  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.
