Variation of the local topological structure of graph embeddings
Abstract
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 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 E' having genus g+ 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 2deg(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.