  Given a graph $G = (V, E)$, a non-empty set $S \subseteq V$ is a defensive
alliance, if for every vertex $v \in S$, the majority of its closed neighbours
are in $S$, that is, $|N_G[v] \cap S| \geq |N_G[v] \setminus S|$. The decision
version of the problem is known to be NP-Complete even when restricted to split
and bipartite graphs. The problem is \textit{fixed-parameter tractable} for the
parameters solution size, vertex cover number and neighbourhood diversity. For
the parameters treewidth and feedback vertex set number, the problem is
W[1]-hard. \\ \hspace*{2em} In this paper, we study the defensive alliance
problem for graphs with bounded degree. We show that the problem is
\textit{polynomial-time solvable} on graphs with maximum degree at most 5 and
NP-Complete on graphs with maximum degree 6. This rules out the fixed-parameter
tractability of the problem for the parameter maximum degree of the graph. We
also consider the problem from the standpoint of parameterized complexity. We
provide an FPT algorithm using the Integer Linear Programming approach for the
parameter distance to clique. We also answer an open question posed in
\cite{AG2} by providing an FPT algorithm for the parameter twin cover.
