Partial domination under other graph transformations

Determine which other graph transformations can be studied in the context of partial domination and establish the corresponding bounds or exact values for the associated parameters.

Background

The paper studies partial domination through the isolation number of middle graphs, proving that the isolation number of the middle graph of a graph G equals the minimum size of a maximal matching in G. The authors conclude by asking whether analogous partial-domination questions can be formulated and solved for other graph transformations, including the determination of corresponding bounds or exact parameter values.