Polarity on $H$-split graphs

Given nonnegative integers, $s$ and $k$, an $(s,k)$-polar partition of a graph $G$ is a partition $(A,B)$ of $V_G$ such that $G[A]$ and $\overline{G[B]}$ are complete multipartite graphs with at most $s$ and $k$ parts, respectively. If $s$ or $k$ is replaced by $\infty$, it means that there is no restriction on the number of parts of $G[A]$ or $\overline{G[B]}$, respectively. A graph admitting a $(1,1)$-polar partition is usually called a split graph. In this work, we present some results related to $(s,k)$-polar partitions on two graph classes generalizing split graphs. Our main results include efficient algorithms to decide whether a graph on these classes admits an $(s,k)$-polar partition, as well as upper bounds for the order of minimal $(s,k)$-polar obstructions on such graph families for any $s$ and $k$ (even if $s$ or $k$ is $\infty$).