  Dual equivalence graphs are a powerful tool in symmetric function theory that
provide a general framework for proving that a given quasisymmetric function is
symmetric and Schur positive. In this paper, we study a larger family of graphs
that includes dual equivalence graphs and define maps that, in certain cases,
transform graphs in this larger family into dual equivalence graphs. This
allows us to broaden the applications of dual equivalence graphs and points the
way toward a broader theory that could solve many important, long-standing
Schur positivity problems.
