Queue number of d-degenerate string graphs

Determine whether d-degenerate string graphs have bounded queue number for every fixed nonnegative integer d.

Background

The paper proves strong product structure results for string graphs of bounded maximum degree but shows that bounded degeneracy alone does not imply bounded row treewidth. It therefore leaves open whether other structural consequences associated with product structure, including bounded queue number, might nevertheless hold throughout the class of d-degenerate string graphs.

References

It is not clear what such a structural description should be. As a guide, one would expect that such a result would lead to positive solutions to the following open problems. Do $d$-degenerate string graphs have bounded queue number?

— String Graphs: Product Structure and Localised Representations  (2511.15156 - Karol, 19 Nov 2025) in Section 7, Conclusion