Genus embeddings of a type of graph

In 2003, Liu set up the joint tree model such that an embedding of a graph on a surface can be represented by a joint tree, further by an associated surface of it. In this paper, by dividing an associated surface into segments layer by layer and doing a sequence of operations on the layer division, we obtain the genus of complete bipartite graphs with a different method as before. Based on it, firstly, the genus of a new type of graph is derived. Then the number of non-homeomorphic genus embeddings of some complete bipartite graph is easily estimated.