  The $2$-cell embeddings of graphs on closed surfaces have been widely
studied. It is well known that ($2$-cell) embedding a given graph $G$ on a
closed orientable surface is equivalent to cyclically ordering the edges
incident to each vertex of $G$. In this paper, we study the following problem:
given a genus $g$ embedding $\mathbb{E}$ of the graph $G$, if we randomly
rearrange the edges around a vertex, i.e., re-embedding, what is the
probability of the resulting embedding $\mathbb{E}'$ having genus $g+\Delta g$?
We give a formula to compute this probability. Meanwhile, some other known and
unknown results are also obtained. For example, we show that the probability of
preserving the genus is at least $\frac{2}{deg(v)+2}$ for re-embedding any
vertex $v$ of degree $deg(v)$ in a one-face embedding; and we obtain a
necessary condition for a given embedding of $G$ to be an embedding with the
minimum genus.
