# Have this subclass of split graphs been studied before?

I am interested in the properties of the following subclass of split graphs:

The class consists of all split graphs $G=(C\cup I)$ where $C$ is a clique and $I$ an independent set, and every pair of vertices in $I$ have at least one common neighbor in $C$.

Does this class of graphs have a special name? Has this class and its properties been studied? If so, what would be some good references for this?

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I don't know if this can help you, but it's easy to see that your class is a superclass of connected threshold graphs. I think that the inclusion is strict, since $S_3$(link) is in your class, but is not a threshold graph.