χ-boundedness for bounded induced chair-packing treewidth

Determine whether the class of graphs with bounded induced \(S_{2,1,1}\)-packing treewidth is \(\chi\)-bounded, where \(S_{2,1,1}\) is the chair (or fork) obtained from \(K_{1,3}\) by subdividing one edge once.

Background

The paper proves that bounded induced packing treewidth implies χ\chi-boundedness when the packed graph is a path or a star. The chair is identified as the smallest tree that is neither a path nor a star.

The question asks whether the same conclusion extends to this next simplest tree. The authors leave this case unresolved after completing the path and star cases.

References

This leads to the following question:

\begin{question} Is it true that graphs of bounded induced $S_{2,1,1}$-packing treewidth form a $\chi$-bounded class? \end{question}

— Induced packing treewidth II. Excluding a clique or a biclique  (2609.21615 - Nikabadi et al., 18 Sep 2026) in Section 2, subsection “Induced star-packing treewidth,” immediately after the proof of Theorem \ref{thm:star-treepi-chi-bounded}