Invariant systems of weighted representatives

It is known that, if removing some $n$ edges from a graph $\mathrm{\Gamma }$ destroys all subgraphs isomorphic to a given finite graph $K$, then all subgraphs isomorphic to $K$ can be destroyed by removing at most $\mathrm{\mid }E(K)\mathrm{\mid }\cdot n$ edges, which form a set invariant with respect to all automorphisms of $\mathrm{\Gamma }$. We construct the first examples of (connected) graphs $K$ for which this estimate is not sharp. Our arguments are based on a ``weighted analogue’’ of an earlier known estimate for the cost of symmetry.