  Motivated by the problem of Hurwitz equivalence of $\Delta ^2$ factorization
in the braid group, we address the problem of Hurwitz equivalence in the
symmetric group, obtained by projecting the $\Delta ^2$ factorizations into
$S_n$. We get $1_{S_n}$ 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.
