Witness Rectangle Graphs

TL;DR

Witness rectangle graphs (WRGs) are a class of proximity graphs defined by points and witnesses, with specific structural properties, characterizations, and bounds on parameters like diameter and domination number, expanding understanding of geometric graph classes.

Contribution

This paper characterizes the structure of WRGs, including their connected components, diameters, and tree representations, and shows how any graph can be represented as a WRG with positive and negative witnesses.

Findings

Any WRG has at most two non-trivial connected components.

A graph is a WRG iff each component is a connected co-interval graph.

A WRG with no isolated vertices has domination number at most four.

Abstract

In a witness rectangle graph (WRG) on vertex point set P with respect to witness points 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 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 point.