Stabilizing the distinguishing number of a graph
Saeid Alikhani, Samaneh Soltani · Communications in Algebra · 2018
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 show that for every natural number k, there exists a graph G such that stD(G)=k. Then we give some upper bounds for stD(G). We also study the edge distinguishing stability number (distinguishing bondage number) of G.