The paper answers the question posed in a joint paper by A. A. Klyachko and N. M. Luneva about the optimality of the estimate for the cost of symmetry in graphs. The original estimate says that if n vertices can be removed from a connected graph so that there is no connected subgraph of isomorphic $Γ$ left in it, then at most $n|V(Γ)|$ vertices that form a set invariant under all automorphisms of the graph so that the graph does not contain a subgraph isomorphic to $Γ$. We will prove that there exists a graph $Γ$ for which this estimate is not optimal.