  Given a set $\mathcal{F}$ of graphs, a graph $G$ is $\mathcal{F}$-free if $G$
does not contain any member of $\mathcal{F}$ as an induced subgraph. Barrus,
Kumbhat, and Hartke [M. D. Barrus, M. Kumbhat, and S. G. Hartke, Graph classes
characterized both by forbidden subgraphs and degree sequences, J. Graph Theory
(2008), no. 2, 131--148] called $\mathcal{F}$ a degree-sequence-forcing (DSF)
set if, for each graph $G$ in the class $\mathcal{C}$ of $\mathcal{F}$-free
graphs, every realization of the degree sequence of $G$ is also in
$\mathcal{C}$. A DSF set is minimal if no proper subset is also DSF. In this
paper, we present new properties of minimal DSF sets, including that every
graph is in a minimal DSF set and that there are only finitely many DSF sets of
cardinality $k$. Using these properties and a computer search, we characterize
the minimal DSF triples.
