Structural properties of the minimum cut of partially-supplied graphs

It is well known that information about the structure of a graph is contained within its minimum cut. Here we investigate how the minimum cut of one graph informs the structure of a second, related graph. We consider pairs of graphs G and H, with respective Laplacian matrices L and M, and call H partially supplied provided that M is a Schur complement of L. Our results show how the minimum cut of H relates to the structure of the larger graph G.