  In 2007 Sami Assaf introduced dual equivalence graphs as a method for
demonstrating that a quasisymmetric function is Schur positive. The method
involves the creation of a graph whose vertices are weighted by Ira Gessel's
fundamental quasisymmetric functions so that the sum of the weights of a
connected component is a single Schur function. In this paper, we improve on
Assaf's axiomatization of such graphs, giving locally testable criteria that
are more easily verified by computers. We further advance the theory of dual
equivalence graphs by describing a broader class of graphs that correspond to
an explicit Schur expansion in terms of Yamanouchi words. Along the way, we
demonstrate several symmetries in the structure of dual equivalence graphs. We
then apply these techniques to give explicit Schur expansions for a family of
Lascoux-Leclerc-Thibon polynomials. This family properly contains the
previously known case of polynomials indexed by two skew shapes, as was
described in a 1995 paper by Christophe Carr\'e and Bernard Leclerc. As an
immediate corollary, we gain an explicit Schur expansion for a family of
modified Macdonald polynomials in terms of Yamanouchi words. This family
includes all polynomials indexed by shapes with at most three cells in the
first row and at most two cells in the second row, providing an extension to
the combinatorial description of the two column case described in 2005 by James
Haglund, Mark Haiman, and Nick Loehr.
