Sphere of influence graphs in general metric spaces

Let χ = {X1…, Xn} be a set of points in a metric space (n ≥ 2). Let ri, denote the minimum distance between Xi and the other points in χ. The sphere of influence at Xi is the open ball with center Xiand radius ri. The sphere of influence graph has vertex set χ with an edge joining a pair of distinct vertices provided the corresponding spheres of influence intersect. This proximity graph was introduced by Toussaint to model computer vision and pattern recognition problems in the Euclidean plane. We discuss the abstract properties of sphere of influence graphs in general metric spaces with emphasis on normed linear spaces. Our work generalizes and simplifies some theorems in the literature on sphere of influence graphs and lays the groundwork for future work.