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.

翻译：无向图上的树宽已知具有许多算法应用。在考虑有向宽度度量时，关于其算法应用的结果则少得多。2022年，第一作者、Rabinovich 和 Wiederrecht 引入了一种新的有向宽度度量 $\vec{\mathcal{S}}$-DAG-宽度，该度量使用有向分离，并为其建立了结构对偶性。2012年，Berwanger 等人证明在具有有界 DAG-宽度的有向图上，奇偶博弈可在多项式时间内求解。通过将这一结果推广到具有有界 $\vec{\mathcal{S}}$-DAG-宽度的有向图，并提供一种计算给定有向图的 $\vec{\mathcal{S}}$-DAG-宽度的算法，我们为该新参数给出了首批算法结果。