Minimal obstructions to $(s,1)$-polarity in cographs

Let $k,l$ be nonnegative integers. A graph $G$ is $(k,l)$-polar if its vertex set admits a partition $(A,B)$ such that $A$ induces a complete multipartite graph with at most $k$ parts, and $B$ induces a disjoint union of at most $l$ cliques with no other edges. A graph is a cograph if it does not contain $P_4$ as an induced subgraph. It is known that $(k,l)$-polar cographs can be characterized through a finite family of forbidden induced subgraphs, for any fixed choice of $k$ and $l$. The problem of determining the exact members of such family for $k = 2 = l$ was posted by Ekim, Mahadev and de Werra, and recently solved by Hell, Linhares-Sales and the second author of this paper. So far, complete lists of such forbidden induced subgraphs are known for $0 \le k,l \le 2$; notice that, in particular, $(1,1)$-polar graphs are precisely split graphs. In this paper, we focus on this problem for $(s,1)$-polar cographs. As our main result, we provide a recursive complete characterization of the forbidden induced subgraphs for $(s,1)$-polar cographs, for every non negative integer $s$. Additionally, we show that cographs having an $(s,1)$-partition for some integer $s$ (here $s$ is not fixed) can be characterized by forbidding a family of four graphs.