Partitions of graphs into one or two independent sets and cliques

It is shown in this note that it can be recognized in polynomial time whether the vertex set of a finite undirected graph can be partitioned into one or two independent sets and one or two cliques. Such graphs generalize bipartite and split graphs and the result also shows that it can be recognized in polynomial time whether a graph can be partitioned into two split graphs. A time bound O(n) is given for the recognition of graphs which can be partitioned into two independent sets and one clique (one independent set and two cliques, resp.), and a time bound O(n ) is given for the recognition of graphs which can be partitioned into two independent sets and two cliques.