A reduction method to find spanning Eulerian subgraphs

We ask, When does a graph G have a subgraph Γ such that the vertices of odd degree in Γ form a specified set S ? V(G), such that G - E(Γ) is connected? If such a subgraph can be found for a suitable choice of S, then this can be applied to problems such as finding a spanning eulerian subgraph of G. We provide a general method, with applications.