Determine complexity of up-color domination weight on graph classes

Determine whether computing $Omega_{uc}(G)$ is NP-complete for trees, bipartite graphs, and interval graphs.

Background

The paper proves that determining Omegauc(G)Omega_{uc}(G) is NP-complete for general graphs and gives a polynomial-time algorithm for the fixed-coloring weight problem on trees. It leaves unresolved whether the graph-level optimization problem remains NP-complete on the listed restricted graph classes.

References

Is determining $\Omega_{uc}(G)$ NP--complete for the following classes of graphs? \begin{itemize} \item Trees. \item Bipartite graphs. \item Interval graphs. \end{itemize}

— Domination on Vertex-weighted Graphs Induce by a Coloring  (2502.07248 - Garrido-Vizuete et al., 11 Feb 2025) in Section 5, Conclusions and open problems, seventh Problem environment