Leaf powers and $k$-leaf powers have been studied for over 20 years, but there are still several aspects of this graph class that are poorly understood. One such aspect is the leaf rank of leaf powers, i.e. the smallest number $k$ such that a graph $G$ is a $k$-leaf power. Computing the leaf rank of leaf powers has proved a hard task, and furthermore, results about the asymptotic growth of the leaf rank as a function of the number of vertices in the graph have been few and far between. We present an infinite family of rooted directed path graphs that are leaf powers, and prove that they have leaf rank exponential in the number of vertices (utilizing a type of subtree model first presented by Rautenbach [Some remarks about leaf roots. Discrete mathematics, 2006]). This answers an open question by Brandst\"adt et al. [Rooted directed path graphs are leaf powers. Discrete mathematics, 2010].

翻译：叶幂图和$k$-叶幂图已被研究超过20年，但这类图结构仍有多个方面尚未完全理解。其中一方面便是叶幂图的叶秩，即使得图$G$成为$k$-叶幂图的最小整数$k$。计算叶幂图的叶秩已被证明是一项困难的任务，而且关于叶秩随图顶点数渐近增长的研究结果甚少。本文构造了一个无穷族的有向根路径图，这些图均为叶幂图，并证明其叶秩随顶点数指数增长（利用Rautenbach首次提出的一类子树模型[Some remarks about leaf roots. Discrete mathematics, 2006]）。这回答了Brandstädt等人提出的一个开放问题[Rooted directed path graphs are leaf powers. Discrete mathematics, 2010]。