Graph Theoretic Method for Determining non Hurwitz Equivalence in the Braid Group and Symmetric group

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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, obtained by projecting the Δ² factorizations into Sₙ. We get 1ₛₙ 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 main result of this paper will help us to compute the "Braid Monodromy Type" invariant. The graph structure gives a weaker but very easy to compute invariant to distinguish between diffeomorphic surfaces which are not deformation of each other.