In this talk I will explain, through examples, how to construct a model for the Fukaya category of punctured Riemann surfaces in terms of a sheaf of dg-categories over a suitable category of ribbon graphs. The main ingredients will be Nadler and Zaslow's work on cotangent bundles, and Kontsevich' recent ideas on the locality of the Fukaya category of a Stein manifold. Further, I will explain applications to homological mirror symmetry for degenerate elliptic curves. This work is joint with David Treumann and Eric Zaslow.