Yes, I think the intention is for E(v) to be one of the inputs of the algorithm (I contemplated E(v) for half an hour to guess this is what he means). I thought

Yes, I think the intention is for E(v) to be one of the inputs of the algorithm (I contemplated E(v) for half an hour to guess this is what he means). I thought maybe the following is possible: I create a new graph with the same vertices as his graph, but instead of having an E(v) I connect each vertex to the members of E(v) via an edge. Afterwards, if two vertices are not connected via an edge in his original graph, I remove this edge from my new graph. I cannot really get to the heart of it, I just had a vague impression this might be equivalent, either way I think the problem could likely be formalized in a more effective way.