### On the connectedness and diameter of a Geometric Johnson Graph

*Crevel Bautista-Santiago, Javier Cano, Ruy Fabila-Monroy, David Flores-Peñaloza, Hernán González-Aguilar, Dolores Lara, Eliseo Sarmiento, Jorge Urrutia*

#### Abstract

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.

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