The evaluation of graphs on surfaces for state-sum models with defects

The evaluation of graphs on 2-spheres is a central ingredient of the Turaev-Viro construction of three-dimensional topological field theories. In this article, we introduce a class of graphs, called extruded graphs, that is relevant for the Turaev-Viro construction with general defect configurations involving defects of various dimensions. We define the evaluation of extruded graphs and show that it is invariant under a set of moves. This ensures the computability and uniqueness of our evaluation.