In 19, the authors give minimal embedded toric resolutions of ADE-singularities in C³ by constructing regular refinements of their dual Newton polyhedrons with the elements of their embedded valuation sets derived from the jet schemes constructed in 18. On the other hand in 1 and 2, the authors represent the Gr\"obner fan of a Newton non-degenerate variety and prove that a regular refinement of the Gr\"obner fan of such a singularity yields an embedded toric resolution. In this paper, we reconstruct embedded toric resolutions of ADE-singularities: We give the explicit constructions of their Gr\"obner fans and refine them using the concept of profile.
Büşra Karadeniz Şen (Tue,) studied this question.