Trees that are sphere-of-influence graphs Original Research Article

Given a collection of points in the plane, an open (closed) disk is drawn about each point with radius equal to the smallest distance from that point to any other in the collection. The sphere-of-influence graph (closed-sphere-of-influence graph) is the intersection graph of these disks. Any graph isomorphic to such a graph is a SIG (CSIG). We characterize trees that are SIGs and CSIGs, and obtain a bound on the number of edges in triangle-free SIGs and CSIGs.