  The class ${\cal L}_k$ of $k$-leaf powers consists of graphs $G=(V,E)$ that
have a $k$-leaf root, that is, a tree $T$ with leaf set $V$, where $xy \in E$,
if and only if the $T$-distance between $x$ and $y$ is at most $k$. Structure
and linear time recognition algorithms have been found for $2$-, $3$-, $4$-,
and, to some extent, $5$-leaf powers, and it is known that the union of all
$k$-leaf powers, that is, the graph class ${\cal L} = \bigcup_{k=2}^\infty
{\cal L}_k$, forms a proper subclass of strongly chordal graphs. Despite from
that, no essential progress has been made lately. In this paper, we use the new
notion of clique arrangements to suggest that leaf powers are a natural special
case of strongly chordal graphs. The clique arrangement ${\cal A}(G)$ of a
chordal graph $G$ is a directed graph that represents the intersections between
maximal cliques of $G$ by nodes and the mutual inclusion of these vertex
subsets by arcs. Recently, strongly chordal graphs have been characterized as
the graphs that have a clique arrangement without bad $k$-cycles for $k \geq
3$. We show that the clique arrangement of every graph of ${\cal L}$ is free of
bad $2$-cycles. The question whether this characterizes the class ${\cal L}$
exactly remains open.

Discrete Mathematics;Combinatorics