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 polynomial time.

中文翻译：

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重述典型序列-计算图形的宽度参数

在这项工作中，我们给出了典型序列合并的结构引理，该引理于1991年引入[Lagergren和Arnborg，Bodlaender和Kloks，均为ICALP 1991]，以获得构造性的线性时间参数化算法，用于树宽和路径宽度。引理解决了这些算法中的运行时瓶颈，但到目前为止，它并没有导致渐近地更快的算法。但是，我们应用引理表明，可以在多项式时间内计算出系列平行二合图的割宽和修正割宽。