  The majorization relation orders the degree sequences of simple graphs into
posets called dominance orders. As shown by Ruch and Gutman (1979) and Merris
(2002), the degree sequences of threshold and split graphs form upward-closed
sets within the dominance orders they belong to, i.e., any degree sequence
majorizing a split or threshold sequence must itself be split or threshold,
respectively. Motivated by the fact that threshold graphs and split graphs have
characterizations in terms of forbidden induced subgraphs, we define a class
$\mathcal{F}$ of graphs to be dominance monotone if whenever no realization of
$e$ contains an element $\mathcal{F}$ as an induced subgraph, and $d$ majorizes
$e$, then no realization of $d$ induces an element of $\mathcal{F}$. We present
conditions necessary for a set of graphs to be dominance monotone, and we
identify the dominance monotone sets of order at most 3.
