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enumerate subgraphs with edge sets


I have been using Sage for the past few months now, and I use the subgraph_search_iterator() heavily. So far, I did not care for the edge set of the subgraph found. But now that I do, it seems that the subgraph_search_iterator() returns only the vertex sets of the corresponding subgraphs.

In particular, I am looking for subdivisions of K3,3 in an input graph. But I want the additional constraint that two fixed vertices (say, u and v) must be branch vertices of the subgraph found - by branch vertices I mean vertices of degree exactly 3 in the subgraph.

Any help would be greatly appreciated. Thanks!