\(\alpha_k\)- and \(\gamma_k\)-stable graphs (Q1970587)

The \(k\)-independence number \(\alpha_k(G)\) of a graph \(G\) is the cardinality of a maximum \(k\)-independent set of \(G\); and the \(k\)-dominating number \(\gamma_k(G)\) of \(G\) is the cardinality of a minimum \(k\)-dominating set of \(G\). A graph \(G\) is \(\alpha_k\)-stable or \(\gamma_k\)-stable if the \(k\)-independence number or the \(k\)-dominating number is unchanged by any single-edge deletion. The authors give conditions under which a graph is \(\alpha_k\)- or \(\gamma_k\)-stable; they also give constructive characterizations of \(\alpha_k\)- and \(\gamma_k\)-stable trees.