A graph $G=(V,E)$ is a $k$-leaf power if there is a tree $T$ whose leaves are the vertices of $G$ with the property that a pair of leaves $u$ and $v$ induce an edge in $G$ if and only if they are distance at most $k$ apart in $T$. For $k\le 4$, it is known that there exists a finite set $F_k$ of graphs such that the class $L(k)$ of $k$-leaf power graphs is characterized as the set of strongly chordal graphs that do not contain any graph in $F_k$ as an induced subgraph. We prove no such characterization holds for $k\ge 5$. That is, for any $k\ge 5$, there is no finite set $F_k$ of graphs such that $L(k)$ is equivalent to the set of strongly chordal graphs that do not contain as an induced subgraph any graph in $F_k$.

翻译：若存在一棵树$T$，其叶子节点与图$G=(V,E)$的顶点一一对应，且满足：$G$中两个叶子节点$u$和$v$之间存在边当且仅当它们在$T$中的距离不超过$k$，则称$G$为$k$-叶幂图。已知当$k\le 4$时，存在有限图集$F_k$，使得$k$-叶幂图类$L(k)$恰好可由那些不包含$F_k$中任何图为导出子图的强弦图构成。本文证明该性质在$k\ge 5$时不成立。换言之，对任意$k\ge 5$，不存在有限图集$F_k$使得$L(k)$等价于所有不包含$F_k$中任何图为导出子图的强弦图集合。