  The $k$-leaf power graph $G$ of a tree $T$ is a graph whose vertices are the
leaves of $T$ and whose edges connect pairs of leaves at unweighted distance at
most~$k$ in $T$. Recognition of the $k$-leaf power graphs for $k \geq 7$ is
still an open problem. In this paper, we provide two algorithms for this
problem for sparse leaf power graphs. Our results shows that the problem of
recognizing these graphs is fixed-parameter tractable when parameterized both
by $k$ and by the degeneracy of the given graph. To prove this, we first
describe how to embed the leaf root of a leaf power graph into a product of the
graph with a cycle graph. We bound the treewidth of the resulting product in
terms of $k$ and the degeneracy of $G$. The first presented algorithm uses
methods based on monadic second-order logic (MSO$_2$) to recognize the
existence of a leaf power as a subgraph of the product graph. Using the same
embedding in the product graph, the second algorithm presents a dynamic
programming approach to solve the problem and provide a better dependence on
the parameters.
