Coverings and homotopy of a graph

Let C be a collection of closed walks of a connected graph Γ. We study coverings and homotopy of Γ under the condition that every member of C can be lifted through covering morphisms, and is contractible. Since they depend on C, we call them C-coverings and C-homotopy. After we review the existence of universal C-covers and their uniqueness modulo isomorphism studied by E. E. Shult and others, we investigate conditions that a finite graph is C-simply connected, i.e., the graph itself is a universal C-cover. As an application, we show that classes of distance-regular graphs and distance-semiregular graphs are C-simply connected when C is the collection of closed paths of minimal length. We also show a finiteness condition of a universal C-cover of a class of connected bipartite graphs when C is the collection of closed paths of minimal length.