\leq k$, and it is also reasonable to assume that $1 \leq q_u \leq p_u$ for every $u \in V$ (as in the above versions). We consider a relation between such $(p,q)$-connectivity versions of [SL]{} and $q$-connectivity versions of [NA]{}, that formally are defined as follows.

Now we define several particular important cases of [NA]{}.

The [$a$-Based Connectivity Augmentation]{} problem was defined in [@KN-aug], where it was also shown to admit an $H(r(D))$-approximation algorithm. The study of this problem in [@KN-aug] is motivated by the following observation.

\[[@KN-aug]\] \[o:aug\] If directed [$a$-Based Connectivity Augmentation]{} admits approximation ratio $\rho$, then directed [Connectivity Augmentation]{} admits the following approximation ratios: $2\rho$ if $Q \neq V$, and $2k \rho$ if $Q=V$.

For an edge-set/graph $J$ let $\delta_J(X)$ denote the set of edges in $J$ from $X$ to $V \setminus X$. This paper is motivated by a recent paper of Fukunaga [@Fuk], that defined the connectivity function $\kappa'$, and observed that [$\kappa'$-SSL]{} is equivalent to the particular case of [$s$-Based Rooted SNA]{} with node-connectivity requirements and [*edge-costs*]{}, and with $\delta_G(s)=\emptyset$. Here, by the same reduction as in Definition \[d:pq\] (for a proof see Section \[s:hard\]), we further observe the following.

\[o:reduction\] For both directed and 