  A set $D$ of vertices of a graph is a \emph{defensive alliance} if, for each
element of $D$, the majority of its neighbours are in $D$. We consider the
notion of local minimality in this paper. We are interested in finding a
locally minimal defensive alliance of maximum size. In Locally Minimal
Defensive Alliance problem, given an undirected graph $G$, a positive integer
$k$, the question is to check whether $G$ has a locally minimal defensive
alliance of size at least $k$. This problem is known to be NP-hard, but its
parameterized complexity remains open until now. We enhance our understanding
of the problem from the viewpoint of parameterized complexity. The main results
of the paper are the following: (1) Locally Minimal Defensive Alliance
restricted to the graphs of minimum degree at least 2 is fixed-parameter
tractable (FPT) when parameterized by the combined parameters solution size
$k$, and maximum degree $\Delta$ of the input graph, (2) Locally Minimal
Defensive Alliance on the graphs of minimum degree at least 2, admits a kernel
with at most $k^{k^{\mathcal{O}(k)}}$ vertices. In particular, the problem
parameterized by $k$ restricted to $C_3$-free and $C_4$-free graphs of minimum
degree at least 2, admits a kernel with at most $k^{\mathcal{O}(k)}$ vertices.
Moreover, we prove that the problem on planar graphs of minimum degree at least
2, admits an FPT algorithm with running time
$\mathcal{O}^{*}(k^{2^{\mathcal{O}(\sqrt{k})}})$. Finally, we prove that (4)
Locally Minimal Defensive Alliance Extension is NP-complete.
