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?



Have this subclass of split graphs been studied before?
Split graphs of diameter 2?


