Computing $\vec{\mathcal{S}}$-DAGs and Parity Games

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.