  The residue of a graph is the number of zeros left after iteratively applying
the Havel-Hakimi algorithm to its degree sequence. Favaron, Mah\'{e}o, and
Sacl\'{e} 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.
