Vertex Partitions of Graphs into Cographs and Stars
Article first published online: 14 FEB 2013
© 2013 Wiley Periodicals, Inc.
Journal of Graph Theory
Volume 75, Issue 1, pages 75–90, January 2014
How to Cite
Dorbec, P., Montassier, M. and Ochem, P. (2014), Vertex Partitions of Graphs into Cographs and Stars. J. Graph Theory, 75: 75–90. doi: 10.1002/jgt.21724
- Issue published online: 22 OCT 2013
- Article first published online: 14 FEB 2013
- Manuscript Revised: 3 DEC 2012
- Manuscript Received: 19 JAN 2012
- cograph partition;
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  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 article, 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 ??? admits a cograph 2-partition such that each component is a star on at most three vertices.