Induced disjoint paths in theta-free graphs

Determine the computational complexity of Induced k-Disjoint Paths in theta-free graphs.

Background

The paper records a polynomial-time algorithm for Induced k-Disjoint Paths in (theta, wheel)-free graphs. Removing the wheel-free assumption leaves the complexity in the broader class of theta-free graphs unresolved.

A theta is any subdivision of K_{2,3}; thus, the problem asks whether the known tractability boundary can be extended from graphs excluding both thetas and wheels to all graphs excluding thetas.

References

Finally, in (theta, wheel)-free graphs, there is a~polynomial-time algorithm for~Induced $k$-Disjoint Paths , while its complexity in theta-free graphs is open.

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