Stabilizing on the distinguishing number of a graph

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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 stD(G) and is the minimum number of vertices whose removal changes the distinguishing number. We obtain a general upper bound stD(G) V(G) -D(G)+1, and a relationships between the distinguishing stabilities of graphs G and G-v, i.e., stD(G) stD(G-v)+1, where v∈ V(G). Also we study the edge distinguishing stability number (distinguishing bondage number) of G.