Enumerate all greedy choices of maximum independent set
Take the graph in the answer you linked to, and add another vertex G that has an edge to every other vertex except C.  Now C has the unique fewest number of neighbours, so it will necessarily be chosen first; then you can choose at most one of A, E, F or the new vertex G since each of these 4 vertices is linked to each of the others by an edge, so the algorithm forces you to produce an independent set of size 2 (either {C, A}, {C, E}, {C, F} or {C, G}).  But the same size-3 independent sets as before still exist (e.g. {B, D, F}).