  Treewidth on undirected graphs is known to have many algorithmic
applications. When considering directed width-measures there are much less
results on their deployment for algorithmic results. In 2022 the first author,
Rabinovich and Wiederrecht introduced a new directed width measure,
$\vec{\mathcal{S}}$-DAG-width, using directed separations and obtained a
structural duality for it. In 2012 Berwanger~et~al.~solved Parity Games in
polynomial time on digraphs of bounded DAG-width. With generalising this result
to digraphs of bounded $\vec{\mathcal{S}}$-DAG-width and also providing an
algorithm to compute the $\vec{\mathcal{S}}$-DAG-width of a given digraphs we
give first algorithmical results for this new parameter.
