Abstract
A graph is a cograph if it is P4-free. A k-polar partition of a graph G is a partition of the set of vertices of H into parts A and B such that the subgraph induced by A is a complete multipartite graph with at most k parts, and the subgraph induced by B is a disjoint union of at most k cliques with no other edges.It is known that k-polar cographs can be characterized by a finite family of forbidden induced subgraphs, for any fixed k. A concrete family of such forbidden induced subgraphs is known for k=1, since 1-polar graphs are precisely split graphs. For larger k such families are not known, and Ekim, Mahadev, and de Werra explicitly asked for the family for k=2. In this paper we provide such a family, and show that the graphs can be obtained from four basic graphs by a natural operation that preserves 2-polarity and also preserves the condition of being a cograph. We do not know such an operation for k>2, nevertheless we believe that the results and methods discussed here will also be useful for higher k.
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