In this paper, we introduce ( \(\mathcal {D\diagup SG}\) ) correspondence to give a full classification to the critical graphs (topology of Stokes complex) of polynomial quadratic differentials \(A\left( z-a\right) \left( z^{2}-1\right) dz^{2}\) on the Riemann sphere \(\widehat{ \mathbb {C}}\) , where \(\left( A,a\right) \in \mathbb {C}^{*}\times \mathbb {C}\) . We prove that the number of short trajectories depends on the location of a in \(\Xi _{\theta }\) , identified as union of certain curves defined in the complex plane as the level sets of some harmonic functions. We point in the "phase transition" on the topology of \(\Xi _{\theta }\) which leads to a double "tree case" for some value of \(\arg A\) .
