Realizability of all round complexities in arbitrary graphs

Determine whether every round-complexity function can be realized by a locally checkable problem in arbitrary graphs when the bounded-maximum-degree restriction is removed.

Background

The paper proves that many previously empty complexity intervals can be populated by explicit locally checkable problems on arbitrary graphs of unbounded degree. Its constructions cover broad families of sublogarithmic, polylogarithmic, polynomial, and refined polynomial-logarithmic complexities.

The conclusion leaves open the full classification question: whether the phenomenon established for trees by Bousquet, Feuilloley, and Pierron extends completely to arbitrary graphs, so that no round-complexity gaps remain in the unbounded-degree setting.

References

And of course, the general question remains whether all round complexities can be obtained in arbitrary graphs, as it was shown to be the case in the restricted case of trees?

— A Simple Construction of Locally Checkable Problems Filling the LOCAL Complexity Gaps in Graphs with Arbitrary Large Degrees  (2608.18684 - Casagrande et al., 19 Aug 2026) in Section 6, Conclusion