  A locating-dominating set $D$ of a graph $G$ is a dominating set of $G$ where
each vertex not in $D$ has a unique neighborhood in $D$, and the
Locating-Dominating Set problem asks if $G$ contains such a dominating set of
bounded size. This problem is known to be $\mathsf{NP-hard}$ even on restricted
graph classes, such as interval graphs, split graphs, and planar bipartite
subcubic graphs. On the other hand, it is known to be solvable in polynomial
time for some graph classes, such as trees and, more generally, graphs of
bounded cliquewidth. While these results have numerous implications on the
parameterized complexity of the problem, little is known in terms of
kernelization under structural parameterizations. In this work, we begin
filling this gap in the literature. Our first result shows that
Locating-Dominating Set, when parameterized by the solution size $d$, admits no
$2^{o(d \log d)}$ time algorithm unless the Exponential Time Hypothesis fails;
as a corollary, we also show that no $n^{o(d)}$ time algorithm exists under
ETH, implying that the naive $\mathsf{XP}$ algorithm is essentially optimal. We
present an exponential kernel for the distance to cluster parameterization and
show that, unless $\mathsf{NP-hard} \subseteq \mathsf{NP-hard}/$\mathsf{poly}$,
no polynomial kernel exists for Locating-Dominating Set when parameterized by
vertex cover nor when parameterized by distance to clique. We then turn our
attention to parameters not bounded by neither of the previous two, and exhibit
a linear kernel when parameterizing by the max leaf number; in this context, we
leave the parameterization by feedback edge set as the primary open problem in
our study.
