# Typical Sequences Revisited --- Computing Width Parameters of Graphs

## TL;DR

This paper revisits the concept of typical sequences to improve the computation of width parameters like cutwidth in series parallel digraphs, providing a structural lemma that enhances understanding and algorithmic efficiency.

## Contribution

It introduces a new structural lemma on merges of typical sequences, enabling quadratic time algorithms for cutwidth and modified cutwidth of series parallel digraphs.

## Key findings

- Cutwidth of series parallel digraphs can be computed in O(n^2) time.
- The structural lemma addresses a key runtime bottleneck in existing algorithms.
- The work connects typical sequences to width parameter computation.

## Abstract

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.