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Table 1 Parameterized results for our problems. Algorithm complexities are given up to polynomial time factors (\(O^*\) notation), \(\Delta\) denotes the maximum number of children in the input tree-decomposition

From: Tree diet: reducing the treewidth to unlock FPT algorithms in RNA bioinformatics

Parameter

Source treewidth

Target treewidth

Difference

Problem

tw

\(tw'\)

\(d=tw-tw'\)

Graph-Diet

FPT

Para-NP-hard

Para-NP-harda

via MSO

\(tw'=2\)

\(d=1\)

Theorem 1

EDP(\(K_4\)) [30]

Theorem 3

Tree-Diet

XP

FPT open

Para-NP-hard

\(tw'=1\)

Theorem 3

Para-NP-hard

\(O^*((6\Delta )^{tw})\)

\(d=1\)

Theorem 7

Theorem 5

Binary-Tree-Diet

FPT

W[1]-hard

Theorem 5

XP open

Path-Diet

\(O^*(12^{tw})\)

XP

Theorem 7

\(O^*(tw^d)\)

Theorem 8

  1. aSee Theorem 2 statement for a more precise formulation