  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 $\epsilon$ of the graph $G$ and a vertex of $G$,
how many different ways of reembedding the vertex such that the resulting
embedding $\epsilon'$ is of genus $g+\Delta g$? We give formulas to compute
this quantity and the local minimal genus achieved by reembedding. In the
process we obtain miscellaneous results. In particular, if there exists a
one-face embedding of $G$, then the probability of a random embedding of $G$ to
be one-face is at least $\prod_{\nu\in V(G)}\frac{2}{deg(\nu)+2}$, where
$deg(\nu)$ denotes the vertex degree of $\nu$. Furthermore we obtain an
easy-to-check necessary condition for a given embedding of $G$ to be an
embedding of minimum genus.
