

Matching points with things

Given an ordered set of points and an ordered set of geometric objects in the plane, we are interested in finding a non-crossing matching between point-object pairs. In this paper, we address the algorithmic problem of determining whether a non-crossing matching exists between a given point-object pair. We show that when the objects we match the points to are finite point sets, the problem is NP-complete in general, and polynomial when the objects are on a line or when their size is at most 2. When the objects are line segments, we show that the problem is NP-complete in general, and polynomial when the segments form a convex polygon or are all on a line. Finally, for objects that are straight lines, we show that the problem of finding a min-max non-crossing matching is NP-complete.

How to cover a point set with a V-shape of minimum width

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 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 cointerval graph, thereby providing a complete characterization of WRGs of this type. We also completely characterize trees drawable as WRGs. 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.

Witness Rectangle Graphs

Given an ordered set of points and an ordered set of geometric objects in the plane, we are interested in finding a non-crossing matching between point-object pairs. We show that when the objects we match the points to are finite point sets, the problem is NP-complete in general, and polynomial when the objects are on a line or when their number is at most 2. When the objects are line segments, we show that the problem is NP-complete in general, and polynomial when the segments form a convex polygon or are all on a line. Finally, for objects that are straight lines, we show that the problem of finding a min-max non-crossing matching is NP-complete.

Mutual witness proximity graphs

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^2logn) 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+@e)-approximation of this V-shape in time O((n/@e)logn+(n/@e^3^/^2)log^2(1/@e)). A much simpler constant-factor approximation algorithm is also described.

Witness proximity graphs and other geometric problems

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.

Witness gabriel graphs

This paper describes one variation on witness proximity graphs called mutual witness proximity graphs. Two witness proximity graphs are said to be mutual when, given two sets of points A and B, A is the vertex set of the first graph and the witness set of the second one, while B is the witness set of the first graph and the vertex set of the second one. We show that in the union of two mutual witness Delaunay graphs, there are always at least @?n-22@? edges, where n=|A|+|B|, which is tight in the worst case. We also show that if two mutual witness Delaunay graphs are complete, then the sets A and B are circularly separable; if two mutual witness Gabriel graphs are complete, then the sets A and B are linearly separable; but two mutual witness rectangle graphs might be complete, with A and B not linearly separable.