Vertex Partitions of Graphs into Cographs and Stars

A cograph is a graph that contains no path on four vertices as an induced subgraph. A cograph k-partition of a graph G = (V,E) is a vertex-partition of G into k sets V1, . . . , Vk ⊂ V so that the graph induced by Vi is a cograph for 1 ≤ i ≤ k. Gimbel and Nešetřil [5] studied the complexity aspects of the cograph k-partitions and raised the following questions: Does there exist a triangle-free planar graph that is not cograph 2-partitionable? If the answer is yes, what is the complexity of the associated decision problem? In this paper, we prove that such an example exists and that deciding whether a triangle-free planar graph admits a cograph 2-partition is NP-complete. We also show that every graph with maximum average degree at most 14 5 admits a cograph 2-partition such that each component is a star on at most three vertices.