Boundedness for monotone guest classes without host restrictions

Determine whether, for every monotone guest class G and every host graph class H without additional restrictions, the induced covering number can be bounded as a function of the corresponding non-induced covering number.

Background

The paper classifies when induced covering numbers are bounded by functions of their non-induced counterparts under structural restrictions on guest and host classes. The overview table indicates one unresolved setting, when the guest class is monotone and no restriction is imposed on the host class. The discussion explicitly identifies this as an unresolved question rather than merely an omitted case.

References

If we do no take the property of bounded chromatic number into account, only for one setting, namely when the guest class~$G$ is monotone, and there is no restriction on~$H$ we do not know whether each such~$H$ is $(\icn{x}{G}{},\icn{x}{G}{})$-bounded, see \cref{tab:overview}.

— Boundedness and Separation Between Induced and Non-Induced Covering Numbers  (2609.09873 - Goetze et al., 9 Sep 2026) in Section 5, subsection “Boundedness and Separation”