  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.
