FPT/XP algorithms for Thinness parameterized by solution size

Determine whether the Thinness problem is fixed-parameter tractable or slice-wise polynomial when parameterized solely by the target thinness value t of the input graph.

Background

The paper studies parameterized algorithms and kernels for computing thinness, introducing positive results under parameters based on restricted modular partitions (interval-modular and cluster-modular cardinality) and several kernelization lower bounds under other width measures. However, the most natural parameterization—by the thinness value itself—remains unresolved.

In particular, the authors achieve linear kernels and FPT algorithms under certain structural parameters, but explicitly note that the existence of FPT or even XP algorithms for Thinness when using the solution size as the parameter is unknown.

References

The existence of FPT or even XP algorithms for Thinness or Simultaneous Interval Number parameterized by the solution size (the numbers t and d, respectively) is open.

— Computing parameters that generalize interval graphs using restricted modular partitions (2512.22975 - Bonomo-Braberman et al., 28 Dec 2025) in Section 1 (Introduction)