It seems not for any general $G$. But if $G$ is 2-connected does the hypothesis hold?
2026-03-31 15:13:01.1774969981
If $f$ is a cut edge in $G-e$, is $e$ a cut edge in $G-f$?
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For a 2-edge connected graph, $G$, if $f$ is a cut-edge of $G-e$ that would mean that ($G-e-f$) is disconnected. Removing of edges is commutative, so $G-f-e$ is also disconnected. Finally, $G-f$ would have $e$ as a cut-edge since $G-f-e$ is disconnected.
As for the general statement if $G$ were 1 edge-connected, there would be a cut-edge, $f$. For example, suppose that you have a graph which is a union of two vertex disjoint $K^{50}$'s bridged by a single edge, $f$. $f$ would be a cut-edge in $G-e$ for any $e$, however $G-f$ wouldn't even be connected, so the definition of "cut-edge" wouldn't even make sense to use in this situation.
However, if $G$ were at least 3 or more edge-connected, there would be no edges $e,f$ satisfying the hypothesis so it is trivially true.