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 s x 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 influence of the set V . The sphere of influence graph on V is defined as the graph where (x; y) is an edge if and only if s x and s y intersect. In this extended abstract, we demonstrate that no Euclidean planar sphere of influence graph (E-SIG) contains more than 15n edges. 1 Introduction In 1980, Godfried Toussaint proposed the sphere of influence graph as a geometric tool for capturing the underlying structures of dot patterns [1, 2, 3]. As is often the case, along with a new graph comes a host of open problems. Toussaint posed the question, "Does there exist a constant c such that a sphere of influence graph in the Euclidean plane (E-SIG) has at most cn edges?" The question remained unanswered for five years, until it was solved by David Avis and Joe Horton ..