Comparing fat graph models of moduli space

Godin introduced the categories of open closed fat graphs $Fat^{oc}$ and admissible fat graphs $Fat^{ad}$ as models of the mapping class group of open closed cobordism. We use the contractibility of the arc complex to give a new proof of Godin's result that $Fat^{ad}$ is a model of the mapping class group of open-closed cobordisms. Similarly, Costello introduced a chain complex of black and white graphs $BW$-Graphs, as a rational homological model of mapping class groups. We use the result on admissible fat graphs to give a new integral proof of Costellos's result that $BW$-Graphs is a homological model of mapping class groups. The nature of this proof also provides a direct connection between both models which were previously only known to be abstractly equivalent. Furthermore, we endow Godin's model with a composition structure which models composition of cobordisms along their boundary and we use the connection between both models to give $BW$-Graphs a composition structure and show that $BW$-Graphs are actually a model for the open-closed cobordism category.