FPT/XP algorithms for Simultaneous Interval Number under modular cardinalities

Investigate whether the Simultaneous Interval Number problem is fixed-parameter tractable or slice-wise polynomial when parameterized by the interval-modular cardinality or by the cluster-modular cardinality of the input graph (without including the solution size in the parameter).

Background

The authors provide an FPT algorithm for Simultaneous Interval Number parameterized by cluster-modular cardinality plus the solution size and derive further corollaries, but the complexity when parameterizing only by modular cardinalities remains unknown.

They explicitly raise this as an open direction in the concluding section.

References

Possible avenues of research that this work leaves open include: Is Simultaneous Interval Number FPT (or even XP) when parameterized by the interval-modular cardinality or the cluster-modular cardinality?

— Computing parameters that generalize interval graphs using restricted modular partitions (2512.22975 - Bonomo-Braberman et al., 28 Dec 2025) in Section 9 (Conclusions and open problems)