The residue of a graph is the number of zeros left after iteratively applying the Havel-Hakimi algorithm to its degree sequence. Favaron, Mahéo, and Saclé showed that the residue is a lower bound on the independence number. The Maxine heuristic reduces a graph to an independent set of size $M$. It has been shown that given a graph $G$, $M$ is bounded between the independence number and the residue of a graph for any application of the Maxine heuristic. We improve upon a forbidden subgraph classification of graphs such that $M$ is equal to the independence number given by Barrus and Molnar in 2015.