# 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 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:

• Unfortunately "split cluster graphs" seems to be in use for a different concept (graphs in which each connected component is split). May 5, 2015 at 18:12