Witness rectangle graphs

Abstract

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 co-interval 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. © 2011 Springer-Verlag.