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?