  The qualitative behavior of a dynamical system can be encoded in a graph.
Each node of the graph is an equivalence class of chain-recurrent points and
there is an edge from node $A$ to node $B$ if, using arbitrary small
perturbations, a trajectory starting from any point of A can be steered to any
point of B. In this article we describe the graph of the logistic map. Our main
result is that the graph is always a tower, namely there is an edge connecting
each pair of distinct nodes. Notice that these graphs never contain cycles. If
there is an edge from node A to node B, the unstable manifold of some periodic
orbit in A contains points that eventually map onto B. For special parameter
values, this tower has infinitely many nodes.
