Let \(k\) be a positive integer and \(G = (V(G), E(G))\) a graph. A subset \(S \subseteq V(G)\) is a \(k\)-dominating set if every vertex of \(V(G)- S\) is adjacent to at least \(k\) vertices of \(S\). The \(k\)-domination number \(\gamma_k(G)\) is the minimum cardinality of a \(k\)-dominating set of \(G\). A graph \(G\) is called \(\gamma_k\)-stable if \(\gamma_{\bar{k}}(G – e) = \gamma_{{k}}(G)\) for every edge \(e\) of \(E(G)\). We first provide a necessary and sufficient condition for \(\gamma_{\bar{k}}\)-stable graphs. Then, for \(k \geq 2\), we offer a constructive characterization of \(\gamma_{\bar{k}}\)-stable trees.