On the size of the euclidean sphere of influence graph

Let V be a set of distinct points in the Euclidean plane. For each point x 2 V , let sx be the ball centered at x with radius equal to the distance from x to its nearest neighbour. We refer to these balls as the spheres of in uence of the set V . The sphere of in uence graph on V is de ned as the graph where (x; y) is an edge if and only if sx and sy intersect. In this extended abstract, we demonstrate that no Euclidean planar sphere of in uence graph (E-SIG) contains more than 15n edges.