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QCSP(K3) on bounded treedepth

Ascertain whether QCSP(K3) is polynomial-time solvable on graph classes of bounded treedepth, thereby resolving its complexity on this structural restriction.

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Background

The paper presents a C23-problem (Long Edge Disjoint Paths) that is NP-hard on some bounded-pathwidth classes but becomes polynomial-time on bounded treedepth. The authors ask whether QCSP(K3) might exhibit similar behavior.

They explicitly identify this as the major open problem arising from their work, linking it to the broader, famous open problem concerning QBF on bounded treedepth.

References

Whether $QCSP(K_3)$ is another such example is an open question. That is, we do not know if $QCSP(K_3)$ is polynomial-time for graph classes of bounded treedepth. This is the major open problem arising from our work.

— Graph Homomorphism, Monotone Classes and Bounded Pathwidth (2403.00497 - Eagling-Vose et al., 1 Mar 2024) in Conclusions