A Halin graph $G$ is a plane graph consisting of a plane embedding of a tree $T$ of order at least 4 containing no vertex of degree 2, and of a cycle $C$ connecting all leaves of $T$. Let $f_h(n,G)$ be the maximum number of copies of $G$ in a Halin graph on $n$ vertices. In this paper, we give exact values of $f_h(n,G)$ when $G$ is a path on $k$ vertices for $2\le k\le 5$. Moreover, we develop a new graph transformation preserving the number of vertices, so that the resulting graph has a monotone behavior with respect to the number of short paths.

中文翻译：

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Halin图中的最大短路径数

哈林图$G$是由树的平面嵌入组成的平面图$\mathrm{时间}$至少为 4 阶且不包含 2 次顶点的环$C$连接所有叶子$\mathrm{时间}$。让$F_H（n,G）$是最大副本数$G$在 Halin 图中$n$顶点。在本文中，我们给出了精确值$F_H（n,G）$什么时候$G$是一条路径$k$顶点为$2\le k\le 5$。此外，我们开发了一种新的图变换，保留了顶点的数量，以便生成的图在短路径的数量方面具有单调行为。