  We present approximation algorithms for network design problems in some
models related to the $(p,q)$-FGC model. Adjiashvili, Hommelsheim and
M\"uhlenthaler introduced the model of Flexible Graph Connectivity that we
denote by FGC. Boyd, Cheriyan, Haddadan and Ibrahimpur introduced a
generalization of FGC. Let $p\geq 1$ and $q\geq 0$ be integers. In an instance
of the $(p,q)$-Flexible Graph Connectivity problem, denoted $(p,q)$-FGC, we
have an undirected connected graph $G = (V,E)$, a partition of $E$ into a set
of safe edges and a set of unsafe edges, and nonnegative costs
$c\in\mathbb{R}_{\geq0}^E$ on the edges. A subset $F \subseteq E$ of edges is
feasible for the $(p,q)$-FGC problem if for any set of unsafe edges, $F'$, with
$|F'|\leq q$, the subgraph $(V, F \setminus F')$ is $p$-edge connected. The
algorithmic goal is to find a feasible edge-set $F$ that minimizes $c(F) =
\sum_{e \in F} c_e$.
