For any graph with distinguishing number 2 and determining number 2, the cost number is at most 4, settling Boutin's open question in this case.
Babai,Asymmetric trees with two prescribed degrees,Acta Math
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On Distinguishing Graphs and Cost Number using Automorphism Representations
For any graph with distinguishing number 2 and determining number 2, the cost number is at most 4, settling Boutin's open question in this case.