and define $S_{u_1,u_2}:=V\setminus\{u_1,u_2\}$. If $u_1\nsim u_2$, then there is $w\in V$ with $u_1,u_2\in N(w)$ since ${\delta}(u_1)+{\delta}(u_2)= 2{\Delta}>|S_{u_1,u_2}|$; in fact, $S_{u_1,u_2}$ is not a defensive (${\Delta}-2$)-alliance in $G$. So, $S_{u_1,u_2}$ may be an exact defensive (${\Delta}-2$)-alliance in $G$, if $u_1\sim u_2$; then there are at most $m$ exact defensive (${\Delta}-2$)-alliances with $n-2$ vertices. Consider now $u_1,\ldots,u_r\in V$ with $3\le r\le {\Delta}-1$ and $u_i\neq u_j$ if $i\neq j$. Note that $S_r:=V\setminus\{u_1,\ldots,u_r\}$ is not a defensive (${\Delta}-2$)-alliance in $G$ if $r>3$, since $N(\overline{S_r},S_r)\ge r({\Delta}-r+1)=2{\Delta}-r+(r-2)({\Delta}-r)> 2{\Delta}+1-r \ge |S_r|$. Besides, if $r=3$ and ${\Delta}\ge5$ (thus ${\Delta}-r\ge2$) we have the same inequality and then $S_r$ is not a defensive (${\Delta}-2$)-alliance in $G$. Note that, if $r=3$ and $n=2{\Delta}$, then $N(\overline{S_r},S_r)\ge 2{\Delta}-r+(r-2)({\Delta}-r)>2{\Delta}-r=n-r\ge |S_r|$ and we also conclude that $S_r$ is not a defensive (${\Delta}-2$)-alliance in $G$. However, if $r=3$, ${\Delta}=4$ and $n=2{\Delta}+1$ (thus, $n=9$), then $S_r$ may be an exact defensive (${\Delta}-2$)-alliance in $G$. But a simple computation gives that these five graphs $G$ verify $A_{2}(G)< 9+18+2$.

We analyze separately the cases $n=2{\Delta}$ and $n=2{\Delta}+1$. Assume first that $n=2{\Delta}$. We only need to compute the possible exact defensive (${\Delta}-2$)-alliances in $G$ with cardinality ${\Delta}$, since every defensive (${\Delta}-2$)-alliance has at least ${\Delta}$ vertices and $n=2{\Delta}$. If $S$ 