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تسجيل مستخدم جديدVariation of the local topological structure of graph embeddings
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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.
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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 $prod_{ uin V(G)}frac{2}{deg( u)+2}$, where $deg( u)$ denotes the vertex degree of $ u$. Furthermore we obtain an easy-to-check necessary condition for a given embedding of $G$ to be an embedding of minimum genus.
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