  Tropicalization is a procedure for associating a polyhedral complex in
Euclidean space to a subvariety of an algebraic torus. We study the question of
which graphs arise from tropicalizing algebraic curves. By using Baker's
specialization of linear systems from curves to graphs, we are able to give a
necessary condition for a balanced weighted graph to be the tropicalization of
a curve. Our condition reproduces a generalization of Speyer's well-spacedness
condition and also gives new conditions. In addition, it suggests a new
combinatorial structure on tropicalizations of algebraic curves.
