have been studied in [@CRST; @CRSTx].

Throughout this paper, $G=(V,E)$ denotes a simple graph (not necessarily connected) of order $|V|=n$ and size $|E|=m$. We denote two adjacent vertices $u$ and $v$ by $u\sim v$. For a nonempty set $X\subseteq V$, and a vertex $v\in V$, $N_X(v)$ denotes the set of neighbors $v$ has in $X$: $N_X(v):=\{u\in X: u\sim v\},$ and the degree of $v$ in $ X$ will be denoted by $\delta_{X}(v)=|N_{X}(v)|$. We denote by ${\delta}$ and ${\Delta}$ the minimum and maximum degree of $G$, respectively. If $G$ is a regular graph with degree ${\Delta}$ we say that $G$ is ${\Delta}$-regular. The subgraph induced by $S\subset V$ will be denoted by $\langle S\rangle $ and the complement of the set $S\in V$ will be denoted by $\overline{S}$.

A nonempty set $S\subseteq V$ with $\langle S\rangle$ connected is a *defensive $k$-alliance* in $G=(V,E)$, $k\in [-\Delta,\Delta]\cap \mathbb{Z}$, if for every $ v\in S$, $$\label{cond-A-Defensiva} \delta _S(v)\ge \delta_{\overline{S}}(v)+k.$$ A vertex $v\in S$ is said to be $k$-*satisfied* by the set $S$, if holds.

We consider the value of $k$ in the set of integers $\mathcal{K}:= [-\Delta,\Delta]\cap \mathbb{Z}$. It may be that there are some values of $k \in\mathcal K$ such that there do 