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 \(\varvec{\Theta _{2}^{\textrm{P}}}\), the class of problems solvable in polynomial time by parallel access to an NP oracle. They studied the common graph parameters \(\varvec{\alpha }\) (the independence number), \(\varvec{\beta }\) (the vertex cover number), \(\varvec{\omega }\) (the clique number), and \(\varvec{\chi }\) (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.

弗雷等人。(J. Comput. Syst. Sci. 123 , 103–121, 2022) 表明，对于某些图参数，稳定性、顶点稳定性和解冻问题对于\(\varvec{\Theta _{2}^{ \textrm{P}}}\)，通过并行访问 NP 预言机可以在多项式时间内解决的一类问题。他们研究了常见的图参数\(\varvec{\alpha }\)（独立数）、\(\varvec{\beta }\)（顶点覆盖数）、\(\varvec{\omega }\)（团数）和\(\varvec{\chi }\)（半音数）。我们通过提供多项式时间算法来补充他们的方法，解决特殊图类的这些问题，即具有有界树宽度或有界团宽度的图。为了进一步改善这些一般时间界限，我们将重点放在树、森林、二分图和联合图上。