Graph theoretic method for determining Hurwitz equivalence in the symmetric group

Motivated by the problem of Hurwitz equivalence of Δ² factorization in the braid group, we address the problem of Hurwitz equivalence in the symmetric group, of 1_Sn factorizations with transposition factors. Looking at the transpositions as the edges in a graph, we show that two factorizations are Hurwitz equivalent if and only if their graphs have the same weighted connected components. The graph structure allows us to compute Hurwitz equivalence in the symmetric group. Using this result, one can compute non-Hurwitz equivalence in the braid group.