ag.algebraic geometry - Local characterization of semistability

It is known that a morphism of schemes $fcolon X to S$ is smooth at a point $x in X$ if and only if there is an open neighborhood $U$ of $x$ and an étale map $g colon U to mathbb A^n_S$ such that $g circ p=f_{|U}$, where $p colon mathbb A^n_S to S$ is the natural projection.

I'm looking for a similar characterization for semistable curves $f colon X to S$. I'm interested in the case $S=Spec(k)$, with $k$ a field, and in the case $S=Spec(V)$, with $V$ a discrete valuation ring, where now $X$ is generically smooth.

In particular my question is: in the second case it is true that we can find $lbrace Spec(R_i)rbrace _{i in I}$, an affine open covering of $X$, such that for each $i$, there is an étale map $V[x,y]/(xy-pi) to R_i$, where $pi$ is a uniformizer of $V$?

Thanks.

Ricky