On the Connectedness and Diameter of a Geometric Johnson Graph

Let P be a set of n points in general position in the plane. A subset I of P is called an island if there exists a convex set C such that I = P ∩C. In this paper we define the generalized island Johnson graph of P as the graph whose vertex consists of all islands of P of cardinality k, two of which are adjacent if their intersection consists of exactly l elements. We show that for large enough values of n, this graph is connected, and give upper and lower bounds on its diameter.