This is a joint work with M. Khovanov. We construct toy models for 2d TQFTs, in which the ‘strings’are graph-like generalizations of an open string. The egdes of a graph are ordinary open strings living in different ‘universes’. The ends of the edges are held together at the vertices of the graph by non-factorizable boundary conditions. We present simple examples of such theories: an A-model with Grassmannian target spaces and a Landau-Ginzburg B-model based on rather simple potentials W. The Hilbert spaces and correlators of the latter model have a complete combinatorial description, and they play an important role in the categorification of the SU(N) HOMFLY polynomial.