In 2011, Gross derived an \(O(n^2)\)-time algorithm to calculate the genus distribution of a given cubic Halin graph. In this paper, with the help of the overlap matrix, we obtain a recurrence relation for the Euler-genus polynomials of cubic caterpillar-Halin graphs. Explicit formulas for the embeddings of the cubic caterpillar-Halin graphs into surfaces with Euler-genus 0, 1 and 2 are also obtained.

中文翻译：

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三次毛虫-Halin图的Euler-属分布

在2011年，格罗斯推导了\（O（n ^ 2）\） -time算法来计算给定三次Halin图的属分布。在本文中，借助重叠矩阵，我们获得了三次履带-Halin图的Euler-genus多项式的递归关系。还获得了将三次履带-Halin图嵌入到Euler属0、1和2的曲面中的显式。