Variation of the local topological structure of graph embeddings

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.