  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 $F_k$.
