Feedback-vertex-set complexity of Vertex-Coloring with weights {1,2}

Determine whether the Vertex-Coloring problem with edge-weight set {1,2} is fixed-parameter tractable or W[1]-hard when parameterized by the feedback vertex set number.

Background

The paper establishes W[1]-hardness for the {0,1} Vertex-Coloring problem under the feedback vertex set parameter and proves W[1]-hardness for both unweighted weight sets under treedepth. For the {1,2} unweighted problem, however, the feedback-vertex-set parameter remains unresolved.

The distinction matters because feedback vertex set and treedepth are incomparable structural parameters, so the treedepth hardness result does not settle the feedback-vertex-set complexity.

References

Finally, we ask whether Vertex-coloring is FPT or W[1]-hard parameterized by the feedback vertex set number.

— Vertex-Coloring Edge-Weighting: Kernelization and Generalization  (2609.27719 - Aute et al., 23 Sep 2026) in Section 7, paragraph titled “Open problems”