A polynomial kernel for $3$-leaf power deletion

For a non-negative integer $\mathrm{\ell }$, the $\mathrm{\ell }$-leaf power of a tree $T$ is a simple graph $G$ on the leaves of $T$ such that two vertices are adjacent in $G$ if and only if their distance in $T$ is at most $\mathrm{\ell }$. We provide a polynomial kernel for the problem of deciding whether we can delete at most $k$ vertices to make an input graph a $3$-leaf power of some tree. More specifically, we present a polynomial-time algorithm for an input instance $(G,k)$ for the problem to output an equivalent instance $(G^{\mathrm{\prime }},k^{\mathrm{\prime }})$ such that $k^{\mathrm{\prime }}\le k$ and $G^{\mathrm{\prime }}$ has at most $O(k^{14})$ vertices.