  A defensive alliance in an undirected graph $G=(V,E)$ is a non-empty set of
vertices $S$ satisfying the condition that every vertex $v\in S$ has at least
as many neighbours (including itself) in $S$ as it has in $V\setminus S$. We
consider the notion of global minimality in this paper. We are interested in
globally minimal defensive alliance of maximum size. 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
by showing that the Globally Minimal Defensive Alliance problem is FPT
parameterized by the neighbourhood diversity of the input graph. The result for
neighborhood diversity implies that the problem is FPT parameterized by vertex
cover number also. We prove that the problem parameterized by the vertex cover
number of the input graph does not admit a polynomial compression unless coNP
$\subseteq$ NP/poly. We show that the problem is W[1]-hard parameterized by a
wide range of fairly restrictive structural parameters such as the feedback
vertex set number, pathwidth, treewidth and treedepth. We also proved that,
given a vertex $r \in V(G)$, deciding if $G$ has a globally minimal defensive
alliance of any size containing vertex $r$ is NP-complete.
