How to guarantee unique values for a decision variable among sub-optimal solutions

I have a network-flow model with binary decision variables for the state of edges in the graph (on or off). I am using the following parameters to generate multiple solutions

"PoolSearchMode" = 2

"PoolSolutions" = 100

I have no problem retrieving the solutions or the variables associated with each. However, upon inspection of the edge-state variables, I am seeing that they are all the same for my solutions, while other variables, e.g., flow allocations and path selections, among the set of edges is different.

Is there are way to write a model such that, for a given decision variable, all of the solutions of the model will have a unique value.

In this case, the decision variable is a vector or set of edges in a graph and I'm interested in finding solutions that have unique sets of edges.

To clarify, I don't mind if the sets overlap some, but I need each set to have at least one edge that is different from all of the other solutions.

I do have a variable to model the similarity between the initial graph, G_0 and the updated graph, G_1. I tried to minimize the similarity as my objective function to find more unique solutions, but in the end I have the same (minimum) similarity value for all of my solutions, and the same set of edges featured in each of them.

I also do not mind if there is just one possible set of edges for any solution that satisfies my constraints, but I would like if the solution pool were only populated with that single solution in such a case.

Please let me know if I can clarify at all.

To summarize, the title is really the distilled version of my request.

Is there a way to guarantee that all solutions in a solution set have a unique value for a given decision variable?

Thanks!

How to guarantee unique values for a decision variable among sub-optimal solutions

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