Abstract Frei et al. (J. Comput. Syst. Sci. 123, 103–121, 2022) show that the stability, vertex stability, and unfrozenness problems with respect to certain graph parameters are complete for $$\varvec{\Theta _{2}^{\textrm{P}}}$$ Θ 2 P , the class of problems solvable in polynomial time by parallel access to an NP oracle. They studied the common graph parameters $$\varvec{\alpha }$$ α (the independence number), $$\varvec{\beta }$$ β (the vertex cover number), $$\varvec{\omega }$$ ω (the clique number), and $$\varvec{\chi }$$ χ (the chromatic number). We complement their approach by providing polynomial-time algorithms solving these problems for special graph classes, namely for graphs with bounded tree-width or bounded clique-width. In order to improve these general time bounds even further, we then focus on trees, forests, bipartite graphs, and co-graphs.