Consider the graph we get by taking the complete graph on $n$ vertices, and then attaching a pendant vertex to each of the $n$ vertices by an edge. Does such a graph have a name, i.e. do such graphs appear somewhere in the literature? Informally, this is like the star graph, but the center is not a single vertex, but a clique.
For example, such graphs are split graphs, but not every split graph is such a graph, so this is not sufficient.