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We start with some preliminary graph-theoretic remarks.   Let us first recall some vocabulary.   Let $G = \tuple{\N, R}$ be a graph and $R^*$ be the transitive and symmetric closure of $R$. A \begin{itemize}  \item[]A  set of nodes $S\subseteq \N$ is \emph{connected} said to be:  \begin{itemize}  \item[]\emph{connected}  if for any $i,j \in \S$: $iR^*j$, \emph{strongly \item[]\emph{strongly  connected} if for any $i,j \in \S$: $iRj$, and \emph{closed} \item[]\emph{closed}  if for any $i\in S$, $j\notin S$, $i\centernot{R}j$.  A it is not the case that $iRj$  \item[]a  \emph{connected component} of $G$ is a set $C\subseteq \N$ such that, if  for any $i,j \in \N$: $iR^*j$ if and only if $i,j\in C$. A set of nodes is said to be ``strongly connected" if there is a directed path from any node to any other node in the set, ``closed" if there is no link from nodes in the set to nodes outside the set, and aperiodic if the greatest common divisor of all directed cycles is $1$  A \end{itemize}  \item[]A  path is a sequence of nodes $$, such that, for all $l\in\{1,\dots,k\}$, $i_lRi_{l+1}$, and a $i_lRi_{l+1}$  \item[]A  cycle is a path of length $k$ such that $i_1=i_k$. \end{itemize}  We observe that a graph which is serial and functional contains exactly one cycle in each of its finite connected components.  \begin{fact} \label{fact:connected}