In our recent paper, we systematized an inverse algorithm to obtain quiver gauge theory living on theM2-branes probing the singularities of a special kind of Calabi-Yau fourfold which were complex cones over toric Fanoℙ3,ℬ1,ℬ2,ℬ3. These quiver gauge theories cannot be given a dimer tiling presentation. We use the method of partial resolution to show that the toric data ofℂ4and Fanoℙ3can be embedded inside the toric data of Fanoℬtheories. This method indirectly justifies that the two-node quiver Chern-Simons theories corresponding toℂ4, Fanoℙ3, and their orbifolds can be obtained by higgsing matter fields of the three-node parent quiver corresponding to Fanoℬ1,ℬ2,ℬ3,ℬ4threefold.