Given a graph, we define a new stability condition for the algebraic and tropical moduli spaces of rational curves. Using the theory of geometric tropicalization, we characterize when the tropical compactification of the compact moduli space agrees with the theory of geometric tropicalization. In this talk, I will give a gentle introduction to the moduli spaces of graphically stable curves and their tropical counterparts, and briefly describe that such a tropicalization occurs only when the graph is complete multipartite.