  Let $P$ be a set of $n$ points in general position in the plane. A subset $I$
of $P$ is called an \emph{island} if there exists a convex set $C$ such that $I
= P \cap C$. In this paper we define the \emph{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.
