Consider the following operation to an undirected graph: one is allowed to take any maximum clique and replace the clique with a single vertex which is attached to every single vertex which has an edge into the clique. Essentially then one is allowed to do a bunch of simultaneous edge contractions but only if they all form a maximum clique. One can then talk about what graphs can be formed from other graphs this way. Note if that a graph A can be formed from a graph B this way, then A is a minor of B, but the converse does not hold. One can think of this also as a restricted form of quotient graph but only allowed to quotient out with subgraphs which are maximum cliques. Is there a term in the literature for this graph relation?
I asked this question earlier on Math Stackexchange but did not get any answers.
