  Taking a Feynman categorical perspective, several key aspects of the geometry
of surfaces are deduced from combinatorial constructions with graphs. This
provides a direct route from combinatorics of graphs to string topology
operations via topology, geometry and algebra.
  In particular, the inclusion of trees into graphs and the dissection of
graphs into aggregates yield a concise formalism for cyclic and modular operads
as well as their polycyclic and surface type generalizations. The latter occur
prominently in two-dimensional topological field theory and in string topology.
The categorical viewpoint allows us to use left Kan extensions of Feynman
operations as an efficient computational tool. The computations involve the
study of certain categories of structured graphs which are expected to be of
independent interest.
