Let $\Gamma_{k,g}$ be the class of $k$-connected cubic graphs of girth at least $g$. For several choices of $k$ and $g$, we determine a set ${\cal O}_{k,g}$ of graph operations, for which, if $G$ and $H$ are graphs in $\Gamma_{k,g}$, $G\not\cong H$, and $G$ contains $H$ topologically, then some operation in ${\cal O}_{k,g}$ can be applied to $G$ to result in a smaller graph $G'$ in $\Gamma_{k,g}$ such that, on one hand, $G'$ is contained in $G$ topologically, and on the other hand, $G'$ contains $H$ topologically.