  Any counterexample to the two-dimensional Jacobian Conjecture gives a
rational map from one projective plane to another. We use some ideas of the
Minimal Model Program to study the combinatorial structure of a rational
surface, that is obtained by resolving this map. Several structural results are
proven, revealing a rather orderly structure of the graph of the curves at
infinity. We also exhibit and discuss a graph that may lead to a counterexample
to the Jacobian Conjecture.
