Liu created joint trees, initiated in his early paper in 1979, of a graph in 2003 which provided a foundation on studying embedding genus distribution.

2003年刘提出的图的联树（1979的文章体现了这种思想）为嵌入亏格分布的研究提供了理论基础。

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This is the problem of embedding genus distribution for a graph.

这就是图的嵌入亏格分布问题。

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In this paper, We obtain the total genus polynomials for two new classes of 4-regular graphs by using the joint tree model of a graph embedding introduced by Yanpei Liu.

在刘彦佩提出的联树法的基础上，通过分类一类新图类的可定向嵌入曲面求出了这类图类的可定向嵌入的亏格分布。

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