  In this work we describe how to systematically implement a full graph
decomposition to set up a linked-cluster expansion for the topological phase of
Kitaev's toric code in a field. This demands to include the non-local effects
mediated by the mutual anyonic statistics of elementary charge and flux
excitations. Technically, we describe how to consistently integrate such
non-local effects into a hypergraph decomposition for single excitations. The
approach is demonstrated for the ground-state energy and the elementary
excitation energies of charges and fluxes in the perturbed topological phase.
