Other resolutions of $\mathscr{U}$ can be obtained by blowing-up the divisors of the special fiber in a different order. Given any order $(i_{1},i_{2},i_{3})$ on $\{1,2,3\}$, we denote

$\begin{array}[]{ccccccccc}\mathscr{V}_{i_{1}i_{2}i_{3}}&\xrightarrow{\begin{subarray}{c}\text{blow-up of }\\ S_{i_{1}i_{3}},S_{i_{2}i_{3}}\end{subarray}}&\mathscr{V}_{i_{1}i_{2}}&\xrightarrow{\begin{subarray}{c}\text{blow-up of }\\ S_{i_{1}i_{2}}\end{subarray}}&\mathscr{U}_{i_{1}i_{2}}&\xrightarrow{\begin{subarray}{c}\text{blow-up of }\\ D_{i_{1}},D_{i_{2}}\end{subarray}}&\mathscr{U}.\\ \cup&&\cup&&\cup&&\\ E_{i_{1}i_{3}},E_{i_{2}i_{3}}&&E_{i_{1}i_{2}}&&S_{i_{1}i_{2}},S_{i_{1}i_{3}},S_{i_{2}i_{3}}&&\end{array}$

The refinement of the fan of $\mathscr{U}$ corresponding to $\mathscr{V}_{i_{1}i_{2}i_{3}}$ is such that the skeleton $\Sk(\mathscr{V}_{i_{1}i_{2}i_{3}})=\Sk(\mathscr{V}_{123})$ as subspaces in the Berkovich space of $\mathscr{U}_{K}$; it is independent on the chosen order so that we simply denote this subspace by $\Sk(\mathscr{V})$. However, the models $\mathscr{V}_{i_{1}i_{2}i_{3}}$ and $\mathscr{V}_{123}$ induce in general different simplicial subdivisions and different retractions onto $<v_{1},v_{2},v_{3}>$. For instance, the only edge in the interior of $\Sk(\mathscr{V}_{i_{1}i_{2}i_{3}})$ is $<v_{i_{2}},v_{i_{1}i_{3}}>$, which indeed depends on the chosen order. Here below we illustrate the skeletons and the Berkovich retractions in a couple of examples.

The blow-up of $\mathscr{V}_{i_{1}i_{2}i_{3}}$ along the toric strata $D_{i_{2}}\cap E_{i_{1}i_{3}}$ and $E_{i_{1}i_{2}}\cap D_{i_{3}}^{\prime}$ yields a refinement of the fan which coincides with the fan of $\mathscr{G}$, constructed at the end of Section 4.3. It follows that the model $\mathscr{G}$ dominates all resolutions $\mathscr{V}_{i_{1}i_{2}i_{3}}$ independently on the order, hence all Berkovich retractions $\rho_{\mathscr{V}_{i_{1}i_{2}i_{3}}}$, $\rho_{\mathscr{V}_{i_{1}i_{2}}}$ and $\rho_{\mathscr{U}_{i_{1}i_{2}}}$ factors through $\rho_{\mathscr{G}}$.

Our goal is to construct a map $\pi$, composing the Berkovich retraction $\rho_{\mathscr{G}}$ with a collapse $\kappa$ of the additional 3-cell $\tau$ and a combinatorial retraction $\rho$

$\begin{array}[]{ccccccccc}&\pi:&\mathscr{U}_{K}^{\an}&\xrightarrow{\rho_{\mathscr{G}}}&\Sk(\mathscr{G})&\xrightarrow[\text{collapse}]{\kappa}&\Sk(\mathscr{V})&\xrightarrow[\text{retraction}]{\rho}&\Sk(\mathscr{U})\\ &\rotatebox[origin={c}]{270.0}{$=$}&\cup&&\cup&&\cup&&\cup\\ \text{over }\Star(v_{j})^{\prime}\text{\hskip 10.0pt}&\rho_{\mathscr{U}_{i_{1}i_{2}}}:&\pi^{-1}(\Star(v_{j})^{\prime})&\xrightarrow{\rho_{\mathscr{G}}}&\Sk(\mathscr{G})&\xrightarrow{\rho_{\mathscr{V}_{i_{1}i_{2}j}}}&\Sk(\mathscr{V}_{i_{1}i_{2}j})&\xrightarrow{\rho_{\mathscr{U}_{i_{1}i_{2}}}}&\Star(v_{j})^{\prime}\end{array}$

such that, given any vertex $v_{j}$ in $\Sk(\mathscr{U})$, the restriction of $\pi$ over $\Star(v_{j})^{\prime}$ (the $\Star$ is taken with respect to the first barycentric subdivision, as in Definition 3.2.1) is $\rho_{\mathscr{U}_{i_{1}i_{2}}}$ for any order $(i_{1},i_{2},j)$ on $\{1,2,3\}$, i.e. any order where the index $j$ is the biggest. This guarantees that around each $v_{j}$, the map $\pi$ is the Berkovich retraction induced by a small resolution $\mathscr{U}_{i_{1}i_{2}}$ where the strict transform of $D_{j}$ is isomorphic to $D_{j}$, so that we are in the set-up of Corollary C.

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  The retraction $\rho$. We identify again the skeleton $\Sk(\mathscr{V})$ with the polyhedron in $\mathbb{R}^{3}$ described in Section 4.3. On the convex hull $P$ of $v_{23},(-1/4,1/4,1),(0,1/3,1),(0,0,0),v_{3}$ and $(0,1/3,0)$, the retraction $\rho$ is given as follows

  (4.4.1)

  $\displaystyle\begin{split}&(x,y,z)\in P\mapsto\begin{cases}\Big(x+\left(1-\frac{t}{2}\right)z,y+\frac{t}{2}z,0\Big)&\text{ if }x+\left(1-\frac{t}{2}\right)z\leqslant 0\\ \Big(0,\frac{y}{x+1},0\Big)&\text{ if }x+\left(1-\frac{t}{2}\right)z\geqslant 0\end{cases}\\ &\text{where }t=\frac{2y}{x+y+1}.\end{split}$

  Here is a pictorial description for certain values of $t$:

  We extend the definition of $\rho$ to $\Sk(\mathscr{V})$ by symmetry along the medians of the triangles $<v_{1},v_{2},v_{3}>$ and $<v_{12},v_{13},v_{23}>$. In particular, we note that the image of $<v_{12},v_{13},v_{23}>$ is the graph in $<v_{1},v_{2},v_{3}>$ of Definition 3.2.1.

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  The combinatorial retraction $\pi^{\prime}$. We define the collapse $\kappa$ as the projection of the additional $3$-cell $\tau$ of $\Sk(\mathscr{G})$ onto $<v_{12},v_{13},v_{23}>$ along the $z$-direction. We call $\pi^{\prime}\coloneqq\rho\circ\kappa$ the combinatorial retraction of the skeleton $\Sk(\mathscr{G})$ onto $\Sk(\mathscr{U})=<v_{1},v_{2},v_{3}>$.

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  Finally, we check that $\pi^{\prime}=\rho_{\mathscr{U}_{i_{1}i_{2}}}$ over $\Star(v_{j})^{\prime}$. As the preimage of $\Star(v_{j})^{\prime}$ is disjoint from $<v_{12},v_{13},v_{23},v^{\prime}_{123}>$, we have to prove that $\rho=\rho_{\mathscr{U}_{i_{1}i_{2}}}$. By symmetry of $\rho$, it is enough to check this for $v_{3}$. Over $\Star(v_{3})^{\prime}$ we have $\rho_{\mathscr{U}_{i_{1}i_{2}}}=\rho_{\mathscr{V}_{i_{1}i_{2}}}$; there, the expression of $\rho_{\mathscr{V}_{i_{1}i_{2}}}$ determined in Eq. 4.3.1 coincides with the definition of $\rho$ in Eq. 4.4.1, hence we conclude.