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.