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R. Ganian, S. Ordyniak, M. Ramanujan:
"On Structural Parameterizations of the Edge Disjoint Paths Problem";
Algorithmica, 83 (2021), 1605 - 1637.

In this paper we revisit 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 focus lies on structural parameterizations for the problem that allow for efficient
(polynomial-time or FPT) algorithms. As our first result, we answer an open
question stated in Fleszar et al. (Proceedings of the ESA, 2016), by showing that
the problem can be solved in polynomial time if the input graph has a feedback vertex
set of size one. We also show that EDP parameterized by the treewidth and the
maximum degree of the input graph is fixed-parameter tractable. Having developed
two novel algorithms for EDP using structural restrictions on the input graph, we
then turn our attention towards the augmented graph, i.e., the graph obtained from
the input graph after adding one edge between every terminal pair. In constrast to
the input graph, where EDP is known to remain NP-hard even for treewidth two, a
result by Zhou et al. (Algorithmica 26(1):3--30, 2000) shows that EDP can be solved
in non-uniform polynomial time if the augmented graph has constant treewidth; we
note that the possible improvement of this result to an FPT-algorithm has remained
open since then. We show that this is highly unlikely by establishing the W[1]-hardness
of the problem parameterized by the treewidth (and even feedback vertex set)
of the augmented graph. Finally, we develop an FPT-algorithm for EDP by exploiting
a novel structural parameter of the augmented graph.

http://dx.doi.org/10.1007/s00453-020-00795-3

https://publik.tuwien.ac.at/files/publik_300152.pdf