In 2007 Sami Assaf introduced dual equivalence graphs as a method fordemonstrating that a quasisymmetric function is Schur positive. The methodinvolves the creation of a graph whose vertices are weighted by Ira Gessel'sfundamental quasisymmetric functions so that the sum of the weights of aconnected component is a single Schur function. In this paper, we improve onAssaf's axiomatization of such graphs, giving locally testable criteria thatare more easily verified by computers. We further advance the theory of dualequivalence graphs by describing a broader class of graphs that correspond toan explicit Schur expansion in terms of Yamanouchi words. Along the way, wedemonstrate several symmetries in the structure of dual equivalence graphs. Wethen apply these techniques to give explicit Schur expansions for a family ofLascoux-Leclerc-Thibon polynomials. This family properly contains thepreviously known case of polynomials indexed by two skew shapes, as wasdescribed in a 1995 paper by Christophe Carr\'e and Bernard Leclerc. As animmediate corollary, we gain an explicit Schur expansion for a family ofmodified Macdonald polynomials in terms of Yamanouchi words. This familyincludes all polynomials indexed by shapes with at most three cells in thefirst row and at most two cells in the second row, providing an extension tothe combinatorial description of the two column case described in 2005 by JamesHaglund, Mark Haiman, and Nick Loehr.
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