Sphere of influence graphs: Edge density and clique size

Abstract

Let X={X 1,…,X n} be a set of n points( n ≥ 2) in the metric space M. Let r i denote the minimum distance between X i and any other point in X . The closed ball B̄ i with center X i and radius r i is the closed sphere of influence at X i . The closed sphere of influence graph CSIG ( M, X ) has vertex set X with distinct vertices X i and X j adjacent provided B i∩ B j≠∅ . The graph G is an M-CSIG provided G is isomorphic to CSIG( M, X ) for some set X of points in M. We prove that, for any metric space M, the clique number is bounded over the class of M-CSIGs if and only if there is a constant C̄, so that the inequality |E| ≤ C|V| holds whenever G = ( V, E) is an M-CSIG. The proof uses Ramsey's Theorem. We also prove that if M = ( R d, ρ) is a d-dimensional Minkowski space, then C ≤ 5 d − 3 2 .