case of $\lambda_G^{p,q}$, but we are also not aware on any work on [*undirected*]{} [$\kappa$-SL]{} (except that it is shown in [@III] that the problem can be solved in polynomial time for $k=\max_{v \in V} d_v \leq 2$ and is NP-hard for $k \geq 3$).

2.  $\hat{\kappa}(S,v)$ is the maximum number of $(S,v)$-paths no two of which have a common node in $V \setminus \{v\}$ if $v \notin S$, and $\hat{\kappa}(S,v)= \infty$ otherwise; equivalently, $\hat{\kappa}(S,v)$ is the minimum size $|C|$ of a cut $C \subseteq E \cup V \setminus (\{v\})$ such that $G \setminus C$ has no $(S,v)$-path.\
    Then $\hat{\kappa}(S,v)=\lambda_G^{p,q}(S,v)$ for $p_u=\infty$ and $q_u=1$ for every $u \in V$.

3.  $\kappa'(S,v)=\hat{\kappa}(S,v)$ if $v \notin S$ and $\kappa'(S,v)=1+ \hat{\kappa}(S \setminus \{v\},v)$ if $v \in S$; equivalently, $\kappa'(S,v)$ is the minimum size $|C|$ of a cut $C \subseteq E \cup V \setminus (\{v\})$ such that $G \setminus C$ has no $(S \setminus \{v\},v)$-path.\
    Then $\kappa'(S,v)=\lambda_G^{p,q}(S,v)$ for $p_u=1$ and $q_u=1$ for every $u \in V$.

Given an instance of [SL]{} or of [NA]{}, let $k$ denote the maximum demand $d_{\max}=\max_{v \in V} d_v$ or the maximum requirement $r_{\max}=\max_{uv \in D}r_{uv}$. Note that in [SL]{} problems with $\psi(S,v)=\lambda_G^{p,q}(S,v)$, we may always assume that $p_u 