  The Havel-Hakimi algorithm iteratively reduces the degree sequence of a graph
to a list of zeroes. As shown by Favaron, Mah\'eo, and Sacl\'e, the number of
zeroes produced, known as the residue, is a lower bound on the independence
number of the graph. We say that a graph has the strong Havel-Hakimi property
if in each of its induced subgraphs, deleting any vertex of maximum degree
reduces the degree sequence in the same way that the Havel-Hakimi algorithm
does. We characterize graphs having this property (which include all threshold
and matrogenic graphs) in terms of minimal forbidden induced subgraphs. We
further show that for these graphs the residue equals the independence number,
and a natural greedy algorithm always produces a maximum independent set.
