Existence of any kernel for Thinness under common width parameters

Determine whether the Thinness problem admits any kernelization (not necessarily polynomial) when parameterized by commonly studied graph width parameters such as treewidth, pathwidth, bandwidth, clique-width, (linear) mim-width, modular-width, or by thinness or simultaneous interval number themselves.

Background

The paper proves that Thinness (and Simultaneous Interval Number) do not admit polynomial kernels under a broad class of parameters via and-cross-composition, but this does not exclude the existence of non-polynomial kernels.

They emphasize that clarifying whether any kernels exist under these parameters remains unresolved.

References

Note that Theorem 11 does not exclude the possibility of a kernel, just of a polynomial kernel. Whether Thinness admits any kernel parameterized by these parameters is still open.

— Computing parameters that generalize interval graphs using restricted modular partitions (2512.22975 - Bonomo-Braberman et al., 28 Dec 2025) in Appendix C (Section: Kernelization lower bounds)