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R. Ganian, S. Ordyniak:
"The Power of Cut‑Based Parameters for Computing Edge‑Disjoint Paths";
Algorithmica, 82 (2020), 1 - 27.

This paper revisits the classical edge-disjoint paths (EDP) problem, where one is
given an undirected graph G and a set of terminal pairs P and asks whether G contains
a set of pairwise edge-disjoint paths connecting every terminal pair in P. Our
aim is to identify structural properties (parameters) of graphs which allow the efficient
solution of EDP without restricting the placement of terminals in P in any way.
In this setting, EDP is known to remain NP-hard even on extremely restricted graph
classes, such as graphs with a vertex cover of size 3. We present three results which
use edge-separator based parameters to chart new islands of tractability in the complexity
landscape of EDP. Our first and main result utilizes the fairly recent structural
parameter tree-cut width (a parameter with fundamental ties to graph immersions
and graph cuts): we obtain a polynomial-time algorithm for EDP on every
graph class of bounded tree-cut width. Our second result shows that EDP parameterized
by tree-cut width is unlikely to be fixed-parameter tractable. Our final, third
result is a polynomial kernel for EDP parameterized by the size of a minimum feedback
edge set in the graph.

http://dx.doi.org/10.1007/s00453-020-00772-w

https://publik.tuwien.ac.at/files/publik_293786.pdf