The Connectivities of Leaf Graphs of Sets of Points in the Plane

Abstract

Let $U$ be a finite set of points in general position in the plane. We consider the following graph $\mathcal{G}$ determined by $U$. A vertex of $\mathcal{G}$ is a spanning tree of $U$ whose edges are straight line segments and do not cross. Two such trees $\mathbf{t}$ and $\mathbf{t}'$ are adjacent if for some vertex $u\in U$, $\mathbf{t}-u$ is connected and coincides with $\mathbf{t}'-u$. We show that $\mathcal{G}$ is 2-connected, which is the best possible result.