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Building Graphs with Specific Properties Using Sage

I would like to use Sage to build graphs with particular properties. I know how to call a program called "nauty" to ask Sage to generate, for example, all graphs on 8 vertices with 16 edges with clique number 4. However, I would like to add more properties.

Is it possible to have Sage generate all graphs with clique number 4 so that all vertices that are contained in a 4-clique satisfy a minimum degree condition? Or is it in general possible to have a particular subset of vertices satisfy a degree condition?

Thank you.