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C2 and C3 status for QCSP(K3) and Sequential 3-Colouring Construction Game

Determine whether QCSP(K3) and Sequential 3-Colouring Construction Game satisfy conditions C2 and C3 of the C123 framework; specifically, ascertain for each problem whether it is computationally hard on subcubic graphs (C2) and whether such hardness is preserved under edge subdivision of subcubic graphs (C3).

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Background

The C123-framework separates problems into tractable versus hard cases on finitely-bounded monotone classes, with hardness established via three conditions (C1: tractable on bounded pathwidth; C2: hard on subcubic graphs; C3: hardness preserved under edge subdivision). The authors prove Pspace-completeness of QCSP(K3) and the Sequential 3-Colouring Construction Game on bounded pathwidth classes, which shows these problems do not satisfy C1.

However, whether these problems satisfy C2 and C3 remains unresolved. Establishing C2 and C3 for these two problems would place them within the C23 category and enable a framework-based classification over finitely-bounded monotone classes similar to other problems studied in the paper.

References

Moreover, we proved hardness for bounded path-width for QCSP$(K_3)$ and {\sc Sequential $3$-Colouring Construction Game}. We do not know if the latter two problems satisfy C2 and C3. We leave this for future work.

— Graph Homomorphism, Monotone Classes and Bounded Pathwidth (2403.00497 - Eagling-Vose et al., 1 Mar 2024) in Conclusions (Section 8)