Smaller modular-width families with unbounded linear mim-width

Determine whether there exist graph families of modular-width strictly smaller than the currently constructed bound that nevertheless have unbounded linear mim-width; equivalently, construct such families or prove their nonexistence.

Background

The paper constructs a family of graphs with bounded modular-width (specifically, using prime nodes of size 7 in the modular decomposition) and unbounded linear mim-width, thereby showing incomparability between modular-width and linear mim-width.

They explicitly ask whether one can lower the modular-width bound in such constructions while maintaining unbounded linear mim-width, leaving this as an open question.

References

We leave as an open question whether there exist families with smaller modular-width and unbounded linear mim-width.

— Computing parameters that generalize interval graphs using restricted modular partitions (2512.22975 - Bonomo-Braberman et al., 28 Dec 2025) in Appendix A (Section: Modular-width vs. linear mim-width)