following.

-   The case of uniform demands (namely, $d_v=k$ for all $v \in V$) in [SSL]{} corresponds to the case of rooted uniform requirements (namely $r_{sv}=k$ for all $v \in V \setminus \{s\}$) in [$s$-Based Rooted SNA]{}.

-   The case of unit costs (namely, $c_v=1$ for all $v \in V$) in [SSL]{} corresponds to the version of [$s$-Based Connectivity Augmentation]{} with node-costs, when we can pick $p_v$ edges from $s$ to $v$ by a cost of $1$.

Directed [Rooted SNA]{} with edge-costs and uniform requirements $r_{sv}=k$ for all $v \in V$ can be solved in polynomial time [@FT]; this easily implies that also undirected [Rooted $s$-Based SNA]{} with edge-costs can be solved in polynomial time. Thus the same holds for [$\kappa'$-SSL]{}, since [$\kappa'$-SSL]{} is a particular case of [$s$-Based Rooted SNA]{} with edge-costs.

Frank [@Frank] showed that [*directed*]{} [$s$-Based Connectivity Augmentation]{} with $\delta_G(s)=\emptyset$ is NP-hard. Using a slight modification of his reduction we can show that the problem is in fact [Set-Cover]{} hard to approximate, and thus is $\Omega(\log n)$-hard to approximate. Given an instance of [Set-Cover]{}, where a family $A$ of sets needs to cover a set $B$ of elements, construct the corresponding directed bipartite graph $G'=(A \cup B,E')$, 