In this work, we give a structural lemma on merges of typical sequences, a notion that was introduced in 1991 [Lagergren and Arnborg, Bodlaender and Kloks, both ICALP 1991] to obtain constructive linear time parameterized algorithms for treewidth and pathwidth. The lemma addresses a runtime bottleneck in those algorithms but so far it does not lead to asymptotically faster algorithms. However, we apply the lemma to show that the cutwidth and the modified cutwidth of series parallel digraphs can be computed in $\mathcal{O}(n^2)$ time.

中文翻译：

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典型序列重温---计算图的宽度参数

在这项工作中，我们给出了一个关于典型序列合并的结构引理，这个概念是在 1991 年引入的 [Lagergren 和 Arnborg，Bodlaender 和 Kloks，都在 ICALP 1991] 以获得用于树宽和路径宽度的建设性线性时间参数化算法。引理解决了这些算法中的运行时瓶颈，但到目前为止它并没有导致渐近更快的算法。然而，我们应用引理来证明串联平行有向图的割宽和修正割宽可以在 $\mathcal{O}(n^2)$ 时间内计算。