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\subsection{Some graph-theoretic preliminaries}  We start with some preliminary graph-theoretic remarks.   A (directed) path in a graph is a sequence of nodes such that each node in the sequence is related to its successor, and a cycle is a path such that the first and last element of the sequence are the same.   We Let us  first note that, in influence graphs, no two cycles belong to the same connected component\footnote{A recall some vocabulary. A  connected component $C$ of a graph $G = \tuple{\N, R}$ is a set of nodes $C\subseteq \N$ such that: for any $i,j\in C$, $iR^+j$, where $R^+$ is the symmetric closure of $R$.}: $R$.   A path in a graph is a sequence of nodes $$, such that, for all $l\in\{1,\dots,k\}$, $i_lRi_{l+1}$, and a cycle is a path of length $k$ such that $i_1=i_k$$.   We observe that our influence graphs are of a special kind with respect to cycles: each connected component contains exactly one cycle.  \begin{fact} \label{fact:connected}  Let $G$ be an influence graph and $C$ be a finite connected component of $G$. Then $C$ contains exactly one cycle.  \end{fact}