  A set $S\subseteq V$ of vertices is an offensive alliance in an undirected
graph $G=(V,E)$ if each $v\in N(S)$ has at least as many neighbours in $S$ as
it has neighbours (including itself) not in $S$. We study the classical and
parameterized complexity of the Offensive Alliance problem, where the aim is to
find a minimum size offensive alliance. Our focus here lies on natural
parameter as well as parameters that measure the structural properties of the
input instance. We enhance our understanding of the problem from the viewpoint
of parameterized complexity by showing that (1) the problem is W[1]-hard
parameterized by a wide range of fairly restrictive structural parameters such
as the feedback vertex set number, treewidth, pathwidth, and treedepth of the
input graph; we thereby resolve an open question stated by Bernhard Bliem and
Stefan Woltran (2018) concerning the complexity of Offensive Alliance
parameterized by treewidth, (2) unless ETH fails, Offensive Alliance problem
cannot be solved in time $\mathcal{O}^{*}(2^{o(k \log k)})$ where $k$ is the
solution size, (3) Offensive Alliance problem does not admit a polynomial
kernel parameterized by solution size and vertex cover of the input graph. On
the positive side we prove that (4) Offensive Alliance can be solved in time
$\mathcal{O}^{*}(\tt{vc(G)}^{\mathcal{O}(\tt{vc(G)})})$ where $\tt{vc(G)}$ is
the vertex cover number of the input graph. In terms of classical complexity,
we prove that (5) Offensive Alliance problem cannot be solved in time
$2^{o(n)}$ even when restricted to bipartite graphs, unless ETH fails, (6)
Offensive Alliance problem cannot be solved in time $2^{o(\sqrt{n})}$ even when
restricted to apex graphs, unless ETH fails. We also prove that (7) Offensive
Alliance problem is NP-complete even when restricted to bipartite, chordal,
split and circle graphs.
