Perverse sheaves and graphs on surfaces

We give an explicit combinatorial description of the category Perv(S,N) of perverse sheaves on an oriented surface S (with boundary) with singularities at a given finite set N. The description is given in terms of any spanning graph K in S with the set of vertices N, so the category is defined entirely in terms of a ribbon graph. This description can be seen as an application, in the theory of perverse sheaves, of the idea of localization on a Lagrangian skeleton.