图嵌入的局部拓扑结构的变化
Variation of the local topological structure of graph embeddings
摘要：
图在闭曲面上的$2$ -胞腔嵌入已经得到了广泛的研究。将给定的图$G$嵌入到闭可定向曲面上的（$2$ -cell）等价于对关联到$G$的每个顶点的边进行循环排序.本文研究如下问题：给定图$G$的亏格$g$嵌入$\mathbb{E}$，如果我们随机地重新排列围绕一个顶点的边，即，重新嵌入，得到的嵌入$\mathbb{E}'$具有亏格$g+Δg$的概率是多少？我们给出一个公式来计算这个概率。同时，也得到了一些已知和未知的结果.例如，我们表明，保留属的概率至少是$\frac{2}{deg(v)+2}$重新嵌入任何顶点$v$度$deg(v)$在一个面嵌入，我们得到了一个必要条件为给定的嵌入$G$是一个嵌入的最小属。
英文摘要：
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 $g+Δg$? 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.
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