Separate complexity of induced linkage and induced flow variants

Determine whether there exists a hereditary graph class in which Induced k-Disjoint Paths is NP-complete while Induced Disjoint S--T Paths with |S|=|T|=k is solvable in polynomial time, with particular attention to the case k=2.

Background

The paper distinguishes Induced k-Disjoint Paths, where prescribed terminal pairs must be connected, from Induced Disjoint S--T Paths, where the endpoints may be matched arbitrarily between two terminal sets. For fixed k, the flow variant can be solved using k! calls to the linkage variant, but this does not determine whether the two problems can have different complexity on a hereditary graph class.

The authors point out that their hardness proofs apply to both formulations, whereas several surveyed polynomial-time results concern the linkage formulation more generally. They therefore ask whether a genuine complexity separation between the two variants is possible.

References

In light of the surveyed polynomial algorithms applying more generally to Induced $k$-Disjoint Paths and the hardness proofs applying more generally to Induced Disjoint $S$--$T$ Paths with $|S|=|T|=k$, we wonder if there is a hereditary graph class in which Induced $k$-Disjoint Paths is NP-complete but Induced Disjoint $S$--$T$ Paths with $|S|=|T|=k$ is polynomial-time solvable; the case $k=2$ is of particular interest.

— Induced Disjoint Paths Without an Induced Minor  (2502.05289 - Aboulker et al., 7 Feb 2025) in Section “Open questions”