Stability of Special Graph Classes
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Frei et al. [6] showed that the problem to decide whether a graph is stable with respect to some graph parameter under adding or removing either edges or vertices is Θ_2^P-complete. They studied the common graph parameters α (independence number), β (vertex cover number), ω (clique number), and χ (chromatic number) for certain variants of the stability problem. We follow their approach and provide a large number of polynomial-time algorithms solving these problems for special graph classes, namely for graphs without edges, complete graphs, paths, trees, forests, bipartite graphs, and co-graphs.
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