Region counting graphs

A new family of proximity graphs, called region counting graphs (RCG) is presented. The RCG for a finite set of points in the plane uses the notion of region counting distance introduced by Demaine et al. to characterize the proximity between two points p and q: the edge pq is in the RCG if and only if there is less than or exactly k vertices in a given geometric neighborhood defined by a region. These graphs generalize many common proximity graphs, such as k-nearest neighbor graphs, β-skeletons or Θ-graphs. This paper concentrates on RCGs that are invariant under translations, rotations and uniform scaling. For k = 0, we give conditions on regions R that define an RCG to ensure a number of properties including planarity, connectivity, triangle freeness, cycle freeness, bipartiteness, and bounded degree. These conditions take form of what we call tight regions : maximal or minimal regions that a region R must contain or be contained in to satisfy a given monotone property.