Table 2.

Table listing pseudoknot classes, corresponding treewidth and resulting complexity of the folding algorithm

			Complexities
Type	Fatgraph	Treewidth	Full Turner	All others
H-type	([)]	4	$O\left(n^5\right)$	$O\left(n^4\right)$(*)
Kissing hairpins	([)(])	4	$O\left(n^5\right)$	$O\left(n^4\right)$
“L” [12]	([{)]}	5	$O\left(n^6\right)$	$O\left(n^6\right)$
“M” [12]	([{)(]})	5	$O\left(n^6\right)$	$O\left(n^6\right)$
4-clique	([{<)]}>	5	$O\left(n^6\right)$	$O\left(n^6\right)$
5-clique	([{<A)]}>a	5	$O\left(n^6\right)$	$O\left(n^6\right)$
5-chain	({[)(][)}]	6	$O\left(n^7\right)$	$O\left(n^7\right)$

For H-type pseudoknots beneath the Turner model, marked as (*), an iterated computation over canonical tree decompositions is required to achieve the complexity (see Theorem 5). For the H-type and kissing hairpins cases, we are in the specific case where the most complex routine is the alignment of a “clique case” helix, which is done in $O(n^4)$ despite a treewidth of 4. These examples are detailed in the Appendix, Fig. 10. The DP equations for each of these examples have been automatically generated by a Python implementation of our pipeline, freely available at https://gitlab.inria.fr/bmarchan/auto-dp