Let ${\cal X}$ be an snc model, $J\subset I_{\cal X}$ a non-empty subset and $Y\subset D_{J}$ a smooth irreducible variety of dimension less or equal than $n$. Let us assume that $Y$ intersects transversally (in $D_{J}$) all subvarieties $D_{J^{\prime}}$ of $D_{J}$ (for $J^{\prime}\supset J$), and that all intersections $Y\cap D_{J}$ are either empty or irreducible. It is obvious that the blow-up ${\cal X}^{\prime}:=Bl_{Y}({\cal X})$ of ${\cal X}$ with the center at $Y$ is again a snc model.

Let us describe the behavior of $S_{\cal X}$ under simple blow-ups. To the set of vertices we add a new vertex corresponding to the divisor $\widetilde{Y}$ obtained from $Y$:

$$I_{{\cal X}^{\prime}}=I_{\cal X}\sqcup\{new\},\,\,\,D_{new}:=\widetilde{Y}\,\,.$$

The degree of the new divisor is (for both the first and the second type)

$$d_{new}:=\sum_{i\in J}d_{j}\,\,.$$

For blow-ups of the first type we have automatically $\#J>1$. Here is the list of faces of $S_{\cal{X}^{\prime}}$:

1) $I^{\prime}$ for $I^{\prime}\in Faces(S_{\cal X}),\,I^{\prime}\not\subset J$;

2) $I^{\prime}\sqcup\{new\}$ for $I^{\prime}\in Faces(S_{\cal X}),\,I^{\prime}\neq J,\,I^{\prime}\cup J\in Faces(S_{\cal X})$;

3) the vertex $\{new\}$.
For blow-ups of the second type the list of faces of $S_{{\cal X}^{\prime}}$ is

1) $I^{\prime}$ for $I^{\prime}\in Faces(S_{\cal X})$;

2) $I^{\prime}\sqcup\{new\}$ for $I^{\prime}\in Faces(S_{\cal X}),\,I^{\prime}\supset J,\,Y\cap D_{I^{\prime}}\neq\emptyset$;

3) the vertex $\{new\}$.

On can deduce from results [AKMW] the following