The distinguishing number D(G) of a graph G is the least integer d such that G has a vertex labeling with d labels that is preserved only by a trivial automorphism. The distinguishing stability of a graph G is denoted by st(D)(G) and is the minimum number of vertices whose removal changes the distinguishing number. We show that for every natural number k, there exists a graph G such that st(D)(G) = k. Then we give some upper bounds for st(D)(G). We also study the edge distinguishing stability number (distinguishing bondage number) of G.
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