Induced disjoint paths in minor-free graph classes

Determine whether Induced k-Disjoint Paths is tractable in every graph class excluding a fixed graph as a minor, potentially by combining irrelevant-vertex techniques with the Robertson–Seymour graph structure theorem.

Background

The paper surveys polynomial-time algorithms for Induced k-Disjoint Paths on planar and bounded-genus graphs and notes that tractability is believed to extend to broader minor-closed classes. Such an extension would require adapting irrelevant-vertex methods to the almost-embeddable and other structural pieces supplied by the Robertson–Seymour structure theorem.

The authors explicitly state that no proof of this general tractability claim is currently known, while also noting that the claim may ultimately fail. This is distinct from the hardness results established in the paper for string graphs, which exclude certain graphs as induced minors rather than as ordinary minors.

References

It is believed that the tractability of Induced $k$-Disjoint Paths (and perhaps even of FO+SDP model checking) holds more generally in classes excluding a~fixed minor, and could be shown by combining the irrelevant-vertex techniques (developed for the planar and bounded-genus cases) with the graph structure theorem of Robertson and Seymour. However, to our knowledge, this has not been proven yet (provided it indeed holds).

— Induced Disjoint Paths Without an Induced Minor  (2502.05289 - Aboulker et al., 7 Feb 2025) in Section 1, Introduction