Isolation numbers of transformation graphs and original graphs

Characterize the relationship between the \mathcal{F}-isolation numbers of different transformation graphs and the \mathcal{F}-isolation number of the original graph, and determine whether these quantities satisfy equalities or inequalities involving other graph-theoretic parameters of the original graph.

Background

The paper establishes a precise relationship between the isolation number of a middle graph and a matching parameter of the original graph. It raises the broader unresolved question of how \mathcal{F}-isolation numbers behave under other graph transformations, both relative to the original graph's \mathcal{F}-isolation number and relative to other graph invariants of the original graph.