Innate degree of freedom of a graph

For a Kekulé structure we consider the smallest number of placements of double bonds such that the full Kekulé structure on the given parent graph is fully determined. These numbers for each Kekulé structure of the parent graph sum to a novel structural invariant F, called the degree of freedom of the graph. Some qualitative characteristics are identified, and it is noted that apparently it behaves differently from a couple of other invariants related to Kekulé structures.