Minimal obstructions to $2$-polar cographs

A graph is a cograph if it is $P_4$-free. A $k$-polar partition of a graph $G$ is a partition of the set of vertices of $G$ 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 explicitely 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 $k$.