Minimal forbidden sets for degree sequence characterizations

Given a set F of graphs, a graph G is F-free if G does not contain any member of F as an induced subgraph. Barrus, Kumbhat, and Hartke [4] called F a degree-sequence-forcing (DSF) set if, for each graph G in the class C of F-free graphs, every realization of the degree sequence of G is also in 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.