Homotopy theory is a well-studied field of mathematics which allows us to determine which spaces can be obtained by a deformation of another. Analogous theories have been developed for the mathematical objects called graphs. In this thesis, I further develop one of these homotopy theories for graphs, called A-homotopy theory, by constructing a covering graph theory, completing the work of my Master’s thesis. Covering graphs allow us to factor graph homomorphisms through other graphs using lifts. Factoring morphisms is an important part of finding structure on a category, and that structure is necessary for us to have a fully developed homotopy theory on the category. An example of such a structure is Hurewicz model structure on the category of spaces, which involves covering spaces. We show that there is no analogous Hurewicz model structure on the category of graphs. Instead, we define a new homotopy equivalence for graphs and a cloven weak factorization system structure on the category of graphs using path objects. This structure allows us to factor any morphism by a strong deformation retract followed by a morphism with the homotopy lifting property. This is a particularly useful factorization, which leads to a complete re-framing of A-homotopy theory.