Name the graph class: Disjoint union of a clique and an independent set Let $G$ be a graph which is the disjoint union of a clique and an independent set, i.e.
$$G = K_{n_1} + \overline{K_{n_2}} = K_{n_1} + I_{n_2} .$$

The graph class of all such graphs is characterized by the forbidden induced subgraphs set $\mathcal{H} = \{2K_2, P_3\}$ and is thus the intersection of a cluster graph and a split (or threshold) graph.

Does this (very simple) graph class have a name?  I was unable to find the graph class on&nbsp;ISGCI, and the papers I know on the topic (e.g. Editing Simple Graphs and On the clique editing problem) do not refer to the class by a name.

Here is a figure of such a graph:
The edge-complement of graphs in your class are complete split graphs: they can be partitioned into an independent set and a clique, such that every vertex in the independent set is adjacent to every vertex in the clique (see, for example, http://www.mathcove.net/petersen/lessons/get-lesson?les=30 ). Hence you could call your graph class co-complete split graphs.