connected and contains a crucial edge. {#SSec:SSCCrucial}

We call an edge $e$ of a strongly connected graph $G$ crucial if $G
\setminus \{e\}$ is not strongly connected. Suppose $e = [ab] \in G$ is a crucial edge.

Denote by $\mathcal R_e^-$ (resp., $\mathcal R_e^+$) the set of all subgraphs $H \subset G$ such that $e \notin H$ (resp., $e \in H$). Let $H
\in \mathcal R_e^-$ be acyclic. Since $G \setminus \{e\}$ is not strongly connected and contains one edge less than $G$, one has by Clause \[SSec:NotSSC\] above $$\label{Eq:SumNoE}
\SumSub(\alpha,\mathcal R_e^-) = \SumSub(\alpha,G \setminus \{e\}) = 0.$$ Let now $H \in \mathcal R_e^+$ be acyclic; such $H$ contains no directed paths joining $b$ with $a$. Since $G \setminus \{e\}$ is not strongly connected, $G \setminus \{e\}$ does not contain a directed path joining $a$ with $b$ either. It means that such path in $H$ will necessarily contain $e$, and therefore the graph $H/e \subset G/e$ (obtained by contraction of the edge $e$) is acyclic. The converse is true for any $e$: if $e \in H$ and $H/e \subset G/e$ is acyclic then $H \subset G$ is acyclic, too. The graph $G/e$ is strongly connected, contains one edge less (and 