subgraph $G_\ell$ by $c = |\mathcal{S}_{G_\ell}|$. The number $c$ of (strongly) connected components in the obligatory subgraph has recently been used as parameter to obtain FPT results [@BBNN17; @sorge2012new]. In this section, we also provide an FPT result with respect to this parameter. More specifically, we will present an algorithm for <span style="font-variant:small-caps;">MinPAC</span> that runs in $O(2^c\cdot c^2 \cdot n + m + 4^c \cdot c^{2c - 3/2})$ time. Our algorithm runs in three phases. In the first phase, it shrinks the graph to a *relevant* subgraph in which each vertex $v$ has at most one arc towards each SCC that does not contain $v$ (). In the second phase, it uses a dynamic programming algorithm to compute the minimum cost to connect each SCC to each subset of other SCCs (). In the last phase, it exhaustively tries all combinations of connecting SCCs to find an optimal solution ().

#### Phase 1.

The following lemma specifies the conditions under which we can remove arcs. It plays a central role in this phase. The basic idea herein is to remove, for each vertex $v \in V(G)$ and each SCC $S$, all but the cheapest arc from $v$ to vertices in 