text:
\begin{document}

\maketitle

\footnotetext[1]{Facult\'e des Sciences A\"{\i}n Chock,
D\'epartement de Math\'ematiques et Informatique, Km 8 route d'El Jadida,
BP 5366 Maarif, Casablanca, Maroc}
\footnotetext[2]{Aix Marseille Univ, CNRS, Centrale Marseille, I2M, Marseille, France}
\footnotetext[4]{{\tt aboussairi@hotmail.com}}
\footnotetext[3]{{\tt cherguibrahim@gmail.com}}
\footnotetext[5]{{\tt pierre.ille@univ-amu.fr}}
\footnotetext[6]{{\tt zaidi.fsac@gmail.com}}

\begin{abstract}
Given a 3-hypergraph $H$,
a subset $M$ of $V(H)$ is a module of $H$ if for each $e\in E(H)$ such that
$e\cap M\neq\emptyset$ and
$e\setminus M\neq\emptyset$, there exists $m\in M$ such that $e\cap M=\{m\}$ and for every $n\in M$, we have
$(e\setminus\{m\})\cup\{n\}\in E(H)$.
For example,
$\emptyset$, $V(H)$ and $\{v\}$, where $v\in V(H)$, are modules of $H$, called trivial.
A 3-hypergraph is prime if all its modules are trivial.
Furthermore, a prime 3-hypergraph is critical if all its induced subhypergraphs, obtained by removing one vertex, are not prime.
We characterize the critical 3-hypergraphs.
\end{abstract}

\medskip

\noindent {\bf Mathematics Subject Classifications (2010):} 05C65, 05C75, 05C76.

\medskip

\noindent {\bf Key words:} 3-hypergraph, module, prime, critical.

\section{Introduction}

Let $H$ be a 3-hypergraph.
A tournament $T$, with the same vertex set as $H$, is a realization of $H$ if the edges of $H$ are exactly the 3-element subsets of the vertex set of $T$ that induce 3-cycles.
In~\cite{BCIZ20}, we characterized the 3-hypergraphs that admit realizations (see \cite[Problem~1]{BILT04}).
To obtain our characterization, we introduced a new notion of a module for hypergraphs.
By using the modular decomposition tree, we demonstrated that a 3-hypergraph is realizable if and only if all its prime (in terms of modular decomposition) induced subhypergraphs are realizable (see \cite[Theorem~13]{BCIZ20}).
Moreover, given a realizable 3-hypergraph $H$, we proved that $H$ is prime if and only if its realizations are prime (see \cite[Theorem~12]{BCIZ20}).
These results lead us to study the prime and induced subhypergraphs of a
prime 3-hypergraph.
Precisely, consider a prime 3-hypergraph $H$.
In \cite[Theorem~13]{BCIZ}, we proved that $H$ admits a prime induced subhypergraph obtained by removing 1 or 2 vertices.
Similar results were obtained for prime digraphs \cite{I97}, for
prime binary relational structures \cite{ST93}, or for prime 2-structures \cite{EHR99}.
Our purpose is to characterize the critical 3-hypergraphs, that is, the prime 3-hypergraphs all the subhypergraphs of which, obtained by removing one vertex, are not prime.

At present, we formalize our presentation.
We consider only finite structures.
A hypergraph $H$ is defined by a vertex set $V(H)$ and an edge set $E(H)$, where
$E(H)\subseteq 2^{V(H)}\setminus\{\emptyset\}$.
Given $k\geq 2$, a hypergraph $H$ is a {\em $k$-hypergraph} if
\begin{equation*}
E(H)\subseteq\binom{V(H)}{k}.
\end{equation*}
Furthermore, a hypergraph $H$ is a {\em $\{k,k+1\}$-hypergraph} if
\begin{equation*}
E(H)\subseteq\binom{V(H)}{k}\cup\binom{V(H)}{k+1}.
\end{equation*}
Let $H$ be a hypergraph.
With each $W\subseteq V(H)$, we associate the {\em subhypergraph} $H[W]$ of $H$ induced by $W$, which is defined on $V(H[W])=W$ by $E(H[W])=\{e\in E(H):e\subseteq W\}$.

\begin{defi}\label{defi_module_hyper}
Let $H$ be a hypergraph.
A subset $M$ of $V(H)$ is a {\em module} of $H$ if for each $e\in E(H)$ such that $e\cap M\neq\emptyset$ and
$e\setminus M\neq\emptyset$, there exists $m\in M$ such that $e\cap M=\{m\}$, and for every $n\in M$, we have
$$(e\setminus\{m\})\cup\{n\}\in E(H).$$
\end{defi}

Let $H$ be a hypergraph.
Clearly, $\emptyset$, $V(H)$ and $\{v\}$, where $v\in V(H)$, are modules of $H$, called {\em trivial modules}.
A hypergraph $H$ is {\em indecomposable} if all its modules are trivial, otherwise it is {\em decomposable}.
A hypergraph $H$ is {\em prime} if it is indecomposable, with $v(H)\geq 3$.
In \cite{BCIZ}, we prove the following result (see \cite[Theorem~9]{BCIZ}).
We need the following notation (see \cite[Notation~5]{BCIZ}).

\begin{nota}\label{nota_p_hyper}
Let $H$ be a hypergraph.
Given $X\subsetneq V(H)$ such that $H[X]$ is prime, consider the following
subsets of $V(H)\setminus X$
\begin{itemize}
\item ${\rm Ext}_H(X)$ denotes the set of $v\in V(H)\setminus X$ such that
$H[X\cup\lbrace v\rbrace ]$ is prime;
\item $\langle X\rangle_H$ denotes the set of $v\in V(H)\setminus X$ such that
$X$ is a module of $H[X\cup\lbrace v\rbrace ]$;
\item for each $y\in X$, $X_H(y)$ denotes the set of $v\in V(H)\setminus X$ such that
$\{y,v\}$ is a module of $H[X\cup\lbrace v\rbrace ]$.
\end{itemize}
The set $\{{\rm Ext}_H(X),\langle X\rangle_H\}\cup\{X_H(y):y\in X\}$ is denoted by $p_{(H,X)}$.
\end{nota}

\begin{thm}\label{prop_Eh-2}
Let $H$ be a 3-hypergraph.
Consider $X\subsetneq V(H)$ such that $H[X]$ is prime.
Set $$\underline{X}=\{y\in X:X_H(y)\neq\emptyset\}.$$
If $H$ is prime, then there exist $v,w\in (V(H)\setminus X)\cup\underline{X}$ such that
$H-\{v,w\}$ is prime.
\end{thm}

The next result follows from Theorem~\ref{prop_Eh-2} (see \cite[Corollary~10]{BCIZ})

\begin{cor}\label{thm1_prime}
Let $H$ be a prime 3-hypergraph.
If $v(H)\geq 4$, then there exist $v,w\in V(H)$ such that $H-\{v,w\}$ is prime.
\end{cor}

Lastly, a prime hypergraph $H$ is {\em critical} if $H-v$ is decomposable for each $v\in V(H)$.
Our purpose is to characterize the critical 3-hypergraphs.

Corollary~\ref{thm1_prime} leads Ille~\cite{I93} to introduce the following auxiliary graph.

\begin{defi}\label{defi_primality_graph}
Let $H$ be a prime 3-hypergraph with $v(H)\geq 5$.
The {\em primality graph} $\mathscr{P}(H)$ associated with $H$ is the graph defined on $V(H)$ as follows.
Given distinct $v,w\in V(H)$, $vw\in E(\mathscr{P}(H))$ if $H-\{v,w\}$ is prime.
When $H$ is critical, it follows from Corollary~\ref{thm1_prime} that $\mathscr{P}(H)$ is nonempty.
\end{defi}

\subsection{Critical and realizable 3-hypergraphs}

Let $T$ be a tournament.
A subset $M$ of $V(T)$ is a {\em module} \cite{S92} of $T$ provided that for any $x,y\in M$ and $v\in V(T)$,
if $xv,vy\in A(T)$, then $v\in M$.
Note that the notions of a module and of a convex subset \cite{HW96} coincide for tournaments.
Moreover, note that the notions of a module and of an interval coincide for linear orders.
Given a tournament $T$, $\emptyset$, $V(T)$ and $\{v\}$, where $v\in V(T)$, are modules of $T$, called {\em trivial modules}.
A tournament is {\em indecomposable} if all its modules are trivial, otherwise it is {\em decomposable}.
A tournament $T$ is {\em prime} if it is indecomposable, with $v(T)\geq 3$.

Lastly, a prime tournament $T$ is {\em critical} if $T-v$ is decomposable for each $v\in V(T)$.
Schmerl and Trotter \cite{ST93} characterized the
critical tournaments.
They obtained the tournaments $T_{2n+1}$, $U_{2n+1}$ and $W_{2n+1}$
defined on $\{0,\ldots, 2n\}$, where $n\geq 1$, as follows.
\begin{itemize}
\item The tournament $T_{2n+1}$ is obtained from $L_{2n+1}$ by reversing all the arcs between even and odd vertices.
\item The tournament $U_{2n+1}$ is obtained from $L_{2n+1}$ by reversing all the arcs between even vertices.
\item The tournament $W_{2n+1}$ is obtained from $L_{2n+1}$ by reversing all the arcs between
$2n$ and the even elements of $\{0,\ldots,2n-1\}$.
\end{itemize}

\begin{thm}[Schmerl and Trotter \cite{ST93}]\label{thm _Critical_ST}
Given a tournament $\tau$, with $v(\tau)\geq 5$,
$\tau$ is critical if and only if $v(\tau)$ is odd and $\tau$ is isomorphic to
$T_{v(\tau)}$, $U_{v(\tau)}$, or $W_{v(\tau)}$.
\end{thm}

A realization of a 3-hypergraph is defined as follows (see ~\cite[Definition~10]{BCIZ20}).
To begin, we associate with each tournament a 3-hypergraph in the following way
(see ~\cite[Definition~9]{BCIZ20}).

\begin{defi}\label{C_3}
The {\em $3$-cycle} is the tournament $C_3=(\{0,1,2\},\{01,12,20\})$.
Given a tournament $T$, the {\em $C_3$-structure} of $T$ is the 3-hypergraph $C_3(T)$ defined on
$V(C_3(T))=V(T)$ by
$$E(C_3(T))=\{X\subseteq V(T):T[X]\ \text{is isomorphic to}\ C_3\}.$$
\end{defi}

\begin{defi}
Given a 3-uniform hypergraph $H$, a tournament $T$, with $V(T)=V(H)$, {\em realizes} $H$ if $H=C_3(T)$.
We say also that $T$ is a {\em realization} of $H$.
\end{defi}

In~\cite{BCIZ20}, we proved the following result (see  \cite[Theorem~12]{BCIZ20})

\begin{thm}\label{thm_same_primality}
Consider a realizable and 3-hypergraph $H$.
For a realization $T$ of $H$, we have $H$ is prime if and only if $T$ is prime.
\end{thm}

The next result follows from Theorems~\ref{thm _Critical_ST} and \ref{thm_same_primality} (see \cite[Theorem~49]{BCIZ20}).

\begin{cor}\label{cor_reali_critical}
Given a realizable 3-hypergraph $H$,
$H$ is critical if and only if $v(H)$ is odd and $H$ is isomorphic to $C_3(T_{v(H)})$, $C_3(T_{v(H)})$, or
$C_3(T_{v(H)})$.
\end{cor}

\begin{defi}\label{defi_reali_critical}
Given a critical 3-hypergraph $H$, we say that $H$ is {\em circular} if $v(H)$ is odd and
$H$ is isomorphic to $C_3(T_{v(H)})$.
\end{defi}

\subsection{A construction of 3-hypergraphs}

The construction is done from the set of the components of a graph.
We use the following notation.

\begin{nota}\label{nota_comp}
Consider a graph $\Gamma$.
The set of the components of $\Gamma$ is denoted by $\mathfrak{C}(\Gamma)$.
Furthermore, set
$\mathfrak{C}_{{\rm even}}(\Gamma)=\{C\in\mathfrak{C}(\Gamma):v(C)\equiv 0 \mod 2\}$ and
$\mathfrak{C}_{{\rm odd}}(\Gamma)=\{C\in\mathfrak{C}(\Gamma):v(C)\equiv 1 \mod 2\}$.
Lastly, set
$\mathfrak{C}_1(\Gamma)=\{C\in\mathfrak{C}(\Gamma):v(C)=1\}$ and
$V_1(\Gamma)=\{v\in V(\Gamma):\{v\}\in\mathfrak{C}_1(\Gamma)\}$.
For each $C\in\mathfrak{C}(\Gamma)$, set
\begin{equation*}
w(C)=\
\begin{cases}
\text{$v(C)/2$ if $C\in\mathfrak{C}_{{\rm even}}(\Gamma)$}\\
\text{or}\\
\text{$(v(C)-1)/2$ if $C\in\mathfrak{C}_{{\rm odd}}(\Gamma)$}.
\end{cases}
\end{equation*}
\end{nota}

We consider a graph $\Gamma$,
\begin{equation}\label{critical_construction_0}
\text{all the components of which are paths.}
\end{equation}
For each $n\geq 1$, recall that the {\em path} $P_n$ is defined on $V(P_n)=\{0,\ldots,n-1\}$ by
\begin{equation*}
E(P_n)=\
\begin{cases}
\text{$\emptyset$ if $n=1$,}\\
\text{$\{k(k+1):0\leq k\leq n-2\}$ if $n\geq 2$.}
\end{cases}
\end{equation*}

We suppose that
\begin{align}\label{critical_construction_1}
\filledstar\ &\mathfrak{C}(\Gamma)\setminus\mathfrak{C}_1(\Gamma)\neq\emptyset,&\nonumber\\
\filledstar\ &\text{for each $C\in\mathfrak{C}_{{\rm odd}}(\Gamma)$, if $V(H)\setminus V(C)\neq\emptyset$, then}\nonumber\\
&\hspace{40mm}\text{
$(\mathfrak{C}(\Gamma)\setminus\mathfrak{C}_1(\Gamma))\setminus\{C\}\neq\emptyset$,}\\
\filledstar\ &\text{for each $C\in\mathfrak{C}_{{\rm even}}(\Gamma)$,
if $|V(H)\setminus V(C)|\geq 2$, then}\nonumber\\
&\hspace{40mm}\text{
$(\mathfrak{C}(\Gamma)\setminus\mathfrak{C}_1(\Gamma))\setminus\{C\}\neq\emptyset$}.\nonumber
\end{align}
Consider also a $\{2,3\}$-hypergraph $\mathbb{H}$ defined on $\mathfrak{C}(\Gamma)$ satisfying
\begin{align}\label{critical_construction_2}
\filledsquare\ &\text{for each $\varepsilon\in E(\mathbb{H})$, $|\varepsilon|=2$ or $3$},\nonumber\\
\filledsquare\ &\text{for each $\varepsilon\in E(\mathbb{H})$, if $|\varepsilon|=2$, then
$\varepsilon\cap\mathfrak{C}_{{\rm even}}(\Gamma)\neq\emptyset$ and
$\varepsilon\cap\mathfrak{C}_{{\rm odd}}(\Gamma)\neq\emptyset$,}\\
\filledsquare\ &\text{for each $\varepsilon\in E(\mathbb{H})$, if $|\varepsilon|=3$, then
$\varepsilon\subseteq\mathfrak{C}_{{\rm odd}}(\Gamma)$.\nonumber}
\end{align}
With $\Gamma$ and $\mathbb{H}$, we associate the 3-hypergraph
$\Gamma\bullet\mathbb{H}$ defined on $V(\Gamma)$ in the following manner.
For each $C\in\mathfrak{C}(\Gamma)$,
we consider an isomorphism $\varphi_C$ from the path $P_{v(C)}$ onto $C$.
With each $C\in\mathfrak{C}_{{\rm odd}}(\Gamma)\setminus\mathfrak{C}_{1}(\Gamma)$,
we associate the set
$$E_C=\varphi_C(E(C_3(U_{v(C)}))).$$
Moreover, consider $\varepsilon\in E(\mathbb{H})$ such that $|\varepsilon|=2$.
There exist $C\in\mathfrak{C}_{{\rm even}}(\Gamma)$ and
$D\in\mathfrak{C}_{{\rm odd}}(\Gamma)$ such that $e=CD$.
Associate with $\varepsilon$ the set
$$E_\varepsilon=\{\varphi_C(2i)\varphi_C(2j+1)\varphi_D(2k):\ 0\leq i\leq j\leq w(C)-1,
 \ 0\leq k\leq w(D)\}.$$
Lastly, consider $\varepsilon\in E(\mathbb{H})$ such that $|\varepsilon|=3$.
There exist $I,J,K\in\mathfrak{C}_{{\rm odd}}(\Gamma)$ such that $\varepsilon=IJK$.
Associate with $\varepsilon$ the set
$$E_\varepsilon=
\{\varphi_{I}(2i)\varphi_{J}(2j)\varphi_{K}(2k):0\leq i\leq w(I),
\ 0\leq j\leq w(J),
\ 0\leq k\leq w(K)\}.$$
The 3-hypergraph $\Gamma\bullet\mathbb{H}$ is defined on $V(\Gamma)$ by
\begin{equation}\label{E1_Rbis_Critical_hyper}
E(\Gamma\bullet\mathbb{H})=(\bigcup_{C\in\mathfrak{C}_{{\rm odd}}(\Gamma)\setminus\mathfrak{C}_{1}(\Gamma)}E_C)\cup
(\bigcup_{\varepsilon\in E(\mathbb{H})}E_\varepsilon).
\end{equation}

\subsection{The main results}

To begin, we prove the following theorem.

\begin{thm}\label{thm_1_main}
Let $H$ be a critical and 3-hypergraph such that $v(H)\geq 5$.
If $H$ is not circular, then there exist a graph $\Gamma$ satisfying \eqref{critical_construction_0} and \eqref{critical_construction_1}, and a
$\{2,3\}$-hypergraph $\mathbb{H}$ defined on $\mathfrak{C}(\Gamma)$ satisfying
\eqref{critical_construction_1} such that $H=\Gamma\bullet\mathbb{H}$.
\end{thm}

In Theorem~\ref{thm_1_main}, the graph $\Gamma$ is the primality graph of $H$
(see Definition~\ref{defi_primality_graph}).
To state, the second main theorem, we consider a graph $\Gamma$ satisfying \eqref{critical_construction_0} and \eqref{critical_construction_1}.
We consider also a $\{2,3\}$-hypergraph $\mathbb{H}$ defined on $\mathfrak{C}(\Gamma)$ satisfying
\eqref{critical_construction_2}.
Furthermore, we use the modules of $\mathbb{H}$ defined as follows.

\begin{defi}
Given $\mathbb{W}\subseteq V(\mathbb{H})$, $\mathbb{W}$ is a module of
$\mathbb{H}$ if $\mathbb{W}$ is a module of
$$(V(\mathbb{H}),E(\mathbb{H})\cap\binom{V(\mathbb{H})}{2})$$ and
$\mathbb{W}$ is a module of
$$(V(\mathbb{H}),E(\mathbb{H})\cap\binom{V(\mathbb{H})}{3}).$$
\end{defi}

\begin{thm}\label{thm_2_main}
Suppose that $v(\Gamma\bullet\mathbb{H})\geq 5$.
The 3-hypergraph $\Gamma\bullet\mathbb{H}$ is critical if and only if the following three assertions hold
\begin{enumerate}
\item[(C1)] $\mathbb{H}$ is connected;
\item[(C2)] for every nontrivial module $\mathbb{M}$ of $\mathbb{H}$, we have
$\mathbb{M}\setminus\mathfrak{C}_{1}(\Gamma)\neq\emptyset$;
\item[(C3)] for each $v\in V_1(\Gamma)$, if
$\mathbb{H}-\{v\}$ is connected, then $\mathbb{H}-\{v\}$ admits a nontrivial module $\mathbb{M}_{\{v\}}$ such that
$\mathbb{M}_{\{v\}}\subseteq\mathfrak{C}_1(\Gamma)\setminus\{\{v\}\}$.
\end{enumerate}
\end{thm}

Before proving Theorem~\ref{thm_2_main}, we establish the next proposition.

\begin{prop}\label{Prop1_Critical_hyper}
The 3-hypergraph $\Gamma\bullet\mathbb{H}$ is prime if and only if the following two assertions hold
\begin{enumerate}
\item[(C1)] $\mathbb{H}$ is connected;
\item[(C2)] for every nontrivial module $\mathbb{M}$ of $\mathbb{H}$, we have
$\mathbb{M}\setminus\mathfrak{C}_{1}(\Gamma)\neq\emptyset$.
\end{enumerate}
\end{prop}

Finally, we improve Theorem~\ref{thm_2_main} by characterizing the 3-hypergraph $\Gamma\bullet\mathbb{H}$ whenever it is critical and satisfies
$\mathscr{P}(\Gamma\bullet\mathbb{H})=\Gamma$ (see Theorem~\ref{thm_3_main} in Section~\ref{sec_improvement}).

\section{The critical tournaments}

The primality graph associated with a prime tournament is defined in the same as that associated with a
prime 3-hypergraph (see Definition~\ref{defi_primality_graph}).

\begin{defi}\label{defi_primality_graph_tournament}
Let $T$ be a prime tournament.
The {\em primality graph} $\mathscr{P}(T)$ of $T$ is defined on $V(T)$ as follows.
Given distinct $v,w\in V(T)$, $vw\in E(\mathscr{P}(T))$ if $T-\{v,w\}$ is prime.
\end{defi}

The basic properties of the primality graph follow.
The next lemma is stated in ~\cite{I93} without a proof. For a proof, see \cite[Lemma~10]{BI09}.

\begin{lem} [Ille~\cite{I93}]\label{lem Boud Ille}
Let $T$ be a critical tournament with $v(T)\geq 5$.
For every $v\in\mathscr{C}(T)$, we have
$d_{\mathscr{P}(T)}(v)\leq 2$.
Moreover, the next two assertions hold.
\begin{enumerate}
\item Given $v\in V(T)$, if $d_{\mathscr{P}(T)}(v)=1$, then
$V(T)\setminus(\{v\}\cup N_{\mathscr{P}(T)}(v))$ is a module of $T-v$.
\item Given $v\in V(T)$, if $d_{\mathscr{P}(T)}(v)=2$, then
$N_{\mathscr{P}(T)}(v)$ is a module of $T-v$.
\end{enumerate}
\end{lem}

For each $n\geq 3$, recall that the {\em cycle} $C_n$ is defined on $V(C_n)=\{0,\ldots,n-1\}$ by
$E(C_n)=E(P_n)\cup\{0(n-1)\}$.
The {\em length} of $C_n$ is $n$.
Given a critical tournament $T$, it follows from Lemma~\ref{lem Boud Ille} that the connected components of $\mathscr{P}(T)$ are paths or cycles.
Boudabbous and Ille \cite{BI09} characterized the critical tournaments from their
primality graphs.
To begin, they examine the connected components of the primality graph associated with a critical tournament.

\begin{lem}[Corollary 17 \cite{BI09}]\label{cor_BI09}
If $T$ is a critical tournament, with $v(T)\geq 5$, then $\mathscr{P}(T)$ satisfies one of the following
\begin{enumerate}
\item $\mathscr{P}(T)$ is a cycle of odd length;
\item $\mathscr{P}(T)$ is a path;
\item $v(T)$ is odd and there is $v\in V(T)$ such that $\mathscr{P}(T)-v$ is a path and
$N_{\mathscr{P}(T)}(v)=\emptyset $.
\end{enumerate}
\end{lem}

For each of the three shapes describe in Lemma~\ref{cor_BI09},
Boudabbous and Ille \cite{BI09} characterized the corresponding critical tournaments.

\begin{prop}[Proposition 18 \cite{BI09}]\label{prop_T2n+1}
Given a tournament such that $v(T)\geq 5$,
$T$ is critical and $\mathscr{P}(T)=C_{v(T)}$ if and only if $v(T)$ is odd and $T=T_{2n+1}$ or
$(T_{2n+1})^\star$.
\end{prop}

\begin{prop}[Proposition 19 \cite{BI09}]\label{prop_U2n+1}
Given a tournament such that $v(T)\geq 5$,
$T$ is critical and $\mathscr{P}(T)=P_{v(T)}$ if and only if $v(T)$ is odd and $T=U_{v(T)}$ or
$(U_{v(T)})^\star$.
\end{prop}

\begin{prop}[Proposition 21 \cite{BI09}]\label{prop_W2n+1}
Given a tournament defined on $\{0,\ldots,2n\}$ with $2n+1\geq 5$,
$T$ is critical, $\mathscr{P}(T)-2n=P_{2n-1}$
and $N_{\mathscr{P}(T)}(2n)=\emptyset$ if and only if $T=W_{2n+1}$ or
$(W_{2n+1})^\star$.
\end{prop}

\section{Proof of Theorem~\ref{thm_1_main}}

The purpose of this section is to characterize the non circular and critical 3-hypergraphs.
We use the following notation.

\begin{nota}\label{equiv}
Let $H$ be a 3-hypergraph.
\begin{equation*}
\text{For}\ e,f\in\binom{V(H)}{3}, \ e\equiv_H f\ \text{means}\
\begin{cases}
e,f\in E(H)\\
\text{or}\\
e,f\not\in E(H).
\end{cases}
\end{equation*}
\end{nota}

An analogue of lemma~\ref{lem Boud Ille} for prime and 3-uniform hypergraphs follows.
For a proof, see \cite[Lemma~10]{BI09}.

\begin{lem}\label{Lcritical}
Let $H$ be a critical and 3-hypergraph with $v(H)\geq 5$.
For every $v\in V(H)$, we have $d_{\mathscr{P}(H)}(v)\leq 2$.
Moreover, the next two assertions hold.
\begin{enumerate}
\item Given $v\in V(H)$, if $d_{\mathscr{P}(H)}(v)=1$, then
$V(H)\setminus(\{v\}\cup N_{\mathscr{P}(H)}(v))$ is a module of $H-v$.
\item Given $v\in V(H)$, if $d_{\mathscr{P}(H)}(v)=2$, then
$N_{\mathscr{P}(H)}(v)$ is a module of $H-v$.
\end{enumerate}
\end{lem}

Given a critical 3-hypergraph $H$, it follows from Lemma~\ref{Lcritical} that the components of
$\mathscr{P}(H)$ are cycles or paths.

\begin{prop}\label{Pcritical_T2n+1}
Let $H$ be a critical 3-hypergraph defined on $\{0,\ldots,p-1\}$, where $p\geq 5$.
If there exists $k\in\{3,\ldots,p\}$ such that $\mathscr{P}(H)[\{0,\ldots,k-1\}]=C_k$, then
$p=2n+1$, $k=p$, and $H=C_3(T_{2n+1})$, where $n\geq 2$.
\end{prop}

\begin{proof}
Suppose that there exists $k\in\{3,\ldots,p\}$ such that $\mathscr{P}(H)[\{0,\ldots,k-1\}]=C_k$.
First, we show that
\begin{equation}\label{E0_Pcritical}
\text{$k$ is odd.}
\end{equation}
Otherwise, there exists $l\geq 2$ such that $\mathscr{P}(H)[\{0,\ldots,2l-1\}]=C_{2l}$.
We verify that $\{0,2\}$ is a module of $H$.
Consider $e\in E(H)$ such that $e\cap\{0,2\}\neq\emptyset$ and
$e\setminus\{0,2\}\neq\emptyset$.
Suppose for a contradiction that $0,2\in e$.
There exists $i\in\{1\}\cup\{3,\ldots,2l-1\}$ such that $e=02i$.
Since $N_{\mathscr{P}(H)}(1)=\{0,2\}$, it follows from Lemma~\ref{Lcritical} that $\{0,2\}$ is a module of $H-1$.
It follows that $i=1$, that is, $e=012$.
It follows from Lemma~\ref{Lcritical} that $01(2l-2)\in E(H)$, which contradicts the fact that
$\{0,2l-2\}$ is a module of $H-(2l-1)$.
Consequently, $|e\cap\{0,2\}|=1$.
For a contradiction, suppose that $1\in e$.
Since $|e\cap\{0,2\}|=1$, there $j\in\{3,\ldots,2l-1\}$ such that $e=01j$ or $12j$.
Denote by $j'$ the unique element of  $\{2l-2,2l-1\}$ such that $j'\equiv j\mod 2$.
It follows from Lemma~\ref{Lcritical} that
$(e\setminus\{j\})\cup\{j'\}\in E(H)$.
Similarly, by denoting by $j''$ the unique element of  $\{3,4\}$ such that $j''\equiv j\mod 2$, we obtain $(e\setminus\{j\})\cup\{j''\}\in E(H)$.
We distinguish the following two cases.
\begin{enumerate}
\item Suppose that $e=01j$.
If $j$ is even, then $01(2l-2)\in E(H)$, which contradicts the fact that $\{0,2l-2\}$ is a module of $H-(2l-1)$.
If $j$ is odd, then $013\in E(H)$, which contradicts the fact that $\{1,3\}$ is a module of $H-2$.
\item Suppose that $e=12j$.
If $j$ is even, then $124\in E(H)$, which contradicts the fact that $\{2,4\}$ is a module of $H-3$.
If $j$ is odd, then $12(2l-1)\in E(H)$, which contradicts the fact that $\{1,2l-1\}$ is a module of $H-0$.
\end{enumerate}
It follows that $1\not\in e$. So, $e\in E(H-1)$.
Since $N_{\mathscr{P}(H)}(1)=\{0,2\}$, it follows from Lemma~\ref{Lcritical} that
$\{0,2\}$ is a module of $H-1$.
Thus, there exists $i\in\{0,2\}$ such that $e\cap\{0,2\}=\{i\}$, and
$(e\setminus\{i\})\cup\{i'\}\in E(H)$ for each $i'\in\{0,2\}$.
Consequently, $\{0,2\}$ is a module of $H$, which contradicts the fact that $H$ is prime.
It follows that \eqref{E0_Pcritical} holds.
Set $k=2n+1$, where $n\geq 1$.

Second, we prove that $\{0,\ldots,2n\}$ is a module of $H$.
Consider $e\in E(H)$ such that $e\cap \{0,\ldots,2n\}\neq\emptyset$ and
$e\setminus \{0,\ldots,2n\}\neq\emptyset$.
We prove that
\begin{equation}\label{E2_Pcritical}
|e\cap \{0,\ldots,2n\}|=1.
\end{equation}
Otherwise, we have $|e\cap \{0,\ldots,2n\}|=2$.
There exist $i,j\in\{0,\ldots,2n\}$, with $i<j$, such that $e\cap \{0,\ldots,2n\}=\{i,j\}$.
Denote by $i'$ the unique element of $\{0,1\}$ such that $i'\equiv i\mod 2$.
As previously, we obtain $(e\setminus\{i\})\cup\{i'\}\in E(H)$.
Denote by $j'$ the unique element of $\{2,3\}$ such that $j'\equiv j\mod 2$.
We obtain $(e\setminus\{i,j\})\cup\{i',j'\}\in E(H)$.
Set $e'=(e\setminus\{i,j\})\cup\{i',j'\}$.
Observe that $e'\setminus\{0,\ldots,2n\}=e\setminus\{0,\ldots,2n\}$, and denote by $v$ the unique element of $e'\setminus\{0,\ldots,2n\}$.
If $i'=0$ and $j'=2$, then $02v\in E(H)$, which contradicts the fact that $\{0,2\}$ is a module of $H-1$.
If $i'=1$ and $j'=3$, then $13v\in E(H)$, which contradicts the fact that $\{1,3\}$ is a module of $H-2$.
Suppose that $i'=0$ and $j'=3$.
We get $03v\in E(H)$, and hence $0(2n-1)v\in E(H)$, which contradicts the fact that $\{0,2n-1\}$ is a module of $H-(2n)$.
Lastly, if $i'=1$ and $j'=4$, then $1(2n)v\in E(H)$, which contradicts the fact that $\{1,2n\}$ is a module of $H-0$.
It follows that \eqref{E2_Pcritical} holds.
Denote by $i$ the unique element of $e\cap\{0,\ldots,2n\}$.
For every $l\in\{1,\ldots,n\}$, we obtain
\begin{equation}\label{E3_Pcritical}
(e\setminus\{i\})\cup\{i+2l\}\in E(H),
\end{equation}
where $i+2l$ is considered modulo $2n+1$.
In particular, we have $(e\setminus\{i\})\cup\{i+2n\}\in E(H)$, that is,
$(e\setminus\{i\})\cup\{i-1\}\in E(H)$.
Since $\{i-1,i+1\}$ is a module of $H-i$, we get $(e\setminus\{i\})\cup\{i+1\}\in E(H)$.
For each $m\in\{0,\ldots,n-1\}$, we obtain
\begin{equation}\label{E4_Pcritical}
(e\setminus\{i\})\cup\{i+2m+1\}\in E(H).
\end{equation}
It follows from \eqref{E3_Pcritical} and \eqref{E4_Pcritical} that
$(e\setminus\{i\})\cup\{i'\}\in E(H)$ for every $i'\in\{0,\ldots,2n\}$.
Consequently, $\{0,\ldots,2n\}$ is a module of $H$.
Since $H$ is prime, we obtain $V(H)=\{0,\ldots,2n\}$.

Third, we prove that $H=C_3(T_{2n+1})$.
We have
\begin{align}\label{E5_Pcritical}
E(C_3(T_{2n+1}))=&\{(2i)(2l+1)(2j):0\leq i\leq l<j\leq n\}\nonumber\\
&\cup\{(2i+1)(2l)(2j+1):0\leq i<p\leq l\leq n-1\}.
\end{align}
For a contradiction, suppose that there are $x,y,z\in\{0,\ldots,2n\}$ such that
$x<y<z$, $x\equiv y\mod 2$, and $xyz\in E(H)$.
It follows from Lemma~\ref{Lcritical} that
$(y-2)yz\in E(H)$, which contradicts the fact that $\{y-2,y\}$ is a module of $H-(y-1)$.
Hence,
$xyz\not\in E(H)$ for $x,y,z\in\{0,\ldots,2n\}$ such that
$x<y<z$ and $x\equiv y\mod 2$.
Similarly,
$xyz\not\in E(H)$ for $x,y,z\in\{0,\ldots,2n\}$ such that
$x<y<z$ and $y\equiv z\mod 2$.
It follows that
\begin{equation}\label{E8_Pcritical}
E(H)\subseteq E(C_3(T)).
\end{equation}
Now, consider $x,y,z\in\{0,\ldots,2n\}$ such that
$x<y<z$, $x\not\equiv y\mod 2$, and $y\not\equiv z\mod 2$.
It follows from Lemma~\ref{Lcritical} that
\begin{equation}\label{E9_Pcritical}
xyz\equiv_Hx(x+1)(x+2).
\end{equation}
Now, we prove that the permutation
\begin{equation*}
\begin{array}{rrcl}
\theta:&\{0,\ldots,2n\}&\longrightarrow&\{0,\ldots,2n\}\\
&x&\longmapsto&x+1\mod 2n+1
\end{array}
\end{equation*}
of $\{0,\ldots,2n\}$ is an automorphism of $H$.
Given $x,y,z\in\{0,\ldots,2n\}$ such that $x<y<z$, we have to verify that
$xyz\equiv_H(x+1)(y+1)(z+1)$.
We have $$xyz\equiv_Hxy(z+2)\equiv_Hx(y+2)(z+2)\equiv_H(x+2)(y+2)(z+2).$$
Thus,
$xyz\equiv_H(x+2n+2)(y+2n+2)(z+2n+2)$, that is,
$$xyz\equiv_H(x+1)(y+1)(z+1).$$
Consequently, $\theta$ is an automorphism of $H$.
Since $H$ is prime, there exist
$u,v,w\in\{0,\ldots,2n\}$ such that $u<v<w$ and
$uvw\in E(H)$.
By \eqref{E8_Pcritical}, $uvw\in E(C_3(T))$.
It follows from \eqref{E5_Pcritical} that $u\not\equiv v\mod 2$ and
$v\not\equiv w\mod 2$.
It follows from \eqref{E9_Pcritical} that
$u(u+1)(u+2)\in E(H)$.
Since $\theta$ is an automorphism of $H$, we obtain
\begin{equation}\label{E10_Pcritical}
012\in E(H).
\end{equation}
Finally, consider $x,y,z\in\{0,\ldots,2n\}$ such that $x<y<z$ and $xyz\in E(C_3(T))$.
By \eqref{E5_Pcritical},
$x\not\equiv y\mod 2$ and $y\not\equiv z\mod 2$.
It follows from \eqref{E9_Pcritical} that $xyz\equiv_Hx(x+1)(x+2)$.
Since $\theta$ is an automorphism of $H$, we obtain $x(x+1)(x+2)\equiv_H012$.
Therefore, $xyz\equiv_H012$.
By \eqref{E10_Pcritical}, $xyz\in E(H)$.
Consequently, we obtain $E(C_3(T))\subseteq E(H)$.
It follows from \eqref{E8_Pcritical} that $H=C_3(T)$.
\end{proof}

The next corollary is an easy consequence of Lemma~\ref{Lcritical} and Proposition~\ref{Pcritical_T2n+1}.

\begin{cor}\label{C_critical_T2n+1}
Given a critical 3-hypergraph $H$, with $v(H)\geq 5$,
$H$ is not circular if and only if all the components of $\mathscr{P}(H)$ are paths.
\end{cor}

\begin{proof}
Suppose that $H$ is circular.
By Definition~\ref{defi_reali_critical}, $H$ is isomorphic to $C_3(T_{v(H)})$.
Hence, $\mathscr{P}(H)$ is isomorphic to $\mathscr{P}(C_3(T_{v(H)}))$.
It follows from Theorem~\ref{thm_same_primality} that
$$\mathscr{P}(C_3(T_{v(H)}))=\mathscr{P}(T_{v(H)}).$$
By Proposition~\ref{prop_T2n+1}, $\mathscr{P}(T_{v(H)})=C_{v(H)}$.
It follows that $\mathscr{P}(H)$ contains a cycle among its components.

Conversely, suppose that $\mathscr{P}(H)$ contains a cycle among its components.
Up to isomorphy, we can assume that $V(H)=\{0,\ldots,p-1\}$, where $p\geq 5$, and there
exists $k\in\{3,\ldots,p\}$ satisfying $\mathscr{P}(H)[\{0,\ldots,k-1\}]=C_k$.
By Proposition~\ref{Pcritical_T2n+1}, $H=C_3(T_{2n+1})$.
Therefore, $H$ is circular.
\end{proof}

\begin{nota}\label{nota_non_circular}
Let $H$ be a non circular and critical 3-hypergraph.
For each $C\in\mathfrak{C}(\mathscr{P}(H))\setminus\mathfrak{C}_1(\mathscr{P}(H))$, $C$ is a path.
Thus, there exists an isomorphism $\varphi_C$ from $P_{v(C)}$ onto $C$.
\end{nota}

\begin{lem}\label{Lcritical_U2n+1_V2n+1}
Let $H$ be a critical and 3-hypergraph with $v(H)\geq 5$.
Suppose that $H$ is not circular.
\begin{enumerate}
\item Let $C\in\mathfrak{C}_{{\rm odd}}(\mathscr{P}(H))\setminus\mathfrak{C}_1(\mathscr{P}(H))$.
For every $e\in E(H)$, if $e\cap V(C)\neq\emptyset$ and $e\setminus V(C)\neq\emptyset$, then there exists
$i\in\{0,\ldots,w(C)\}$ (see Notation~\ref{nota_comp}) such that $e\cap V(C)=\{\varphi_C(2i)\}$.
\item Let $C\in\mathfrak{C}_{{\rm even}}(\mathscr{P}(H))$.
For every $e\in E(H)$, if $e\cap V(C)\neq\emptyset$ and $e\setminus V(C)\neq\emptyset$, then there exist
$i,j\in\{0,\ldots,w(C)-1\}$ such that $i\leq j$ and
$e\cap V(C)=\{\varphi_C(2i),\varphi_C(2j+1)\}$.
\end{enumerate}
\end{lem}

\begin{proof}
For the first assertion, consider $C\in\mathfrak{C}_{{\rm odd}}(\mathscr{P}(H))\setminus\mathfrak{C}_1(\mathscr{P}(H))$.
Let $e\in E(H)$ such that $e\cap V(C)\neq\emptyset$ and
$e\setminus V(C)\neq\emptyset$.
We prove that
\begin{equation}\label{E0_Pcritical_U2n+1}
e\cap\{\varphi_C(2i+1):i\in\{0,\ldots,w(C)-1\}\}=\emptyset.
\end{equation}
Otherwise, there exists $i\in\{0,\ldots,w(C)-1\}$ such that
$\varphi_C(2i+1)\in e$.
We distinguish the following two cases.
\begin{itemize}
\item Suppose that there exists $j\in\{0,\ldots,i\}$ such that $\varphi_C(2j)\in e$.
Since $e\setminus V(C)\neq\emptyset$, $e\cap V(C)=\{\varphi_C(2j),\varphi_C(2i+1)\}$.
By denoting by $v$ the unique element of $e\setminus V(C)$, we obtain
$e=\varphi_C(2j)\varphi_C(2i+1)v$.
It follows from Lemma~\ref{Lcritical} that $\varphi_C(2j)\varphi_C(2w(C)-1)v\in E(H)$, which contradicts the fact that $V(H)\setminus\{\varphi_C(2w(C)-1),\varphi_C(2w(C))\}$ is a module of $H-\varphi_C(2w(C))$.
\item Suppose that $e\cap\{\varphi_C(2j):j\in\{0,\ldots,i\}\}=\emptyset$.
It follows from Lemma~\ref{Lcritical} that there exists $f\in E(H)$ such that $\varphi_C(1)\in f$,
$\varphi_C(0)\not\in f$, and $f\setminus V(C)\neq\emptyset$, which contradicts the fact that
$V(H)\setminus\{\varphi_C(0),\varphi_C(1)\}$ is a module of
$H-\varphi_C(0)$.
\end{itemize}
Consequently, \eqref{E0_Pcritical_U2n+1} holds.

Now, we prove hat there exists $i\in\{0,\ldots,w(C)\}$ such that
\begin{equation}\label{E3_Pcritical_U2n+1}
e\cap V(C)=\{\varphi_C(2i)\}.
\end{equation}
Otherwise, it follows from \eqref{E0_Pcritical_U2n+1} that there exist distinct $i,j\in\{0,\ldots,w(C)\}$ such that $0\leq i<j\leq w(C)$ such that $e\cap V(C)=\{\varphi_C(2i),\varphi_C(2j)\}$.
It follows from Lemma~\ref{Lcritical} that there exists $f\in E(H)$ such that $f\cap V(C)=\{\varphi_C(2i),\varphi_C(2i+2)\}$ and $f\setminus V(C)\neq\emptyset$, which contradicts the fact that
$\{\varphi_C(2i),\varphi_C(2i+2)\}$ is a module of
$H-\varphi_C(2i+1)$.
Consequently, \eqref{E3_Pcritical_U2n+1} holds.

For the second assertion, consider $C\in\mathfrak{C}_{{\rm even}}(\mathscr{P}(H))$.
Let $e\in E(H)$ such that $e\cap V(C)\neq\emptyset$ and
$e\setminus V(C)\neq\emptyset$.
Set $$p=\min(\{i\in\{0,\ldots,2w(C)-1\}:\varphi_C(i)\in e\}).$$
For a contradiction, suppose that $p$ is odd.
It follows from Lemma~\ref{Lcritical} that $(e\setminus\{\varphi_C(p)\})\cup\{\varphi_C(1)\}\in E(H)$, which contradicts the fact that
$V(C)\setminus\{\varphi_C(0),\varphi_C(1)\}$ is a module of $H-\varphi_C(0)$.
It follows that $p$ is even.
Thus, there exists $i\in\{0,\ldots,w(C)-1\}$ such that
$p=\varphi_C(2i)$.
Similarly, $\max(\{i\in\{0,\ldots,2w(C)-1\}:\varphi_C(i)\in e\})$ is odd.
Thus there exists $j\in\{0,\ldots,w(C)-1\}$ such that
$\max(\{i\in\{0,\ldots,2w(C)-1\}:\varphi_C(i)\in e\})=\varphi_C(2j+1)$.
We obtain that $i\leq j$ and
$e\cap V(C)=\{\varphi_C(2i),\varphi_C(2j+1)\}$.
\end{proof}

\begin{prop}\label{Pcritical_U2n+1}
Let $H$ be a critical and 3-hypergraph with $v(H)\geq 5$.
Suppose that $H$ is not circular.
Consider $C\in\mathfrak{C}_{{\rm odd}}(\mathscr{P}(H))\setminus\mathfrak{C}_1(\mathscr{P}(H))$.
\begin{enumerate}
\item Suppose that $V(C)\subsetneq V(H)$. There exists $e\in E(H)$, such that $e\cap V(C)\neq\emptyset$ and
$e\setminus V(C)\neq\emptyset$.
Moreover, one of the following two assertions holds
\begin{itemize}
\item there exist distinct
$D,D'\in\mathfrak{C}_{{\rm odd}}(\mathscr{P}(H))\setminus\{C\}$ such that
$e\subseteq V(C)\cup V(D)\cup V(D')$,
$e\cap V(D)\neq\emptyset$, and $e\cap V(D')\neq\emptyset$;
\item there exists
$D\in\mathfrak{C}_{{\rm even}}(\mathscr{P}(H))\setminus\{C\}$ such that
$e\subseteq V(C)\cup V(D)$, and
$e\cap V(D)\neq\emptyset$.
\end{itemize}
Furthermore, there exists $i\in\{0,\ldots,w(C)\}$ such that
$e\cap V(C)=\{\varphi_C(2i)\}$.
Lastly, for each $j\in\{0,\ldots,w(C)\}$, $(e\setminus\{\varphi_C(2i)\})\cup\{\varphi_C(2j)\}\in E(H)$.
\item The function $\varphi_C$ is an isomorphism from $C_3(U_{v(C)})$ onto $H[V(C)]$.
\item If $V(C)\subsetneq V(H)$, then there exists $C'\in\mathfrak{C}(\mathscr{P}(H))\setminus\mathfrak{C}_1(\mathscr{P}(H))$ such that $C'\neq C$.
\end{enumerate}
\end{prop}

\begin{proof}
For the first assertion, suppose that $V(C)\subsetneq V(H)$.
Since $V(C)$ is not a module of $H$, there exists $e\in E(H)$ such that
$e\cap V(C)\neq\emptyset$ and $e\setminus V(C)\neq\emptyset$.
It follows from Lemma~\ref{Lcritical_U2n+1_V2n+1} that
there exists
$i\in\{0,\ldots,w(C)\}$ such that $e\cap V(C)=\{\varphi_C(2i)\}$.
It follows from Lemma~\ref{Lcritical} that
$(e\setminus\{\varphi_C(2i)\})\cup\{\varphi_C(2j)\}\in E(H)$ for each
$j\in\{0,\ldots,w(C)\}$.
Moreover, since $|e\cap V(C)|=1$, there exists $D\in\mathfrak{C}(\mathscr{P}(H))\setminus\{C\}$ such that $e\cap V(D)\neq\emptyset$.
We distinguish the following two cases.
\begin{itemize}
\item Suppose that $D\in\mathfrak{C}_{{\rm even}}(\mathscr{P}(H))$.
By Lemma~\ref{Lcritical_U2n+1_V2n+1}, $|e\cap V(D)|=2$.
Thus $e\subseteq V(C)\cup V(D)$.
\item Suppose that $D\in\mathfrak{C}_{{\rm odd}}(\mathscr{P}(H))$.
By Lemma~\ref{Lcritical_U2n+1_V2n+1}, $|e\cap V(D)|=1$.
Therefore, there exists $D'\in\mathfrak{C}(\mathscr{P}(H))\setminus\{C,D\}$ such that $e\cap V(D')\neq\emptyset$.
It follows from Lemma~\ref{Lcritical_U2n+1_V2n+1} that
$D'\in\mathfrak{C}_{{\rm odd}}(\mathscr{P}(H))$.
Thus $e\subseteq V(C)\cup V(D)\cup V(D')$.
\end{itemize}

For the second assertion, suppose for a contradiction that for every $e\in E(H)$ such that
$e\cap V(C)\neq\emptyset$, we have $e\setminus V(C)\neq\emptyset$.
By Lemma~\ref{Lcritical_U2n+1_V2n+1}, there exists
$i\in\{0,\ldots,w(C)\}$ such that
$e\cap V(C)=\{\varphi_C(2i)\}$.
Thus, for every $e\in E(H)$, we have
$e\cap\{\varphi_C(2i+1):i\in\{0,\ldots,w(C)-1\}\}=\emptyset$.
Therefore, $\{\varphi_C(2i+1):i\in\{0,\ldots,w(C)-1\}\}$ is a module of $H$, and hence $H$ is decomposable.
Consequently, there exists $e\in V(H)$ such that $e\subseteq V(C)$.
Set $$p=\min(\{i\in\{0,\ldots,2w(C)\}:\varphi_C(i)\in e\}).$$
For a contradiction, suppose that $p$ is odd.
It follows from Lemma~\ref{Lcritical} that
$(e\setminus\{\varphi_C(p)\})\cup\{\varphi_C(1)\}\in E(H)$,
which contradicts the fact that
$V(C)\setminus\{\varphi_C(0),\varphi_C(1)\}$ is a module of $H-\varphi_C(0)$.
Thus, there exists $i\in\{0,\ldots,w(C)\}$ such that
$p=\varphi_C(2i)$.
Similarly, there exists $k\in\{0,\ldots,w(C)\}$ such that
$\max(\{i\in\{0,\ldots,2w(C)\}:\varphi_C(i)\in e\})=\varphi_C(2k)$.
since $e\subseteq V(C)$, $i<k$.
Consider $q\in\{0,\ldots,2w(C)\}$ such that $e=\varphi_C(2i)\varphi_C(q)\varphi_C(2k)$.
For a contradiction, suppose that $q$ is even.
We obtain $2i<q<2k$.
It follows from Lemma~\ref{Lcritical} that
$\varphi_C(2i)\varphi_C(2i+2)\varphi_C(2k)\in E(H)$,
which contradicts the fact that
$\{\varphi_C(2i),\varphi_C(2i+2)\}$ is a module of $H-\varphi_C(2i+1)$.
It follows that $p$ is odd.
Hence, there exists $j\in\{0,\ldots,w(C)-1\}$ such that
$q=\varphi_C(2j+1)$.
We obtain $e=\varphi_C(2i)\varphi_C(2j+1)\varphi_C(2k)$, where
$0\leq i\leq j<k\leq w(C)$.
It follows that
\begin{equation}\label{E6bis_Pcritical_U2n+1}
E(H[V(C)])\subseteq\{\varphi_C(2i')\varphi_C(2j'+1)\varphi_C(2k'):0\leq i'\leq j'<k'\leq w(C)\}.
\end{equation}
It follows from Lemma~\ref{Lcritical} that
\begin{equation}\label{E7_Pcritical_U2n+1}
\varphi_C(2i')\varphi_C(2j'+1)\varphi_C(2k')\equiv_H\varphi_C(0)\varphi_C(1)\varphi_C(2)
\end{equation}
for $0\leq i'\leq j'<k'\leq w(C)$.
Since $e=\varphi_C(2i)\varphi_C(2j+1)\varphi_C(2k)$, we obtain $\varphi_C(0)\varphi_C(1)\varphi_C(2)$.
It follows from \eqref{E6bis_Pcritical_U2n+1} and \eqref{E7_Pcritical_U2n+1} that
$$E(H[V(C)])=\{\varphi_C(2i')\varphi_C(2j'+1)\varphi_C(2k'):0\leq i'\leq j'<k'\leq w(C)\}.$$
In other words, $\varphi_C$ is an isomorphism from $C_3(U_{v(C)})$ onto $H[V(C)]$.

For the third assertion, suppose that $V(C)\subsetneq V(H)$.
By Theorem~\ref{thm _Critical_ST}, $U_{v(C)}$ is prime.
Hence, $C_3(U_{v(C)})$ is prime by Theorem~\ref{thm_same_primality}.
It follows from the second assertion above that $H[V(C)]$ is prime.
By Theorem~\ref{prop_Eh-2}, there exist $v,w\in (V(H)\setminus V(C))\cup\underline{V(C)}$ such that $H-\{v,w\}$ is prime.
Since $H$ is critical, we have $v\neq w$.
Therefore, there exists
$C'\in\mathfrak{C}(\mathscr{P}(H))\setminus\mathfrak{C}_1(\mathscr{P}(H))$ such that $v,w\in C'$.
Lastly, suppose for a contradiction that $\underline{V(C)}\neq\emptyset$.
There exist $c\in V(C)$ and $u\in V(H)\setminus V(C)$ such that $\{c,u\}$ is a module of $H[V(C)\cup\{u\}]$.
Since $C$ is isomorphic to $C_3(U_{v(C)})$ by the second assertion above,
there exists $e\in E(C)$ such that $c\in e$.
Since $\{c,u\}$ is a module of $H[V(C)\cup\{u\}]$, we get
$(e\setminus\{c\})\cup\{u\}\in E(H)$, which contradicts Lemma~\ref{Lcritical_U2n+1_V2n+1}.
It follows that $\underline{V(C)}=\emptyset$.
Thus $v,w\not\in V(C)$, so $C'\neq C$.
\end{proof}

\begin{prop}\label{Pcritical_V2n+1}
Let $H$ be a critical and 3-hypergraph such that $v(H)\geq 5$.
Suppose that $H$ is not circular.
Consider $C\in\mathfrak{C}_{{\rm even}}(\mathscr{P}(H))$.
\begin{enumerate}
\item There exists $D\in\mathfrak{C}_{{\rm odd}}(\mathscr{P}(H))$ such that
$\varphi_C(2i)\varphi_C(2j+1)\varphi_D(2k)\in E(H)$, where $i,j\in\{0,\ldots,w(C)-1\}$, with $i\leq j$, and $k\in\{0,\ldots,w(D)\}$;
\item For each $k\in\{0,\ldots,w(D)\}$,
the extension $\psi_C^{2k}:\{0,\ldots,v(C)\}\longrightarrow
V(C)\cup\{\varphi_D(2k)\}$ of $\varphi_C$ defined by
\begin{equation*}
\begin{cases}
(\psi_C^{2k})_{\restriction\{0,\ldots,v(C)-1\}}=
\varphi_{C}\\
\text{and}\\
\psi_C^{2k}(v(C))=\varphi_{D}(2k),
\end{cases}
\end{equation*}
is an isomorphism from
$C_3(W_{v(C)+1})$ onto
$H[V(C)\cup\{\varphi_D(2k)\}]$.

\item If $|V(H)\setminus V(C)|\geq 2$, then there exists $C'\in\mathfrak{C}(\mathscr{P}(H))\setminus\mathfrak{C}_1(\mathscr{P}(H))$ such that $C'\neq C$.
\end{enumerate}
\end{prop}

\begin{proof}
For the first assertion, since $V(C)$ is not a module of $H$, there exists $e\in E(H)$ such that
$e\cap V(C)\neq\emptyset$ and $e\setminus V(C)\neq\emptyset$.
By Lemma~\ref{Lcritical_U2n+1_V2n+1}, there exist $i,j\in\{0,\ldots,w(C)-1\}$ such that
$i\leq j$ and $e\cap V(C)=\{\varphi_C(2i),\varphi_C(2j+1)\}$.
Since $|e\cap V(C)|=2$, there exists $D\in\mathfrak{C}(\mathscr{P}(H))\setminus\{C\}$ such that $e\cap V(D)\neq\emptyset$.
It follows from Lemma~\ref{Lcritical_U2n+1_V2n+1} that
$D\in\mathfrak{C}_{{\rm odd}}(\mathscr{P}(H))$ and $e\cap V(D)=\{\varphi_D(2k)\}$, where $k\in\{0,\ldots,w(D)\}$.

For the second assertion, it follows from the first assertion above that
there exists $D\in\mathfrak{C}_{{\rm odd}}(\mathscr{P}(H))$ such that
\begin{equation}\label{E0_Pcritical_V2n+1}
\varphi_C(2i_0)\varphi_C(2j_0+1)\varphi_D(2k_0)\in E(H),
\end{equation}
where $i_0,j_0\in\{0,\ldots,w(C)-1\}$, with $i_0\leq j_0$, and $k_0\in\{0,\ldots,w(D)\}$.
It follows from Lemma~\ref{Lcritical} that
\begin{equation}\label{E2_Pcritical_V2n+1}
\varphi_C(2i_0)\varphi_C(2j_0+1)\varphi_D(2k)\in E(H)
\end{equation}
for each $k\in\{0,\ldots,w(D)\}$.
Let $k\in\{0,\ldots,w(D)\}$.
We prove that
\begin{align}\label{E3_Pcritical_V2n+1}
E(H[V(C)\cup\{\varphi_D(2k)\}])=\{&\varphi_C(2i)\varphi_C(2j+1)\varphi_D(2k):\\
&0\leq i\leq j\leq w(C)-1\}.\nonumber
\end{align}
Consider $i,j\in\{0,\ldots,w(C)-1\}$ such that $i\leq j$.
It follows from Lemma~\ref{Lcritical} that
\begin{equation}\label{E4_Pcritical_V2n+1}
\varphi_C(2i)\varphi_C(2j+1)\varphi_D(2k)\equiv_H
\varphi_C(0)\varphi_C(1)\varphi_D(2k).
\end{equation}

By \eqref{E2_Pcritical_V2n+1} and \eqref{E4_Pcritical_V2n+1}, we have
$\varphi_C(0)\varphi_C(1)\varphi_D(2k)\in E(H)$.
It follows from \eqref{E4_Pcritical_V2n+1} that
$\varphi_C(2i)\varphi_C(2j+1)\varphi_D(2k)\in E(H)$.
Therefore,
\begin{align}\label{E5_Pcritical_V2n+1}
E(H[V(C)\cup\{\varphi_D(2k)\}])\supseteq\{&\varphi_C(2i)\varphi_C(2j+1)\varphi_D(2k):\\
&0\leq i\leq j\leq w(C)-1\}.\nonumber
\end{align}
Conversely, consider $e\in E(H[V(C)\cup\{\varphi_D(2k)\}])$.
For a contradiction, suppose that $\varphi_D(2k)\not\in e$.
There exist $p,q,r\in\{0,\ldots,2w(C)-1\}$, with $p<q<r$, such that
$e=\varphi_C(p)\varphi_C(q)\varphi_D(r)$.
If $p$ is odd, then $\varphi_C(1)\varphi_C(q)\varphi_D(r)\in E(H)$, which contradicts the fact that $V(H)\setminus\{\varphi_C(0),\varphi_C(1)\}$ is a module of
$H-\varphi_C(0)$.
Hence, $p=2i$, where $i\in\{0,\ldots,w(C)-1\}$.
Similarly, we have $r=2l+1$, where $l\in\{i,\ldots,w(C)-1\}$.
If $q$ is even, then $\varphi_C(2i)\varphi_C(2i+2)\varphi_D(2l+1)\in E(H)$, which contradicts the fact that $\{\varphi_C(2i),\varphi_C(2i+2)\}$ is a module of
$H-\varphi_C(2i+1)$.
If $q$ is odd, then $\varphi_C(2i)\varphi_C(2l-1)\varphi_D(2l+1)\in E(H)$, which contradicts the fact that $\{\varphi_C(2l-1),\varphi_C(2l+1)\}$ is a module of
$H-\varphi_C(2l)$.
Consequently, we obtain
$\varphi_D(2k)\in e$.
It follows from Lemma~\ref{Lcritical_U2n+1_V2n+1} that there exist $i,j\in\{0,\ldots,w(C)-1\}$ such that
$i\leq j$ and $e\cap V(C)=\{\varphi_C(2i),\varphi_C(2j+1)\}$.
Thus,
\begin{align}\label{E6_Pcritical_V2n+1}
E(H[V(C)\cup\{\varphi_D(2k)\}])\subseteq\{&\varphi_C(2i)\varphi_C(2j+1)\varphi_D(2k):\\
&0\leq i\leq j\leq w(C)-1\}.\nonumber
\end{align}
It follows from \eqref{E5_Pcritical_V2n+1} and \eqref{E6_Pcritical_V2n+1} that \eqref{E3_Pcritical_V2n+1} holds.
Therefore, the extension $\psi_C^{2k}$ of $\varphi_C$ is an isomorphism from
$C_3(W_{v(C)+1})$ onto $H[V(C)\cup\{\varphi_D(2k)\}]$.

For the third assertion, suppose that $|V(H)\setminus V(C)|\geq 2$.
Let $k\in\{0,\ldots,w(D)\}$.
Set $Y=V(C)\cup\{\varphi_D(2k)\}$.
We have $Y\subsetneq V(H)$.
By Theorem~\ref{thm _Critical_ST}, $W_{v(C)+1}$ is prime.
Hence, $C_3(W_{v(C)+1})$ is prime by Theorem~\ref{thm_same_primality}.
It follows from the second assertion above that $H[Y]$ is prime.
By Proposition~\ref{prop_Eh-2}, there exist $v,w\in (V(H)\setminus Y)\cup\underline{Y}$ such that $H-\{v,w\}$ is prime.
Since $H$ is critical, we have $v\neq w$.
Therefore, there exists
$C'\in\mathfrak{C}(\mathscr{P}(H))\setminus\mathfrak{C}_1(\mathscr{P}(H))$ such that $v,w\in C'$.
Lastly, suppose for a contradiction that $V(C)\cap\underline{Y}\neq\emptyset$.
There exist $c\in V(C)$ and $u\in V(H)\setminus Y$ such that $\{c,u\}$ is a module of
$H[Y\cup\{u\}]$.
By the second assertion above,
there exists $d\in V(C)\setminus\{c\}$ such that $cd\varphi_D(2k)\in E(H[Y])$.
Since $\{c,u\}$ is a module of $H[V(C)\cup\{u\}]$, we get
$ud\varphi_D(2k)\in E(H[Y])$, which contradicts Lemma~\ref{Lcritical_U2n+1_V2n+1}.
It follows that $\underline{Y}\subseteq V(H)\setminus V(C)$.
Thus $v,w\not\in V(C)$, so $C'\neq C$.
\end{proof}

\begin{proof}[Proof of Theorem~\ref{thm_1_main}]
Let $H$ be a non circular and critical and 3-hypergraph such that $v(H)\geq 5$.
By Corollary~\ref{C_critical_T2n+1}, all the components of $\mathscr{P}(H)$ are paths.
We associate with $H$ the hypergraph $\underline{H}$ defined on
$\mathfrak{C}(\mathscr{P}(H))$ as  follows
\begin{enumerate}
\item given distinct $C,D\in\mathfrak{C}(\mathscr{P}(H))$, $CD\in E(\underline{H})$ if
$C\in\mathfrak{C}_{{\rm even}}(\mathscr{P}(H))$,
$D\in\mathfrak{C}_{{\rm odd}}(\mathscr{P}(H))$, and
there exists $e\in E(H)$ such that $|e\cap V(C)|=2$ and $|e\cap V(D)|=1$;
\item given distinct $I,J,K\in\mathfrak{C}(\mathscr{P}(H))$, $IJK\in E(\underline{H})$ if
$I,J,K\in\mathfrak{C}_{{\rm odd}}(\mathscr{P}(H))$ and
$|e\cap V(I)|=|e\cap V(J)|=|e\cap V(K)|=1$.
\end{enumerate}
It follows from Propositions~\ref{Pcritical_U2n+1} and \ref{Pcritical_V2n+1} that
$H=\mathscr{P}(H)\bullet\underline{H}$.
\end{proof}

\section{Proof of Proposition~\ref{Prop1_Critical_hyper}}

We use the following preliminary result (see \cite[Fact~20]{BCIZ}

\begin{lem}\label{fact1_same_block}
Let $H$ be a 3-hypergraph. Consider $X\subsetneq V(H)$ such that $H[X]$ is prime.
Let $M$ be a module of $H$. We have $M\cap X=\emptyset$, $M\supseteq X$ or $M\cap X=\{y\}$, where $y\in X$. Moreover, the following assertions hold.
\begin{enumerate}
\item If $M\cap X=\emptyset$, then all the elements of $M$ belong to the same block of $p_{(H,X)}$.
\item If $M\supseteq X$, then all the elements of $V(H)\setminus M$ belong to $\langle X\rangle_H$.
\item If $M\cap X=\{y\}$, where $y\in X$, then all the elements of
 $M\setminus\{y\}$ belong to $X_H(y)$.
\end{enumerate}
\end{lem}

In this section and the next one, we consider a graph $\Gamma$ satisfying \eqref{critical_construction_0} and \eqref{critical_construction_1}.
We consider also a $\{2,3\}$-hypergraph $\mathbb{H}$ defined on $\mathfrak{C}(\Gamma)$ satisfying
\eqref{critical_construction_2}.
We use the following notation.

\begin{nota}
For $\mathbb{W}\subseteq V(\mathbb{H})$, set
$\overline{\mathbb{W}}=\bigcup_{C\in\mathbb{W}}V(C)$.
Conversely, for $W\subseteq V(H_{(\Gamma,\mathbb{H})})$, set
$W/\mathfrak{C}(\Gamma)=\{C\in\mathfrak{C}(\Gamma):V(C)\cap W\neq\emptyset\}$.
\end{nota}

Both next remarks are useful.

\begin{rem}\label{v(C)_odd}
Let $C\in\mathfrak{C}_{{\rm odd}}(\Gamma)\setminus\mathfrak{C}_1(\Gamma)$.
First, suppose that $w(C)=1$.
It follows from the definition of $\Gamma\bullet\mathbb{H}$ that
$E((\Gamma\bullet\mathbb{H})[V(C)])=\{V(C)\}$.
Therefore,
$(\Gamma\bullet\mathbb{H})[V(C)]$ is prime.
Since $v(C)=3$, the next three assertions are obvious
\begin{enumerate}
\item $V(C)\setminus\{\varphi_C(0),\varphi_C(1)\}$
is a module of $(\Gamma\bullet\mathbb{H})[V(C)]-\varphi_C(0)$;
\item $V(C)\setminus\{\varphi_C(1),\varphi_C(2)\}$
is a module of $(\Gamma\bullet\mathbb{H})[V(C)]-\varphi_C(2)$;
\item $\{\varphi_C(0),\varphi_C(2)\}$ is a module of $(\Gamma\bullet\mathbb{H})[V(C)]-\varphi_C(1)$.
\end{enumerate}

Second, suppose that $w(C)\geq 2$.
It follows from the definition of $\Gamma\bullet\mathbb{H}$ that $\varphi_C$ is an isomorphism from $C_3(U_{v(C)})$ onto $(\Gamma\bullet\mathbb{H})[V(C)]$.
By Theorem~\ref{thm _Critical_ST}, $U_{v(C)}$ is critical.
By Theorem~\ref{thm_same_primality}, $C_3(U_{v(C)})$ is critical and
$\mathscr{P}(C_3(U_{v(C)}))=\mathscr{P}(U_{v(C)})$.
Furthermore,
we have $\mathscr{P}(U_{v(C)})=P_{v(C)}$ by Proposition~\ref{prop_U2n+1}.
Since $\varphi_C$ is an isomorphism from $C_3(U_{v(C)})$ onto $(\Gamma\bullet\mathbb{H})[V(C)]$,
$(\Gamma\bullet\mathbb{H})[V(H)]$ is critical and
$$\mathscr{P}((\Gamma\bullet\mathbb{H})[V(C)])=(V(C),\{\varphi_C(i)\varphi_C(i+1):0\leq i<2w(C)\}).$$
\end{rem}

\begin{rem}\label{v(C)_even}
Let $C\in\mathfrak{C}_{{\rm even}}(\Gamma)$ such that $w(C)\geq 1$.
Suppose that there exists $e\in E(\mathbb{H})$ such that $C\in e$.
We have $e=CD$, where $D\in\mathfrak{C}_{{\rm odd}}(\Gamma)$.
Let $k\in\{0,\ldots,w(D)\}$.
First, suppose that $w(C)=1$.
It follows from the definition of $\Gamma\bullet\mathbb{H}$ that
$E((\Gamma\bullet\mathbb{H})[V(C)\cup\{\varphi_D(2k)\}])=\{V(C)\cup\{\varphi_D(2k)\}\}$.
Therefore,
$(\Gamma\bullet\mathbb{H})[V(C)\cup\{\varphi_D(2k)\}]$ is prime.

Second, suppose that $w(C)\geq 2$.
Let $\psi_C^{2k}:\{0,\ldots,v(C)\}\longrightarrow
V(C)\cup\{\varphi_D(2k)\}$ satisfying $(\psi_C^{2k})_{\restriction\{0,\ldots,v(C)-1\}}=\varphi_{C}$ and
$\psi_C^{2k}(v(C))=\varphi_{D}(2k)$.
By \eqref{E1_Rbis_Critical_hyper}, $\psi_C^{2k}$ is an isomorphism from
$C_3(W_{v(C)+1})$ onto
$(\Gamma\bullet\mathbb{H})[V(C)\cup\{\varphi_D(2k)\}]$.
By Theorem~\ref{thm _Critical_ST}, $W_{v(C)+1}$ is critical.
By Theorem~\ref{thm_same_primality}, $C_3(W_{v(C)+1})$ is critical and
$\mathscr{P}(C_3(W_{v(C)+1}))=\mathscr{P}(W_{v(C)+1})$.
Furthermore, it follows from Proposition~\ref{prop_W2n+1} that $\mathscr{P}(W_{v(C)+1})-2n=P_{v(C)}$ and
$N_{\mathscr{P}(W_{v(C)+1})}(2n)=\emptyset$.
Since $\psi_C^{2k}$ is an isomorphism from
$C_3(W_{v(C)+1})$ onto
$(\Gamma\bullet\mathbb{H})[V(C)\cup\{\varphi_D(2k)\}]$,
$(\Gamma\bullet\mathbb{H})[V(C)\cup\{\varphi_D(2k)\}]$ is critical and we have
\begin{align*}
\mathscr{P}((\Gamma\bullet\mathbb{H})[V(C)\cup\{\varphi_D(2k)\}])&-\varphi_D(2k)=\\
&(V(C),\{\varphi_C(i)\varphi_C(i+1):0\leq i<2w(C)-1\})
\end{align*}
and
$N_{\mathscr{P}((\Gamma\bullet\mathbb{H})[V(C)\cup\{\varphi_D(2k)\}])}(\varphi_D(2k))=\emptyset$.
\end{rem}

If $\mathbb{H}$ is disconnected, then $\Gamma\bullet\mathbb{H}$ is decomposable, whence the necessity of  Assertion (C1) in Proposition~\ref{Prop1_Critical_hyper}.
Indeed, we have

\begin{lem}\label{lem1bis_Crticial_hyper}
If $\mathbb{C}$ is a component of $\mathbb{H}$, then
$\overline{V(\mathbb{C})}$ and
$V(\Gamma\bullet\mathbb{H})\setminus\overline{V(\mathbb{C})}$ are modules of
$\Gamma\bullet\mathbb{H}$.
\end{lem}

\begin{proof}
For a contradiction, suppose that there exists $e\in E(\Gamma\bullet\mathbb{H})$ such that
$e\cap\overline{V(\mathbb{C})}\neq\emptyset$
and
$e\cap(V(\Gamma\bullet\mathbb{H})\setminus\overline{V(\mathbb{C})})\neq\emptyset$.
There exist $C\in V(\mathbb{C})$ and $D\not\in V(\mathbb{C})$ such that $e\cap V(C)\neq\emptyset$ and $e\cap V(D)\neq\emptyset$.
It follows from the definition of $\Gamma\bullet\mathbb{H}$ that there exists
$\varepsilon\in E(\mathbb{H})$ such that $C,D\in\varepsilon$, which contradicts the fact that
$\mathbb{C}$ is a component of $\mathbb{H}$.
Consequently, for each $e\in E(\Gamma\bullet\mathbb{H})$, we have $e\subseteq\overline{V(\mathbb{C})}$ or
$e\subseteq(V(\Gamma\bullet\mathbb{H})\setminus\overline{V(\mathbb{C})})\neq\emptyset$.
It follows that $\overline{V(\mathbb{C})}$ and
$V(\Gamma\bullet\mathbb{H})\setminus\overline{V(\mathbb{C})}$ are modules of
$\Gamma\bullet\mathbb{H}$.
\end{proof}

Each edge of $\mathbb{H}$ induces a prime subhypergraph of $\Gamma\bullet\mathbb{H}$.
Precisely, we have

\begin{lem}\label{Cl0_Rbis_Critical_hyper}
For every $\varepsilon\in E(\mathbb{H})$,
$(\Gamma\bullet\mathbb{H})[\overline{\varepsilon}]$ is prime.
\end{lem}

\begin{proof}
Let $\varepsilon\in E(\mathbb{H})$.
First, suppose that $|\varepsilon|=2$.
There exist $C\in\mathfrak{C}_{{\rm even}}(\Gamma)$ and
$D\in\mathfrak{C}_{{\rm odd}}(\Gamma)$ such that
$\varepsilon=CD$.
Hence, we have to show that
$(\Gamma\bullet\mathbb{H})[V(C)\cup V(D)]$ is prime.
If $w(D)=0$, then
$(\Gamma\bullet\mathbb{H})[V(C)\cup V(D)]$ is prime by Remark~\ref{v(C)_even}.
Suppose that $w(D)\geq 1$.
Set $X=V(D)$.
By Remark~\ref{v(C)_odd}, $(\Gamma\bullet\mathbb{H})[X]$ is prime.
We have $V(C)\subseteq\langle X\rangle_{\Gamma\bullet\mathbb{H}}$
(see Notation~\ref{nota_p_hyper}).
Let $M$ be a module of
$(\Gamma\bullet\mathbb{H})[X\cup V(C)]$ such that $|M|\geq 2$.
We have to show that $M=X\cup V(C)$.
By Lemma~\ref{fact1_same_block}, we have
$M\cap X=\emptyset$, $M\supseteq X$, or $M\cap X=\{y\}$, where $y\in X$.
It follows from the third assertion of Lemma~\ref{fact1_same_block} that $|M\cap X|\neq 1$.
Now, suppose that $M\cap X=\emptyset$.
Hence, we have $M\subseteq V(C)$.
By Remark~\ref{v(C)_even}, $(\Gamma\bullet\mathbb{H})[V(C)\cup\{\varphi_{D}(0)\}]$ is prime.
Since $|M|\geq 2$ and $M\subseteq V(C)$, we obtain
$M=V(C)\cup\{\varphi_{D}(0)\}$, which contradicts
$M\cap X=\emptyset$.
Therefore, we have $M\supseteq X$.
Let $i,j\in\{0,\ldots,w(C)-1\}$ with $i\leq j$.
We have $\varphi_{C}(2i)\varphi_{C}(2j+1)\varphi_{D}(0)\in E(H)$.
Since $M$ is a module of
$(\Gamma\bullet\mathbb{H})[X\cup V(C)]$ and
$\varphi_{C}(2i)\varphi_{C}(2j+1)\varphi_{D}(1)\not\in E(H)$, we obtain
$\varphi_{C}(2i),\varphi_{C}(2j+1)\in M$.
It follows that $M=X\cup V(C)$.

Second, suppose that $|e|=3$.
There exist $I,J,K\in\mathfrak{C}_{{\rm odd}}(\Gamma)$ such that
$e=IJK$.
Hence, we have to show that
$(\Gamma\bullet\mathbb{H})[V(I)\cup V(J)\cup V(K)]$ is prime.
If $w(I)=w(J)=w(K)=0$, then $|V(I)\cup V(J)\cup V(K)|=3$, and hence
$E((\Gamma\bullet\mathbb{H})[V(I)\cup V(J)\cup V(K)])=
\{\varphi_{I}(0)\varphi_{J}(0)\varphi_{K}(0)\}$.
It follows that
$(\Gamma\bullet\mathbb{H})[V(I)\cup V(J)\cup V(K)]$ is prime.
Now, suppose that $w(I)>0$, $w(J)>0$, or $w(K)>0$.
For instance, assume that $w(I)>0$.
By Remark~\ref{v(C)_odd}, $(\Gamma\bullet\mathbb{H})[V(I)]$ is prime.
Set $X=V(I)$.
We have $V(I)\cup V(J)\cup V(K)\subseteq\langle X\rangle_{\Gamma\bullet\mathbb{H}}$.
Let $M$ be a module of $(\Gamma\bullet\mathbb{H})[X\cup V(J)\cup V(K)]$ such that
$|M|\geq 2$.
We have to show that
$M=X\cup V(J)\cup V(K)$.
It follows from Lemma~\ref{fact1_same_block} that $M\cap X=\emptyset$ or $X\subseteq M$.

For a contradiction, suppose that $M\cap X=\emptyset$.
For $0\leq j\leq w(J)$ and $0\leq k\leq w(K)$, we have
$\varphi_{I}(0)\varphi_{J}(2j)\varphi_{K}(2k)\in E(\Gamma\bullet\mathbb{H})$.
It follows that
$M\cap\{\varphi_{J}(2j):0\leq j\leq w(J)\}=\emptyset$
or
$M\cap\{\varphi_{K}(2k):0\leq k\leq w(K)\}=\emptyset$.
For instance, assume that
\begin{equation}\label{E2_Cl0_Rbis_Critical_hyper}
M\cap\{\varphi_{J}(2j):0\leq j\leq w(J)\}=\emptyset.
\end{equation}
We distinguish the following two cases.
\begin{enumerate}
\item Suppose that
$M\cap\{\varphi_{J}(2j+1):0\leq j\leq w(J)-1\}
\neq\emptyset$.
Hence, there exists $j\in\{0,\ldots,w(J)-1\}$ such that
$\varphi_{J}(2j+1)\in M$.
We have
$\varphi_{J}(2j)\varphi_{J}(2j+1)\varphi_{J}(2j+2)\in E(H)$.
By \eqref{E2_Cl0_Rbis_Critical_hyper}, $\varphi_{J}(2j),
\varphi_{J}(2j+2)\not\in M$.
Since $M$ is a module of $(\Gamma\bullet\mathbb{H})[X\cup V(J)\cup V(K)]$, we obtain
$\varphi_{J}(2j)x\varphi_{J}(2j+2)\in E(H)$ for every $x\in M$.
For each $v\in V(K)$, we have
$\varphi_{J}(2j)v\varphi_{J}(2j+2)\not\in E(H)$.
It follows that $M\cap V(K)=\emptyset$.
Therefore, $M\subseteq V(J)$, and hence
$M$ is a module of $(\Gamma\bullet\mathbb{H})[X\cup V(J)]$.
Since $|M|\geq 2$, $w(J)\geq 1$.
By Remark~\ref{v(C)_odd},
$(\Gamma\bullet\mathbb{H})[V(J)]$ is prime.
Therefore, $M=V(J)$, which contradicts \eqref{E2_Cl0_Rbis_Critical_hyper}.
\item Suppose that
$M\cap\{\varphi_{J}(2j+1):0\leq j\leq w(J)-1\}=\emptyset$.
By \eqref{E2_Cl0_Rbis_Critical_hyper}, $M\cap V(J)=\emptyset$, and hence $M\subseteq V(K)$.
As previously for $J$, we obtain $w(k)\geq 1$ and $M=V(K)$, which is impossible because
$\varphi_{I}(0)\varphi_{J}(0)\varphi_{K}(0)\in E(H)$ and
$\varphi_{I}(0)\varphi_{J}(0)\varphi_{K}(1)\not\in E(H)$.
\end{enumerate}

It follows that $X\subseteq M$.
Recall that $\varphi_{I}(0)\varphi_{J}(2j)\varphi_{K}(2k)\in E(\Gamma\bullet\mathbb{H})$ for
$0\leq j\leq w(J)$ and $0\leq k\leq w(K)$.
Let $j\in\{0,\ldots,w(J)\}$ and
$k\in\{0,\ldots,w(J)\}$.
We have $\varphi_{I}(0)\in M$ because $X\subseteq M$.
Since
$\varphi_{I}(1)\varphi_{J}(2j)\varphi_{K}(2k)\not\in E(\Gamma\bullet\mathbb{H})$ and $M$ is a module of
$(\Gamma\bullet\mathbb{H})[X\cup V(J)\cup V(K)]$, we obtain
$\varphi_{J}(2j),\varphi_{K}(2k)\in M$.
It follows that $X\cup\{\varphi_{J}(2j):0\leq j\leq w(J)\}\cup\{\varphi_{K}(2k):0\leq k\leq w(K)\}\subseteq M$.
Clearly, $V(J)\subseteq M$ when
$w(J)=0$.
Suppose that $w(J)\geq 1$.
By Remark~\ref{v(C)_odd},
$(\Gamma\bullet\mathbb{H})[V(J)]$ is prime.
Since $M\cap V(J)$ is a module of $(\Gamma\bullet\mathbb{H})[V(J)]$ such that
$\{\varphi_{J}(2j):0\leq j\leq w(J)\}\subseteq
M\cap V(J)$, we obtain $V(J)\subseteq M$.
Similarly, $V(K)\subseteq M$.
Consequently, $M=X\cup V(J)\cup V(K)$.
\end{proof}

In the next four results, we compare the module of $\mathbb{H}$ with those of $\Gamma\bullet\mathbb{H}$.

\begin{lem}\label{lem1_Crticial_hyper}
Suppose that $\mathbb{H}$ is connected.
For each nontrivial module $M$ of $\Gamma\bullet\mathbb{H}$, we have
$M=\overline{M/\mathfrak{C}(\Gamma)}$,
$|M/\mathfrak{C}(\Gamma)|\geq 2$, and
$M/\mathfrak{C}(\Gamma)$ is a module of $\mathbb{H}$.
\end{lem}

\begin{proof}
First, we prove that $V(C)\subseteq M$ for every $C\in M/\mathfrak{C}(\Gamma)$.
To begin, suppose that $C\in\mathfrak{C}_{{\rm odd}}(\Gamma)$.
If $w(C)=0$, then we clearly have $V(C)\subseteq M$.
Hence, suppose that $w(C)\geq 1$.
By Remark~\ref{v(C)_odd}, $(\Gamma\bullet\mathbb{H})[V(C)]$ is prime.
For a contradiction, suppose that $|V(C)\cap M|=1$.
Denote by $c$ the unique element of $V(C)\cap M$.
Since $(\Gamma\bullet\mathbb{H})[V(C)]$ is prime, there exist distinct $d,d'\in V(C)\setminus M$ such that
$cdd'\in E(\Gamma\bullet\mathbb{H})$.
Since $|M|\geq 2$, there exists $v\in M\setminus V(C)$.
Since $M$ is a module of $\Gamma\bullet\mathbb{H}$, we obtain
$vdd'\in E(H_{(\Gamma,\mathbb{H})})$, which contradicts the definition of
$\Gamma\bullet\mathbb{H}$.
It follows that $|V(C)\cap M|\geq 2$.
Since $(\Gamma\bullet\mathbb{H})[V(C)]$ is prime, $V(C)\subseteq M$.
Now, suppose that $C\in\mathfrak{C}_{{\rm even}}(\Gamma)$.
Since $\mathbb{H}$ is connected, there exists
$D\in\mathfrak{C}_{{\rm odd}}(\Gamma)$ such that $CD\in E(\mathbb{H})$.
For a contradiction, suppose that $|V(C)\cap M|=1$.
Denote by $c$ the unique element of $V(C)\cap M$.
There exists $c'\in V(C)\setminus M$ such that
$cc'\varphi_D(0)\in E(\Gamma\bullet\mathbb{H})$.
By Remark~\ref{v(C)_even},
$(\Gamma\bullet\mathbb{H})[V(C)\cup\{\varphi_D(0)\}]$ is prime.
Hence, if $|(V(C)\cup\{\varphi_D(0)\})\cap M|\geq 2$, then
$V(C)\cup\{\varphi_D(0)\}\subseteq M$, which contradicts $|V(C)\cap M|=1$.
Therefore, $|(V(C)\cup\{\varphi_D(0)\})\cap M|=1$.
Since $|M|\geq 2$, there exists $v\in M\setminus(V(C)\cup\{\varphi_D(0)\})$.
Since $M$ is a module of $\Gamma\bullet\mathbb{H}$, we obtain
$vc'\varphi_D(0)\in E(\Gamma\bullet\mathbb{H})$, which contradicts the definition of
$\Gamma\bullet\mathbb{H}$.
It follows that $|V(C)\cap M|\geq 2$.
Since $(\Gamma\bullet\mathbb{H})[V(C)\cup\{\varphi_D(0)\}]$ is prime, we obtain
$V(C)\cup\{\varphi_D(0)\}\subseteq M$.

Second, we prove that $|M/\mathfrak{C}(\Gamma)|\geq 2$.
If $v(C)=1$ for each $C\in M/\mathfrak{C}(\Gamma)$, then $|M/\mathfrak{C}(\Gamma)|\geq 2$ because $|M|\geq 2$.
Hence, consider $C\in M/\mathfrak{C}(\Gamma)$ such that $v(C)\geq 2$.
Since $\mathbb{H}$ is connected, there exists $\varepsilon\in E(\mathbb{H})$ such that $C\in\varepsilon$.
By Lemma~\ref{Cl0_Rbis_Critical_hyper},
$(\Gamma\bullet\mathbb{H})[\overline{\varepsilon}]$ is prime.
Since $V(C)\subseteq M$ and $v(C)\geq 2$, we get $\overline{\varepsilon}\subseteq M$.
Thus $e\subseteq (M/\mathfrak{C}(\Gamma))$, so
$|M/\mathfrak{C}(\Gamma)|\geq 2$.

Third, we prove that $M/\mathfrak{C}(\Gamma)$ is a module of $\mathbb{H}$.
Let $\varepsilon\in E(\mathbb{H})$ such that $\varepsilon\cap(M/\mathfrak{C}(\Gamma))\neq\emptyset$ and
$\varepsilon\setminus(M/\mathfrak{C}(\Gamma))\neq\emptyset$.
We distinguish the following two cases.
\begin{itemize}
\item Suppose that $|\varepsilon|=2$.
There exist $C\in\mathfrak{C}_{{\rm even}}(\Gamma)$ and
$D\in\mathfrak{C}_{{\rm odd}}(\Gamma)$ such that $\varepsilon=CD$.
Thus, there exist distinct $c,c'\in V(C)$ and $d\in V(D)$ such that
$cc'd\in E(\Gamma\bullet\mathbb{H})$.
For a contradiction, suppose that $C\in M/\mathfrak{C}(\Gamma)$.
Since $\varepsilon\setminus(M/\mathfrak{C}(\Gamma))\neq\emptyset$, we have
$D\not\in M/\mathfrak{C}(\Gamma)$.
By the first point above, $V(C)\subseteq M$ and $V(D)\cap M=\emptyset$.
We obtain $cc'd\in E(\Gamma\bullet\mathbb{H})$, with $c,c'\in M$ and $d\not\in M$, which contradicts the fact that $M$ is a module of $\Gamma\bullet\mathbb{H}$.
It follows that $D\in M/\mathfrak{C}(\Gamma)$ and $C\not\in M/\mathfrak{C}(\Gamma)$.
By the first point above, $V(D)\subseteq M$ and $V(C)\cap M=\emptyset$.
We obtain $cc'd\in E(\Gamma\bullet\mathbb{H})$, with $c,c'\not\in M$ and $d\in M$.
Since $M$ is a module of $\Gamma\bullet\mathbb{H}$, we obtain
\begin{equation}\label{E1_lem1_Crticial_hyper}
\text{$cc'x\in E(H_{(\Gamma,\mathbb{H})})$ for every $x\in M$.}
\end{equation}
Consider $D'\in M/\mathfrak{C}(\Gamma)$.
By the first point above, $V(D')\subseteq M$.
Let $d'\in V(D')$.
By \eqref{E1_lem1_Crticial_hyper},
$cc'd'\in E(\Gamma\bullet\mathbb{H})$.
It follows from the definition of $\Gamma\bullet\mathbb{H}$ that
$CD'\in E(\mathbb{H})$.
\item Suppose that $|\varepsilon|=3$.
There exist distinct $I,J,K\in\mathfrak{C}_{{\rm odd}}(\Gamma)$ such that
$\varepsilon=IJK$.
Moreover, we can assume that
$I\in M/\mathfrak{C}(\Gamma)$ and $K\not\in M/\mathfrak{C}(\Gamma)$.
By the first point above, $V(I)\subseteq M$ and $V(K)\cap M=\emptyset$.
By definition of $\Gamma\bullet\mathbb{H}$, we obtain
$\varphi_I(0)\varphi_{J}(0)\varphi_{K}(0)\in E(\Gamma\bullet\mathbb{H})$.
Since $M$ is a module of $\Gamma\bullet\mathbb{H}$, we have $\varphi_{J}(0)\not\in M$.
It follows from the first point above that $V(J)\cap M=\emptyset$.
Thus, $\varphi_I(0)\varphi_{J}(0)\varphi_{K}(0)\in E(\Gamma\bullet\mathbb{H})$, with
$\varphi_I(0)\in M$ and $\varphi_{J}(0),\varphi_{K}(0)\not\in M$.
Since $M$ is a module of $\Gamma\bullet\mathbb{H}$, we obtain
\begin{equation}\label{E2_lem1_Crticial_hyper}
\text{$x\varphi_{J}(0)\varphi_{K}(0)\in E(\Gamma\bullet\mathbb{H})$ for every
$x\in M$.}
\end{equation}
Consider any $L\in M/\mathfrak{C}(\Gamma)$.
By the first point above, $V(L)\subseteq M$
Let $d\in V(L)$.
By \eqref{E2_lem1_Crticial_hyper},
$d\varphi_{J}(0)\varphi_{K}(0)\in E(\Gamma\bullet\mathbb{H})$.
Since $J,K,L$ are distinct element of $\mathfrak{C}(\Gamma)$, it
follows from the definition of $\Gamma\bullet\mathbb{H}$ that
$JKL\in E(\mathbb{H})$.
\end{itemize}
Consequently, $M/\mathfrak{C}(\Gamma)$ is a module of $\mathbb{H}$.
\end{proof}

\begin{lem}\label{lem2_Crticial_hyper}
Suppose that $\mathbb{H}$ is connected.
For a nontrivial module $\mathbb{M}$ of $\mathbb{H}$, we have
$\mathbb{M}\subseteq\mathfrak{C}_{{\rm even}}(\Gamma)$ or
$\mathbb{M}\subseteq\mathfrak{C}_{{\rm odd}}(\Gamma)$.
\end{lem}

\begin{proof}
Suppose that $\mathbb{M}\setminus\mathfrak{C}_{{\rm odd}}(\Gamma)\neq\emptyset$.
We have to show that $\mathbb{M}\subseteq\mathfrak{C}_{{\rm even}}(\Gamma)$.
Since $\mathbb{M}\setminus\mathfrak{C}_{{\rm odd}}(\Gamma)\neq\emptyset$, there exists $C\in\mathbb{M}\cap\mathfrak{C}_{{\rm even}}(\Gamma)$.
Since $\mathbb{H}$ is connected and $\mathbb{M}\subsetneq V(\mathbb{H})$, there exists $\varepsilon\in E(\mathbb{H})$ such that $\varepsilon\cap\mathbb{M}\neq\emptyset$ and $\varepsilon\setminus\mathbb{M}\neq\emptyset$.
Furthermore, since $\mathbb{M}$ is a module of $\mathbb{H}$, we have $|\varepsilon\cap\mathbb{M}|=1$.
It follows that $(\varepsilon\setminus\mathbb{M})\cup\{C\}\in E(\mathbb{H})$.
Since $C\in\mathbb{M}\cap\mathfrak{C}_{{\rm even}}(\Gamma)$, we obtain
$|\varepsilon\setminus\mathbb{M}|=1$ and
$\varepsilon\setminus\mathbb{M}\subseteq\mathfrak{C}_{{\rm odd}}(\Gamma)$.
Lastly, consider $D\in\mathbb{M}$.
Since $\mathbb{M}$ is a module of $\mathbb{H}$, we have
$(\varepsilon\setminus\mathbb{M})\cup\{D\}\in E(\mathbb{H})$.
Since $|(\varepsilon\setminus\mathbb{M})\cup\{D\}|=2$ and
$(\varepsilon\setminus\mathbb{M})\subseteq\mathfrak{C}_{{\rm odd}}(\Gamma)$, we obtain
$D\in\mathbb{M}\cap\mathfrak{C}_{{\rm even}}(\Gamma)$.
Consequently,
$\mathbb{M}\subseteq\mathfrak{C}_{{\rm even}}(\Gamma)$.
\end{proof}

The next result follows from Lemmas~\ref{lem1_Crticial_hyper} and \ref{lem2_Crticial_hyper}.

\begin{cor}\label{cor1_Crticial_hyper}
Suppose that $\mathbb{H}$ is connected.
For each nontrivial module $M$ of $\Gamma\bullet\mathbb{H}$, we have
$M\subseteq V_1(\Gamma)$.
\end{cor}

\begin{proof}
It follows from Lemma~\ref{lem1_Crticial_hyper} that $M/\mathfrak{C}(\Gamma)$ is a nontrivial module of $\mathbb{M}$.
By Lemma~\ref{lem2_Crticial_hyper},
$M/\mathfrak{C}(\Gamma)\subseteq\mathfrak{C}_{{\rm even}}(\Gamma)$ or
$M/\mathfrak{C}(\Gamma)\subseteq\mathfrak{C}_{{\rm odd}}(\Gamma)$.

For a contradiction, suppose that
$M/\mathfrak{C}(\Gamma)\subseteq\mathfrak{C}_{{\rm even}}(\Gamma)$.
Since $\mathbb{H}$ is connected, there exists $\varepsilon\in E(\mathbb{H})$ such that
$\varepsilon\cap(M/\mathfrak{C}(\Gamma))\neq\emptyset$ and
$\varepsilon\setminus(M/\mathfrak{C}(\Gamma))\neq\emptyset$.
Since $M/\mathfrak{C}(\Gamma)\subseteq\mathfrak{C}_{{\rm even}}(\Gamma)$, there exist $C\in M/\mathfrak{C}(\Gamma)$ and $D\in\mathfrak{C}_{{\rm odd}}(\Gamma)$ such that $\varepsilon=CD$.
By Lemma~\ref{lem1_Crticial_hyper}, $M=\overline{M/\mathfrak{C}(\Gamma)}$.
Therefore, we obtain $V(C)\subseteq M$ and $V(D)\cap M=\emptyset$.
It follows that $V(C)$ is a nontrivial module of $(\Gamma\bullet\mathbb{H})[V(C)\cup V(D)]$, which contradicts Lemma~\ref{Cl0_Rbis_Critical_hyper}.
Consequently, we have
$M/\mathfrak{C}(\Gamma)\subseteq\mathfrak{C}_{{\rm odd}}(\Gamma)$.

For a contradiction, suppose that $M\setminus V_1(\Gamma)\neq\emptyset$.
There exists
$D\in(M/\mathfrak{C}(\Gamma))\cap(\mathfrak{C}_{{\rm odd}}(\Gamma)\setminus\mathfrak{C}_{1}(\Gamma))$.
Since $\mathbb{H}$ is connected, there exists $\varepsilon\in E(\mathbb{H})$ such that
$\varepsilon\cap(M/\mathfrak{C}(\Gamma))\neq\emptyset$ and
$\varepsilon\setminus(M/\mathfrak{C}(\Gamma))\neq\emptyset$.
Since $M/\mathfrak{C}(\Gamma)$ is a module of $\mathbb{M}$, there exists
$C\in M/\mathfrak{C}(\Gamma)$ such that $\varepsilon\cap(M/\mathfrak{C}(\Gamma))=\{C\}$ and
$(\varepsilon\setminus\{C\})\cup\{D\}\in E(\mathbb{H})$.
By Lemma~\ref{lem1_Crticial_hyper}, $V(D)\subseteq M$.
We obtain that $V(D)$ is a nontrivial module of
$(\Gamma\bullet\mathbb{H})[\overline{(e\setminus\{C\})\cup\{D\}}]$, which contradicts Lemma~\ref{Cl0_Rbis_Critical_hyper}.
Consequently, $M\subseteq V_1(\Gamma)$.
\end{proof}

\begin{lem}\label{lem3_Crticial_hyper}
Given a module $\mathbb{M}$ of $\mathbb{H}$, if
$\mathbb{M}\subseteq\mathfrak{C}_1(\Gamma)$, then
$\overline{\mathbb{M}}$ is a module of $\Gamma\bullet\mathbb{H}$.
\end{lem}

\begin{proof}
Let $e\in E(\Gamma\bullet\mathbb{H})$ such that
$e\cap\overline{\mathbb{M}}\neq\emptyset$ and
$e\setminus\overline{\mathbb{M}}\neq\emptyset$.
There exist $c\in e\cap\overline{\mathbb{M}}$ and
$x\in e\setminus\overline{\mathbb{M}}$.
We have $\{c\}\in\mathfrak{C}(\Gamma)$.
Consider $C_x\in\mathfrak{C}(\Gamma)$ such that $x\in C_x$.
For a contradiction, suppose that $e=cdx$, where $d\in\overline{\mathbb{M}}$.
We have $\{d\}\in\mathfrak{C}(\Gamma)$.
Since $cdx\in E(\Gamma\bullet\mathbb{H})$, we obtain $C_cC_dC_x\in E(\mathbb{H})$, which contradicts the fact that $\mathbb{M}$ is a module of $\mathbb{H}$.
It follows that $e=cxy$, where
$y\in V(\Gamma\bullet\mathbb{H})\setminus\overline{\mathbb{M}}$.
Consider $C_y\in\mathfrak{C}(\Gamma)$ such that $y\in C_y$.
Given $d\in\overline{\mathbb{M}}$, we have to verify that
$dxy\in E(\Gamma\bullet\mathbb{H})$.
We have $\{d\}\in\mathfrak{C}(\Gamma)$.
We distinguish the following two cases.
\begin{itemize}
\item Suppose that $C_x=C_y$.
Since $cxy\in E(\Gamma\bullet\mathbb{H})$, we obtain
$C_x\in\mathfrak{C}_{{\rm even}}(\Gamma)$,
$\{x,y\}=\{\varphi_{C_x}(2i),\varphi_{C_x}(2j+1)\}$, where $0\leq i\leq j\leq w(C_x)-1$, and $\{c\}C_x\in E(\mathbb{H})$.
Since $\mathbb{M}$ is a module of $\mathbb{H}$, $\{d\}C_x\in E(\mathbb{H})$.
Since $C_x\in\mathfrak{C}_{{\rm even}}(\Gamma)$ and
$\{x,y\}=\{\varphi_{C_x}(2i),\varphi_{C_x}(2j+1)\}$, where $0\leq i\leq j\leq w(C_x)-1$, we obtain $dxy\in E(H_{(\Gamma,\mathbb{H})})$.
\item Suppose that $C_x\neq C_y$.
Since $cxy\in E(H_{(\Gamma,\mathbb{H})})$, we obtain
$C_x,C_y\in\mathfrak{C}_{{\rm odd}}(\Gamma)$,
$\{x,y\}=\{\varphi_{C_x}(2i),\varphi_{C_y}(2j)\}$, where $0\leq i\leq w(C_x)$ and
$0\leq j\leq w(C_y)$, and $\{c\}C_xC_y\in E(\mathbb{H})$.
Since $\mathbb{M}$ is a module of $\mathbb{H}$, $\{d\}C_xC_y\in E(\mathbb{H})$.
Since $C_x,C_y\in\mathfrak{C}_{{\rm odd}}(\Gamma)$ and
$\{x,y\}=\{\varphi_{C_x}(2i),\varphi_{C_y}(2j)\}$, where $0\leq i\leq w(C_x)$ and
$0\leq j\leq w(C_y)$, we obtain $dxy\in E(H_{(\Gamma,\mathbb{H})})$. \qedhere
\end{itemize}
\end{proof}

\begin{proof}[Proof of Proposition~\ref{Prop1_Critical_hyper}]
To begin, suppose that $\Gamma\bullet\mathbb{H}$ is decomposable.
If $\mathbb{H}$ is disconnected, then Assertion~(C1) does not hold.
Hence, suppose that $\mathbb{H}$ is connected.
Consider a nontrivial module $M$ of $\Gamma\bullet\mathbb{H}$.
It follows from Lemma~\ref{lem1_Crticial_hyper} that
$M/\mathfrak{C}(\Gamma)$ is a nontrivial module of $\mathbb{H}$.
Furthermore, $M\subseteq V_1(\Gamma)$ by Corollary~\ref{cor1_Crticial_hyper}.
Thus, $M/\mathfrak{C}(\Gamma)\subseteq\mathfrak{C}_{1}(\Gamma)$.
Consequently, Assertion~(C2) does not hold.

Conversely, if $\mathbb{H}$ is disconnected, then
it follows from Lemma~\ref{lem1bis_Crticial_hyper} that
$\Gamma\bullet\mathbb{H}$ is decomposable.
Hence, suppose that $\mathbb{H}$ is connected.
Moreover, suppose that there exists a nontrivial module $\mathbb{M}$ of $\mathbb{H}$
such that $\mathbb{M}\subseteq\mathfrak{C}_{1}(\Gamma)$.
It follows from Lemma~\ref{lem3_Crticial_hyper} that
$\overline{\mathbb{M}}$ is a nontrivial module of $\Gamma\bullet\mathbb{H}$.
Hence, $\Gamma\bullet\mathbb{H}$ is decomposable.
\end{proof}

\section{Proof of Theorem~\ref{thm_2_main}}

In the next remark, we describe a simple way to obtain automorphisms of $\Gamma\bullet\mathbb{H}$.

\begin{rem}\label{Auto_critical_hyper}
For every $C\in\mathfrak{C}(\Gamma)$, consider the function
\begin{equation*}
\begin{array}{rccl}
F_C:&V(\Gamma\bullet\mathbb{H})&\longrightarrow&V(\Gamma\bullet\mathbb{H})\\
&\varphi_C(m)\ (0\leq m\leq v(C)-1)&\longmapsto&\varphi_C(v(C)-1-m)\\
&v\not\in V(C)&\longmapsto&v.
\end{array}
\end{equation*}
Clearly, $F_C$ is an automorphism of $\Gamma$.
It is easy to verify that $F_C$ is an automorphism of $\Gamma\bullet\mathbb{H}$.
Moreover, consider a permutation $F$ of $V(\Gamma\bullet\mathbb{H})$ such that $F(v)=v$ for each $v\in V_1(\Gamma)$.
If for every $C\in\mathfrak{C}(\Gamma)$, $F_{\restriction V(C)}=(F_C)_{\restriction V(C)}$ or ${\rm Id}_{V(C)}$, then $F$ is an automorphism of $\Gamma\bullet\mathbb{H}$.
\end{rem}

The next three results are useful to study the criticality of $\Gamma\bullet\mathbb{H}$.
They allow us to prove Theorem~\ref{thm_2_main}.

\begin{lem}\label{lem4_Crticial_hyper}
Given $C\in\mathfrak{C}_{{\rm odd}}(\Gamma)\setminus\mathfrak{C}_{1}(\Gamma)$, the following assertions hold
\begin{enumerate}
\item $V(\Gamma\bullet\mathbb{H})\setminus\{\varphi_C(0),\varphi_C(1)\}$ is a module of
$(\Gamma\bullet\mathbb{H})-\varphi_C(0)$;
\item $V(\Gamma\bullet\mathbb{H})\setminus\{\varphi_C(2w(C)-1),\varphi_C(2w(C))\}$ is a module of
$(\Gamma\bullet\mathbb{H})-\varphi_C(2w(C))$;
\item if $w(C)\geq 1$, then for $m\in\{1,\ldots,2w(C)-1\}$, we have
$\{\varphi_C(m-1),\varphi_C(m+1)\}$ is a module of $(\Gamma\bullet\mathbb{H})-\varphi_C(m)$.
\end{enumerate}
\end{lem}

\begin{proof}
We begin with the first assertion.
It follows from Remark~\ref{v(C)_odd} that $V(C)\setminus\{\varphi_C(0),\varphi_C(1)\}$
is a module of $((\Gamma\bullet\mathbb{H})[V(C)])-\varphi_C(0)$.
For every $e\in E(\Gamma\bullet\mathbb{H})$, if $\varphi_C(1)\in e$, then $e\subseteq V(C)$ and $\varphi_C(0)\in e$.
Therefore, $V(C)\setminus\{\varphi_C(0),\varphi_C(1)\}$
is a module of $(\Gamma\bullet\mathbb{H})-\varphi_C(0)$.

For the second assertion, recall that the function $F_C$ is an automorphism of
$\Gamma\bullet\mathbb{H}$ (see Remark~\ref{Auto_critical_hyper}).
It follows from the first assertion above that $F_C(V(C)\setminus\{\varphi_C(0),\varphi_C(1)\})$
is a module of $(\Gamma\bullet\mathbb{H})-F_C(\varphi_C(0))$, that is, $V(C)\setminus\{\varphi_C(2w(C)-1),\varphi_C(2w(C))\}$
is a module of $(\Gamma\bullet\mathbb{H})-\varphi_C(2w(C))$.

For the third assertion, suppose that $w(C)\geq 1$.
Let $i\in\{0,\ldots,w(C)-1\}$.
It follows from Remark~\ref{v(C)_odd} that $\{\varphi_C(2i),$ $\varphi_C(2i+2)\}$ is a module of
$((\Gamma\bullet\mathbb{H})[V(C)])-\varphi_C(2i+1)$.
Let $e\in E(\Gamma\bullet\mathbb{H})$ such that $e\cap\{\varphi_C(2i),\varphi_C(2i+2)\}\neq\emptyset$ and $e\setminus V(C)\neq\emptyset$.
By definition of $\Gamma\bullet\mathbb{H}$,
$|e\cap\{\varphi_C(2i),\varphi_C(2i+2)\}|=1$, $|e\setminus V(C)|=2$, and
$(e\setminus V(C))\cup\{\varphi_C(2i)\},(e\setminus V(C))\cup\{\varphi_C(2i+2)\}\in E(H_{(\Gamma,\mathbb{H})})$.
Thus, $\{\varphi_C(2i),\varphi_C(2i+2)\}$ is a module of
$(\Gamma\bullet\mathbb{H})-\varphi_C(2i+1)$.
Lastly, suppose that $w(C)\geq 2$, and consider $i\in\{1,\ldots,w(C)-1\}$.
It follows from Remark~\ref{v(C)_odd} that
$\{\varphi_C(2i-1),\varphi_C(2i+1)\}$ is a module of
$((\Gamma\bullet\mathbb{H})[V(C)])-\varphi_C(2i)$.
For every $e\in E(\Gamma\bullet\mathbb{H})$, if $e\cap\{\varphi_C(2i-1),\varphi_C(2i+1)\}\neq\emptyset$, then $e\subseteq V(C)$.
Therefore,  $\{\varphi_C(2i-1),\varphi_C(2i+1)\}$ is a module of
$(\Gamma\bullet\mathbb{H})-\varphi_C(2i)$.
\end{proof}

\begin{lem}\label{lem5_Crticial_hyper}
Given $C\in\mathfrak{C}_{{\rm even}}(\Gamma)$, the following assertions hold
\begin{enumerate}
\item $V(\Gamma\bullet\mathbb{H})\setminus\{\varphi_C(0),\varphi_C(1)\}$ is a module of
$(\Gamma\bullet\mathbb{H})-\varphi_C(0)$;
\item $V(\Gamma\bullet\mathbb{H})\setminus\{\varphi_C(2w(C)-2),\varphi_C(2w(C)-1)\}$ is a module of $(\Gamma\bullet\mathbb{H})-\varphi_C(2w(C)-1)$;
\item if $w(C)\geq 2$, then for $m\in\{1,\ldots,2w(C)-2\}$, we have
$\{\varphi_C(m-1),\varphi_C(m+1)\}$ is a module of $(\Gamma\bullet\mathbb{H})-\varphi_C(m)$.
\end{enumerate}
\end{lem}

\begin{proof}
For the first assertion, consider $e\in E(\Gamma\bullet\mathbb{H})$ such that
$\varphi_C(1)\in e$.
By definition of $\Gamma\bullet\mathbb{H}$, there exists
$D\in\mathfrak{C}_{{\rm odd}}(\Gamma)$ such that
$e=\varphi_C(0)\varphi_C(1)\varphi_D(2k)$, where $0\leq k\leq w(D)$.
Consequently, there does not exist $e\in E((\Gamma\bullet\mathbb{H})-\varphi_C(0))$ such that $\varphi_C(1)\in e$.
It follows that $V(\Gamma\bullet\mathbb{H})\setminus\{\varphi_C(0),\varphi_C(1)\}$ is a module of $(\Gamma\bullet\mathbb{H})-\varphi_C(0)$.

As in the proof of the second assertion of Lemma~\ref{lem4_Crticial_hyper}, the second assertion is deduced from the first one by using Remark~\ref{Auto_critical_hyper}.

For the third assertion, suppose that $w(C)\geq 2$.
Consider $i\in\{1,\ldots,w(C)-1\}$.
We prove that $\{\varphi_C(2i-1),\varphi_C(2i+1)\}$ is a module of
$(\Gamma\bullet\mathbb{H})-\varphi_C(2i)$.
Let $e\in E((\Gamma\bullet\mathbb{H})-\varphi_C(2i))$ such that
$e\cap\{\varphi_C(2i-1),\varphi_C(2i+1)\}\neq\emptyset$ and
$e\setminus\{\varphi_C(2i-1),\varphi_C(2i+1)\}\neq\emptyset$.
By definition of $\Gamma\bullet\mathbb{H}$, there exist
$D\in\mathfrak{C}_{{\rm odd}}(\Gamma)$ and $k\in\{0,\ldots,w(D)\}$ such that
\begin{equation*}
\begin{cases}
\text{$e=\varphi_C(2j)\varphi_C(2i-1)\varphi_D(2k)$, where $j\in\{0,\ldots,i-1\}$}\\
\text{or}\\
\text{$e=\varphi_C(2j)\varphi_C(2i+1)\varphi_D(2k)$, where $j\in\{0,\ldots,i\}$}.
\end{cases}
\end{equation*}
In the second instance, we have $j\in\{0,\ldots,i-1\}$ because $\varphi_C(2i)\not\in e$.
By definition of $\Gamma\bullet\mathbb{H}$, we have
$\varphi_C(2j)\varphi_C(2i-1)\varphi_D(2k),\varphi_C(2j)\varphi_C(2i+1)\varphi_D(2k)\in
E((\Gamma\bullet\mathbb{H})-\varphi_C(2i))$.
Thus, $\{\varphi_C(2i-1),\varphi_C(2i+1)\}$ is a module of
$(\Gamma\bullet\mathbb{H})-\varphi_C(2i)$.
Lastly, consider $i\in\{0,\ldots,w(C)-2\}$.
We prove that $\{\varphi_C(2i),\varphi_C(2i+2)\}$ is a module of
$(\Gamma\bullet\mathbb{H})-\varphi_C(2i+1)$.
Let $e\in E((\Gamma\bullet\mathbb{H})-\varphi_C(2i+1))$ such that
$e\cap\{\varphi_C(2i),\varphi_C(2i+2)\}\neq\emptyset$ and
$e\setminus\{\varphi_C(2i),\varphi_C(2i+2)\}\neq\emptyset$.
By definition of $\Gamma\bullet\mathbb{H}$, there exist
$D\in\mathfrak{C}_{{\rm odd}}(\Gamma)$ and $k\in\{0,\ldots,w(D)\}$ such that
\begin{equation*}
\begin{cases}
\text{$e=\varphi_C(2i)\varphi_C(2j+1)\varphi_D(2k)$, where $j\in\{i,\ldots,w(C)-1\}$}\\
\text{or}\\
\text{$e=\varphi_C(2i+2)\varphi_C(2j+1)\varphi_D(2k)$, where $j\in\{i+1,\ldots,w(C)-1\}$}.
\end{cases}
\end{equation*}
In the first instance, we have $j\in\{i+1,\ldots,w(C)-1\}$ because $\varphi_C(2i+1)\not\in e$.
By definition of $\Gamma\bullet\mathbb{H}$, we have
$\varphi_C(2i)\varphi_C(2j+1)\varphi_D(2k),\varphi_C(2i+2)\varphi_C(2j+1)\varphi_D(2k)\in
E((\Gamma\bullet\mathbb{H})-\varphi_C(2i+1))$.
Thus, $\{\varphi_C(2i),\varphi_C(2i+2)\}$ is a module of
$(\Gamma\bullet\mathbb{H})-\varphi_C(2i+1)$.
\end{proof}

The next result is an immediate consequence of Lemmas~\ref{lem4_Crticial_hyper}
and \ref{lem5_Crticial_hyper}.

\begin{cor}\label{cor2_Crticial_hyper}
Let $C\in\mathfrak{C}(\Gamma)\setminus\mathfrak{C}_{1}(\Gamma)$.
For every $c\in V(C)$, we have $(\Gamma\bullet\mathbb{H})-c$ is decomposable.
Moreover, we have
\begin{enumerate}
\item $N_{\mathscr{P}(\Gamma\bullet\mathbb{H})}(\varphi_C(0))\subseteq\{\varphi_C(1)\}$;
\item $N_{\mathscr{P}(\Gamma\bullet\mathbb{H})}(\varphi_C(v(C)-1))\subseteq\{\varphi_C(v(C)-2)\}$;
\item if $v(C)\geq 3$, then for every $m\in\{1,\ldots,v(C)-2\}$, we have
$N_{\mathscr{P}(\Gamma\bullet\mathbb{H})}(\varphi_C(m))\subseteq\{\varphi_C(m-1),\varphi_C(m+1)\}$.
\end{enumerate}
\end{cor}

\begin{proof}[Proof of Theorem~\ref{thm_2_main}]
To begin, suppose that $\Gamma\bullet\mathbb{H}$ is critical.
In particular, $\Gamma\bullet\mathbb{H}$ is prime.
By Proposition~\ref{Prop1_Critical_hyper}, Assertions (C1) and (C2) hold.
For Assertion (C3), consider $v\in V_1(\Gamma)$.
Since $\Gamma\bullet\mathbb{H}$ is critical, $(\Gamma\bullet\mathbb{H})-v$ is decomposable.
Furthermore, since $v\in V_1(\Gamma)$, we have $\{v\}\in\mathfrak{C}_{1}(\Gamma)$.
It follows that $(\Gamma\bullet\mathbb{H})-v=(\Gamma-v)\bullet(\mathbb{H}-\{v\})$.
By Proposition~\ref{Prop1_Critical_hyper} applied to $(\Gamma\bullet\mathbb{H})-v$, $\mathbb{H}-\{v\}$ is disconnected or
$\mathbb{H}-\{v\}$ is connected, and there exists a nontrivial module
$\mathbb{M}_{\{v\}}$ of $\mathbb{H}-\{v\}$ such that
$\mathbb{M}_{\{v\}}\subseteq\mathfrak{C}_1(\Gamma-v)$.
Since $\mathfrak{C}_1(\Gamma-v)=\mathfrak{C}_1(\Gamma)\setminus\{\{v\}\}$,
$\mathbb{M}_{\{v\}}\subseteq\mathfrak{C}_1(\Gamma)\setminus\{\{v\}\}$.

Conversely, suppose that Assertions (C1), (C2) and (C3) hold.
Since Assertions (C1) and (C2) hold, it follows from Proposition~\ref{Prop1_Critical_hyper},
$\Gamma\bullet\mathbb{H}$ is prime.
Furthermore, it follows from Assertion (C3) and Proposition~\ref{Prop1_Critical_hyper} that
$(\Gamma\bullet\mathbb{H})-v$ is decomposable for each
$v\in V_1(\Gamma)$.
Lastly, consider $v\in V(\Gamma)\setminus V_1(\Gamma)$.
There exists $C\in\mathfrak{C}(\Gamma)\setminus\mathfrak{C}_{1}(\Gamma)$ such that $v\in V(C)$.
By Corollary~\ref{cor2_Crticial_hyper}, $(\Gamma\bullet\mathbb{H})-v$ is decomposable.
Consequently, $\Gamma\bullet\mathbb{H}$ is critical.
\end{proof}

\section{An improvement of Theorem~\ref{thm_2_main}}\label{sec_improvement}

The purpose of this section is to demonstrate the following result.

\begin{thm}\label{thm_3_main}
Suppose that $v(\Gamma\bullet\mathbb{H})\geq 5$.
The 3-hypergraph $\Gamma\bullet\mathbb{H}$ is critical and
$\mathscr{P}(\Gamma\bullet\mathbb{H})=\Gamma$
if and only if the following six assertions hold
\begin{enumerate}
\item[(C1)] $\mathbb{H}$ is connected;
\item[(C2)] for every nontrivial module $\mathbb{M}$ of $\mathbb{H}$, we have
$\mathbb{M}\setminus\mathfrak{C}_{1}(\Gamma)\neq\emptyset$;
\item[(C3)] for each $v\in V_1(\Gamma)$, if
$\mathbb{H}-\{v\}$ is connected, then $\mathbb{H}-\{v\}$ admits a nontrivial module $\mathbb{M}_{\{v\}}$ such that $\mathbb{M}_{\{v\}}\subseteq\mathfrak{C}_1(\Gamma)\setminus\{\{v\}\}$;
\item[(C4)] for each $C\in\mathfrak{C}_{{\rm even}}(\Gamma)$ such that $v(C)=2$, we have
\begin{itemize}
\item $\mathbb{H}-C$ is connected,
\item for every nontrivial module $\mathbb{M}_C$ of $\mathbb{H}-C$,
we have $\mathbb{M}_C\setminus\mathfrak{C}_{1}(\Gamma)\neq\emptyset$;
\end{itemize}
\item[(C5)] for each $C\in\mathfrak{C}_{{\rm odd}}(\Gamma)$ such that $v(C)=3$, we have
\begin{itemize}
\item $\mathbb{H}_C$ is connected, where $\mathbb{H}_C$ is obtained from $\mathbb{H}$ by replacing $C$ by $\{\varphi_C(0)\}$,
\item for every nontrivial module $\mathbb{M}_C$ of $\mathbb{H}_C$,
we have $\mathbb{M}_C\setminus(\mathfrak{C}_{1}(\Gamma)\cup\{\varphi_C(0)\})\neq\emptyset$;
\end{itemize}
\item[(C6)] for each $C\in\mathfrak{C}_1(\Gamma)$, if there exists $D\in\mathfrak{C}_1(\Gamma)\setminus\{C\}$ such that $D$ is an isolated vertex of $\mathbb{H}-C$ and
$\mathbb{H}\setminus\{C,D\}$ is connected, then $\mathbb{H}\setminus\{C,D\}$ admits a nontrivial module $\mathbb{M}$ such that
$\mathbb{M}\subseteq\mathfrak{C}_1(\Gamma)\setminus\{C,D\}$.
\end{enumerate}
\end{thm}

We use the next four results to prove Theorem~\ref{thm_3_main}.

\begin{prop}\label{Prop2_Critical_hyper}
Suppose that $v(\Gamma\bullet\mathbb{H})\geq 5$.
If the 3-hypergraph $\Gamma\bullet\mathbb{H}$ is critical, then
for each $C\in\mathfrak{C}(\Gamma)$ such that $v(C)\geq 4$, we have
\begin{equation}\label{E1bis_thm1bis_Critical_hyper}
\begin{cases}
\mathscr{P}(\Gamma\bullet\mathbb{H})[V(C)]=C,\\
\text{and}\\
\text{for each $c\in V(C)$,
$N_{\mathscr{P}(\Gamma\bullet\mathbb{H})}(c)\subseteq V(C).$}
\end{cases}
\end{equation}
\end{prop}

\begin{proof}
By Corollary~\ref{cor2_Crticial_hyper}, for each $C\in\mathfrak{C}(\Gamma)\setminus\mathfrak{C}_{1}(\Gamma)$ and for every $c\in V(C)$, we have
$N_{\mathscr{P}(\Gamma\bullet\mathbb{H})}(c)\subseteq V(C)$.
Now, consider $C\in\mathfrak{C}(\Gamma)$ such that $v(C)\geq 4$.
For instance, suppose that $C\in\mathfrak{C}_{{\rm odd}}(\Gamma)$.
We have $w(C)\geq 2$.
Consider $i\in\{1,\ldots,w(C)-1\}$.
We verify that
\begin{equation}\label{E2_thm1_Critical_hyper}
N_{\mathscr{P}(\Gamma\bullet\mathbb{H})}(\varphi_C(2i))=
\{\varphi_C(2i-1),\varphi_C(2i+1)\}.
\end{equation}
We show that $(\Gamma\bullet\mathbb{H})-\{\varphi_C(2i-1),\varphi_C(2i)\}$ is prime.
To define $\Gamma\bullet\mathbb{H}$, we consider an isomorphism from $P_{2w(C)+1}$ onto $C$.
It follows from the definition of $\Gamma\bullet\mathbb{H}$ that $\varphi_C$ is an isomorphism from
$C_3(U_{2w(C)+1})$ onto $(\Gamma\bullet\mathbb{H})[V(C)]$.
The bijection
\begin{equation*}
\begin{array}{rccl}
\psi:&\{0,\ldots,2w(C)-2\}&\longrightarrow&\{0,\ldots,2w(C)\}\setminus\{2i-1,2i\}\\
&0\leq j\leq 2i-2&\longmapsto&j\\
&2i-1\leq j\leq 2w(C)-1&\longmapsto&j+2,
\end{array}
\end{equation*}
is an isomorphism from $C_3(U_{2w(C)-1})$ onto $C_3(U_{2w(C)+1})-\{2i-1,2i\}$.
Therefore, $((\varphi_C)_{\restriction\{0,\ldots,2w(C)+1\}\setminus\{i-1,i\}})\circ\psi$ is an isomorphism from
$C_3(U_{2w(C)-1})$ onto $(\Gamma\bullet\mathbb{H})[V(C)\setminus\{\varphi_C(2i-1),\varphi_C(2i)\}]$.
Observe that $\psi$ is also an isomorphism from $P_{2w(C)-1}$ onto the path $C^-$ defined on
$V(C^-)=\{0,\ldots,2w(C)+1\}\setminus\{i-1,i\}$ by
$E(C^-)=(E(C)\setminus\{\varphi_C(2i-2)\varphi_C(2i-1),\varphi_C(2i-1)\varphi_C(2i),
\varphi_C(2i)\varphi_C(2i+1)\})\cup\{\varphi_C(2i-2)\varphi_C(2i+1)\}$.
Now, consider the graph $\Gamma^-$ obtained from $\Gamma$ by replacing the path $C$ by $C^-$.
Moreover, consider the hypergraph $\mathbb{H}^-$ obtained from
$\mathbb{H}$ by renaming the vertex $C$ by $C^-$.
Hence, the bijection
\begin{equation*}
\begin{array}{rrcl}
\theta:&V(\mathbb{H})&\longrightarrow&(V(\mathbb{H})\setminus\{C\})\cup\{C^-\}\\
&D\neq C&\longmapsto&D\\
&C&\longmapsto&C^-,
\end{array}
\end{equation*}
is an isomorphism from $\mathbb{H}$ onto $\mathbb{H}^-$.
Since $\psi$ is increasing and preserves the parity, we obtain
$(\Gamma\bullet\mathbb{H})-\{\varphi_C(2i-1),\varphi_C(2i)\}=
\Gamma^-\bullet\mathbb{H}^-$.
Since $\mathbb{H}$ is connected, $\mathbb{H}^-$ is as well.
Furthermore, consider any nontrivial module $\mathbb{M}^-$ of
$\mathbb{H}^-$.
We obtain that $\theta^{-1}(\mathbb{M}^-)$ is a nontrivial module of $\mathbb{H}$.
Since $\Gamma\bullet\mathbb{H}$ is critical, it is prime.
By Assertion~(C2) of Proposition~\ref{Prop1_Critical_hyper},
$\theta^{-1}(\mathbb{M}^-)\setminus\mathfrak{C}_{1}(\Gamma)\neq\emptyset$.
Therefore, we obtain
$(\mathbb{M}^-)\setminus\theta(\mathfrak{C}_{1}(\Gamma))\neq\emptyset$.
Clearly, we have $\theta(\mathfrak{C}_{1}(\Gamma))=
\mathfrak{C}_{1}(\Gamma)=\mathfrak{C}_{1}(\Gamma^-)$.
It follows that
$(\mathbb{M}^-)\setminus\mathfrak{C}_{1}(\Gamma^-)\neq\emptyset$.
By Proposition~\ref{Prop1_Critical_hyper},
$\Gamma^-\bullet\mathbb{H}^-$, that is,
$(\Gamma\bullet\mathbb{H})-\{\varphi_C(2i-1),\varphi_C(2i)\}$ is prime.
Consequently,
$\varphi_C(2i-1)\in N_{\mathscr{P}(\Gamma\bullet\mathbb{H})}(\varphi_C(2i))$.
Similarly, $\varphi_C(2i+1)\in N_{\mathscr{P}(\Gamma\bullet\mathbb{H})}(\varphi_C(2i))$.
Therefore, \eqref{E2_thm1_Critical_hyper} follows from Lemma~\ref{Lcritical}.

Now, we verify that
\begin{equation}\label{E3_thm1_Critical_hyper}
N_{\mathscr{P}(H_{(\Gamma,\mathbb{H})})}(\varphi_C(0))=
\{\varphi_C(1)\}.
\end{equation}
As previously, we consider the graph $C^-$ defined on $V(C)\setminus\{\varphi_C(0),\varphi_C(1)\}$ by
$E(C^-)=E(C)\setminus\{\varphi_C(0)\varphi_C(1),\varphi_C(1)\varphi_C(2)\}$.
Now, consider the graph $\Gamma^-$ obtained from $\Gamma$ by replacing the path $C$ by $C^-$.
Moreover, consider the hypergraph $\mathbb{H}^-$ obtained from
$\mathbb{H}$ by renaming the vertex $C$ by $C^-$.
We obtain
$(\Gamma\bullet\mathbb{H})-\{\varphi_C(0),\varphi_C(1)\}=
\Gamma^-\bullet\mathbb{H}^-$.
It follows from Proposition~\ref{Prop1_Critical_hyper} applied with
$\Gamma^-$ and $\mathbb{H}^-$ that
$(\Gamma\bullet\mathbb{H})-\{\varphi_C(0),\varphi_C(1)\}$ is prime.
Hence,
$\varphi_C(1)\in N_{\mathscr{P}(\Gamma\bullet\mathbb{H})}(\varphi_C(0))$.
Set $X=V(\Gamma\bullet\mathbb{H})\setminus\{\varphi_C(0),\varphi_C(1)\}$.
Since $(\Gamma\bullet\mathbb{H})[X]$ is prime and
$\varphi_C(1)\in\langle X\rangle_{\Gamma\bullet\mathbb{H}}$, it follows from
Lemma~\ref{Lcritical} that \eqref{E3_thm1_Critical_hyper} holds.
It follows from Remark~\ref{Auto_critical_hyper} that
\begin{equation}\label{E4_thm1_Critical_hyper}
N_{\mathscr{P}(\Gamma\bullet\mathbb{H})}(\varphi_C(2w(C)))=
\{\varphi_C(2w(C)-1)\}.
\end{equation}

Lastly, we prove that
\begin{equation}\label{E5_thm1_Critical_hyper}
N_{\mathscr{P}(\Gamma\bullet\mathbb{H})}(\varphi_C(2i+1))=
\{\varphi_C(2i),\varphi_C(2i+2)\}
\end{equation}
for each $i\in\{0,\ldots,w(C)-1\}$.
It follows from \eqref{E2_thm1_Critical_hyper} that
$\varphi_C(2i)\in N_{\mathscr{P}(\Gamma\bullet\mathbb{H})}(\varphi_C(2w(C)+1))$ and
$\varphi_C(2i+2)\in N_{\mathscr{P}(\Gamma\bullet\mathbb{H})}(\varphi_C(2w(C)+1))$.
By Lemma~\ref{Lcritical}, \eqref{E5_thm1_Critical_hyper} holds.

Consequently, it follows from \eqref{E2_thm1_Critical_hyper}, \eqref{E3_thm1_Critical_hyper}, \eqref{E4_thm1_Critical_hyper}, \eqref{E5_thm1_Critical_hyper}, and from Lemma~\ref{Lcritical} that
$\mathscr{P}(\Gamma\bullet\mathbb{H})[V(C)]=C$.
\end{proof}

\begin{lem}\label{lem_6_Critical_hyper}
Suppose that $\Gamma\bullet\mathbb{H}$ is critical with $v(\Gamma\bullet\mathbb{H})\geq 5$.
Given $C\in\mathfrak{C}(\Gamma)$ such that $v(C)=2$, $C$ satisfies \eqref{E1bis_thm1bis_Critical_hyper} if and only if the following two assertions hold
\begin{itemize}
\item $\mathbb{H}-C$ is connected;
\item for every nontrivial module $\mathbb{M}_C$ of $\mathbb{H}-C$,
we have $\mathbb{M}_C\setminus\mathfrak{C}_{1}(\Gamma)\neq\emptyset$;
\end{itemize}
\end{lem}

\begin{proof}
By Corollary~\ref{cor2_Crticial_hyper}, for every $c\in V(C)$, we have
$N_{\mathscr{P}(\Gamma\bullet\mathbb{H})}(c)\subseteq V(C)$.
Consider an isomorphism $\varphi_C$ from $P_2$ onto $C$.
We have $(\Gamma\bullet\mathbb{H})-\{\varphi_C(0),\varphi_C(1)\}=
(\Gamma-V(C))\bullet(\mathbb{H}-C)$.
Moreover, we have
$\mathfrak{C}_{1}(\Gamma-V(C))=\mathfrak{C}_{1}(\Gamma)$.
To conclude, it suffices to apply Proposition~\ref{Prop1_Critical_hyper} to
$(\Gamma-V(C))\bullet(\mathbb{H}-C)$.
\end{proof}

\begin{lem}\label{lem_7_Critical_hyper}
Suppose that $\Gamma\bullet\mathbb{H})$ is critical with
$v(\Gamma\bullet\mathbb{H}))\geq 5$.
Given $C\in\mathfrak{C}(\Gamma)$ such that $v(C)=3$, $C$ satisfies \eqref{E1bis_thm1bis_Critical_hyper} if and only if the following two assertions hold
\begin{itemize}
\item $\mathbb{H}_C$ is connected, where $\mathbb{H}_C$ is obtained from $\mathbb{H}$ by replacing $C$ by $\{\varphi_C(0)\}$;
\item for every nontrivial module $\mathbb{M}_C$ of $\mathbb{H}_C$,
we have $\mathbb{M}_C\setminus(\mathfrak{C}_{1}(\Gamma)\cup\{\varphi_C(0)\})\neq\emptyset$.
\end{itemize}
\end{lem}

\begin{proof}
By Corollary~\ref{cor2_Crticial_hyper}, for every $c\in V(C)$, we have
$N_{\mathscr{P}(\Gamma\bullet\mathbb{H})}(c)\subseteq V(C)$.
Furthermore, consider an isomorphism $\varphi_C$ from $P_3$ onto $C$.
Denote by $\mathbb{H}_C$ the $\{2,3\}$-hypergraph obtained from $\mathbb{H}$ by replacing $C$ by
$\{\varphi_C(0)\}$.
We have
\begin{equation}\label{E1_lem_7_Critical_hyper}
\begin{cases}
(\Gamma\bullet\mathbb{H})-\{\varphi_C(1),\varphi_C(2)\}=
(\Gamma-\{\varphi_C(1),\varphi_C(2)\})\bullet\mathbb{H}_C\\
\text{and}\\
\mathfrak{C}_{1}(\Gamma-\{\varphi_C(1),\varphi_C(2)\})=\mathfrak{C}_{1}(\Gamma)\cup\{\varphi_C(0)\}.
\end{cases}
\end{equation}

To begin, suppose that $C$ satisfies \eqref{E1bis_thm1bis_Critical_hyper}.
We obtain that $(\Gamma\bullet\mathbb{H})-\{\varphi_C(1),$ $\varphi_C(2)\}$ is prime.
By Proposition~\ref{Prop1_Critical_hyper} applied to
$H_{(\Gamma-\{\varphi_C(1),\varphi_C(2)\},\mathbb{H}_C)}$, both assertions above hold.

Conversely, suppose that both assertions above hold.
Since \eqref{E1_lem_7_Critical_hyper} holds, it follows from Proposition~\ref{Prop1_Critical_hyper} that
$(\Gamma\bullet\mathbb{H})-\{\varphi_C(1),\varphi_C(2)\}$ is prime.
By Corollary~\ref{cor2_Crticial_hyper}, $\{\varphi_C(0),\varphi_C(2)\}$ is a module of
$(\Gamma\bullet\mathbb{H})-\varphi_C(1)$.
Therefore, $(\Gamma\bullet\mathbb{H})-\{\varphi_C(0),\varphi_C(1)\}$ and
$(\Gamma\bullet\mathbb{H})-\{\varphi_C(1),\varphi_C(2)\}$ are isomorphic.
It follows that $(\Gamma\bullet\mathbb{H})-\{\varphi_C(0),\varphi_C(1)\}$ is prime.
Lastly, it follows from Corollary~\ref{cor2_Crticial_hyper} that
$V(\Gamma\bullet\mathbb{H})\setminus\{\varphi_C(0),\varphi_C(1)\}$ is a module of
$(\Gamma\bullet\mathbb{H})-\varphi_C(0)$.
Thus, $V(\Gamma\bullet\mathbb{H})\setminus\{\varphi_C(0),\varphi_C(1),\varphi_C(2)\}$ is a non trivial module of $(\Gamma\bullet\mathbb{H})-\{\varphi_C(0),\varphi_C(2)\}$.
Hence, $(\Gamma\bullet\mathbb{H})-\{\varphi_C(0),\varphi_C(2)\}$ is decomposable.
It follows that
$\mathscr{P}(\Gamma\bullet\mathbb{H})[$ $V(C)]=C$.
It follows that $C$ satisfies \eqref{E1bis_thm1bis_Critical_hyper}.
\end{proof}

\begin{lem}\label{lem_8_Critical_hyper}
Suppose that $\Gamma\bullet\mathbb{H}$ is critical with $v(\Gamma\bullet\mathbb{H})\geq 5$.
For every $v\in V_1(\Gamma)$, we have $d_{\mathscr{P}(\Gamma\bullet\mathbb{H})}=1$ if and only if there exists $w\in V_1(\Gamma)$ satisfying
\begin{itemize}
\item $\{w\}$ is an isolated vertex of $\mathbb{H}-\{v\}$;
\item $\mathbb{H}-\{\{v\},\{w\}\}$ is connected;
\item for every nontrivial module $\mathbb{M}_{vw}$ of $\mathbb{H}-\{\{v\},\{w\}\}$,
we have $\mathbb{M}_{vw}\setminus(\mathfrak{C}_{1}(\Gamma)\setminus\{\{v\},\{w\}\})\neq\emptyset$.
\end{itemize}
\end{lem}

\begin{proof}
To begin, suppose that $d_{\mathscr{P}(\Gamma\bullet\mathbb{H})}(v)=1$.
Denote by $w$ the unique neighbour of $v$ in $\mathscr{P}(\Gamma\bullet\mathbb{H})$.
It follows from Proposition~\ref{Prop2_Critical_hyper}, Lemma~\ref{lem_6_Critical_hyper}, and
Lemma~\ref{lem_7_Critical_hyper} that $w\in V_1(\Gamma)$.
We have
\begin{equation}\label{E1_lem_8_Critical_hyper}
\begin{cases}
(\Gamma\bullet\mathbb{H})-\{v,w\}=
(\Gamma-\{v,w\})\bullet(\mathbb{H}-\{\{v\},\{w\}\})\\
\text{and}\\
\mathfrak{C}_{1}(\Gamma-\{v,w\})=\mathfrak{C}_{1}(\Gamma)\setminus\{\{v\},\{w\}\}.
\end{cases}
\end{equation}
Since $N_{\mathscr{P}(\Gamma\bullet\mathbb{H})}(v)=\{w\}$, $(\Gamma\bullet\mathbb{H})-\{v,w\}$ is prime.
By Proposition~\ref{Prop1_Critical_hyper} applied to
$(\Gamma-\{v,w\})\bullet(\mathbb{H}-\{\{v\},\{w\}\})$,
$\mathbb{H}-\{\{v\},\{w\}\}$ is connected, and
for every nontrivial module $\mathbb{M}_{vw}$ of $\mathbb{H}-\{\{v\},\{w\}\}$,
$\mathbb{M}_{vw}\setminus(\mathfrak{C}_{1}(\Gamma)\setminus\{\{v\},\{w\}\})\neq\emptyset$.
Moreover, since $N_{\mathscr{P}(\Gamma\bullet\mathbb{H})}(v)=\{w\}$, it follows from Lemma~\ref{Lcritical} that $V(\Gamma\bullet\mathbb{H})\setminus\{v,w\}$ is a module of $(\Gamma\bullet\mathbb{H})-v$.
We show that
\begin{equation}\label{E2_lem_8_Critical_hyper}
\text{for each $\varepsilon\in E(\mathbb{H})$, if $\{w\}\in \varepsilon$, then $\{v\}\in \varepsilon$.}
\end{equation}
Indeed, consider $\varepsilon\in E(\mathbb{H})$ such that $\{w\}\in \varepsilon$.
For a contradiction, suppose that $|\varepsilon|=2$.
There exists $C\in\mathfrak{C}_{{\rm even}}(\Gamma)$ such that $\varepsilon=\{w\}C$.
We obtain $C\subseteq V(\Gamma\bullet\mathbb{H})\setminus\{v,w\}$ and
$w\varphi_C(0)\varphi_C(1)\in E(\Gamma\bullet\mathbb{H})$, which contradicts the fact that
$V(\Gamma\bullet\mathbb{H})\setminus\{v,w\}$ is a module of $(\Gamma\bullet\mathbb{H})-v$.
Consequently, we have $|\varepsilon|=3$.
Thus, there exist distinct $C,D\in\mathfrak{C}_{{\rm odd}}(\Gamma)$ such that $\varepsilon=\{w\}CD$.
If $C\neq\{v\}$ and $D\neq\{v\}$, then
$\varphi_C(0),\varphi_D(0)\in V(\Gamma\bullet\mathbb{H})\setminus\{v,w\}$ and
$w\varphi_C(0)\varphi_D(0)\in E(\Gamma\bullet\mathbb{H})$, which contradicts the fact that
$V(\Gamma\bullet\mathbb{H})\setminus\{v,w\}$ is a module of $(\Gamma\bullet\mathbb{H})-v$.
Therefore, $C=\{v\}$ or $D=\{v\}$.
Hence, \eqref{E2_lem_8_Critical_hyper} holds.
It follows that $\{w\}$ is an isolated vertex of $\mathbb{H}-\{v\}$.

Conversely, suppose that there exists $w\in V_1(\Gamma)$ such that the three assertions above hold.
Since \eqref{E1_lem_8_Critical_hyper} holds, it follows from Proposition~\ref{Prop1_Critical_hyper} applied to
$(\Gamma-\{v,w\})\bullet(\mathbb{H}-\{\{v\},\{w\}\})$
that $(\Gamma\bullet\mathbb{H})-\{v,w\}$ is prime.
Thus, we have
\begin{equation}\label{E3_lem_8_Critical_hyper}
w\in N_{\mathscr{P}(\Gamma\bullet\mathbb{H})}(v).
\end{equation}
Furthermore, since $\{w\}$ is an isolated vertex of $\mathbb{H}-\{v\}$, $\mathbb{H}[\{\{w\}\}]$
is a component of $\mathbb{H}-\{v\}$.
By Lemma~\ref{lem1bis_Crticial_hyper},
$V(\Gamma\bullet\mathbb{H})\setminus\{v,w\}$ is a module of $(\Gamma\bullet\mathbb{H})-v$.
Consider any $u\in V(\Gamma\bullet\mathbb{H})\setminus\{v,w\}$.
We obtain that
$V(\Gamma\bullet\mathbb{H})\setminus\{u,v,w\}$ is a module of $(\Gamma\bullet\mathbb{H})-\{u,v\}$.
Hence, we obtain $u\not\in N_{\mathscr{P}(\Gamma\bullet\mathbb{H})}(v)$.
It follows from \eqref{E3_lem_8_Critical_hyper} that
$N_{\mathscr{P}(\Gamma\bullet\mathbb{H})}(v)=\{w\}$.
\end{proof}

\begin{proof}[Proof of Theorem~\ref{thm_3_main}]
To begin, suppose that $\Gamma\bullet\mathbb{H}$ is critical and
$\mathscr{P}(\Gamma\bullet\mathbb{H})=\Gamma$.
Since $\Gamma\bullet\mathbb{H}$ is critical, it follows from Theorem~\ref{thm_2_main} that Assertions (C1), (C2), and (C3) hold.
For Assertion (C4), consider $C\in\mathfrak{C}_{{\rm even}}(\Gamma)$ such that $v(C)=2$.
By Corollary~\ref{cor2_Crticial_hyper}, for every $c\in V(C)$, we have
$N_{\mathscr{P}(\Gamma\bullet\mathbb{H})}(c)\subseteq V(C)$.
Therefore, since $\mathscr{P}(\Gamma\bullet\mathbb{H})=\Gamma$,
$C$ satisfies \eqref{E1bis_thm1bis_Critical_hyper}.
By Lemma~\ref{lem_6_Critical_hyper}, $C$ satisfies Assertion (C4).

For Assertion (C5), consider $C\in\mathfrak{C}_{{\rm odd}}(\Gamma)$
such that $v(C)=3$.
By Corollary~\ref{cor2_Crticial_hyper}, for every $c\in V(C)$, we have
$N_{\mathscr{P}(\Gamma\bullet\mathbb{H})}(c)\subseteq V(C)$.
Therefore, since $\mathscr{P}(\Gamma\bullet\mathbb{H})=\Gamma$,
$C$ satisfies \eqref{E1bis_thm1bis_Critical_hyper}.
By Lemma~\ref{lem_7_Critical_hyper}, $C$ satisfies Assertion (C5).

For Assertion (C6), consider distinct
$C,D\in\mathfrak{C}_1(\Gamma)$ such that $D$ is an isolated vertex of $\mathbb{H}-C$ and $\mathbb{H}\setminus\{C,D\}$ is connected.
Since $\mathscr{P}(\Gamma\bullet\mathbb{H})=\Gamma$, we have
$d_{\mathscr{P}(\Gamma\bullet\mathbb{H})}(C)=0$.
It follows from Lemma~\ref{lem_8_Critical_hyper} that there exists a
nontrivial module $\mathbb{M}$ of $\mathbb{H}-\{C,D\}$ such that
$\mathbb{M}\subseteq\mathfrak{C}_{1}(\Gamma)\setminus\{C,D\}$.

Conversely, suppose that Assertions (C1),...,(C6) hold.
By Theorem~\ref{thm_2_main}, $\Gamma\bullet\mathbb{H}$ is critical.
It remains to show that
\begin{equation}\label{E1_thm2_Critical_hyper}
\mathscr{P}(\Gamma\bullet\mathbb{H})=\Gamma.
\end{equation}
It follows from Proposition~\ref{Prop2_Critical_hyper} that
for each $C\in\mathfrak{C}(\Gamma)$ such that $v(C)\geq 4$, $C$ satisfies \eqref{E1bis_thm1bis_Critical_hyper}.
Furthermore, since Assertion (C4) holds, it follows from Lemma~\ref{lem_6_Critical_hyper} that
\eqref{E1bis_thm1bis_Critical_hyper} is satisfied by each
$C\in\mathfrak{C}_{{\rm even}}(\Gamma)$ such that $v(C)=2$.
Similarly, since Assertion (C5) holds, it follows from Lemma~\ref{lem_7_Critical_hyper} that
\eqref{E1bis_thm1bis_Critical_hyper} is satisfied by each
$C\in\mathfrak{C}_{{\rm even}}(\Gamma)$ such that $v(C)=3$.
It follows that \eqref{E1bis_thm1bis_Critical_hyper} is satisfied by every
$C\in\mathfrak{C}(\Gamma)\setminus\mathfrak{C}_{1}(\Gamma)$.
To prove that \eqref{E1_thm2_Critical_hyper} holds, it remains to prove that
$d_{\mathscr{P}(\Gamma\bullet\mathbb{H})}(v)=0$ for each $v\in V_1(\Gamma)$.
Set $\mathcal{N}=\{v\in V_1(\Gamma):d_{\mathscr{P}(\Gamma\bullet\mathbb{H})}(v)\neq 0\}$.
For a contradiction, suppose that $\mathcal{N}\neq\emptyset$.
Since \eqref{E1bis_thm1bis_Critical_hyper} is satisfied by every
$C\in\mathfrak{C}(\Gamma)\setminus\mathfrak{C}_{1}(\Gamma)$, we get
\begin{equation}\label{E3_thm2_Critical_hyper}
\text{for each $v\in\mathcal{N}$,
$N_{\mathscr{P}(\Gamma\bullet\mathbb{H})}(v)\subseteq\mathcal{N}$.}
\end{equation}
For a contradiction, suppose that there exists $v\in\mathcal{N}$ such that
$d_{\mathscr{P}(\Gamma\bullet\mathbb{H})}(v)=1$.
By Lemma~\ref{lem_8_Critical_hyper},
there exists $w\in V_1(\Gamma)$ satisfying
\begin{itemize}
\item $\{w\}$ is an isolated vertex of $\mathbb{H}-\{v\}$;
\item $\mathbb{H}-\{\{v\},\{w\}\}$ is connected;
\item for every nontrivial module $\mathbb{M}_{vw}$ of $\mathbb{H}-\{\{v\},\{w\}\}$,
we have $\mathbb{M}_{vw}\setminus(\mathfrak{C}_{1}(\Gamma)\setminus\{\{v\},\{w\}\})\neq\emptyset$.
\end{itemize}
Therefore, Assertion (C6) is not satisfied for $C=\{v\}$ and $D=\{w\}$.
It follows from Lemma~\ref{Lcritical} that $d_{\mathscr{P}(\Gamma\bullet\mathbb{H})}(v)=2$
for every $v\in\mathcal{N}$.
Moreover, by \eqref{E3_thm2_Critical_hyper}, $N_{\mathscr{P}(\Gamma\bullet\mathbb{H})}(v)\subseteq\mathcal{N}$ for every $v\in\mathcal{N}$.
Therefore, $\mathscr{P}(\Gamma\bullet\mathbb{H})$ admits a component $\mathcal{C}$ such that
$v(\mathcal{C})\geq 3$,
$V(\mathcal{C})\subseteq\mathcal{N}$, and
$\mathscr{P}(\Gamma\bullet\mathbb{H})[V(\mathcal{C})]$ is a cycle.
Since $\mathfrak{C}(\Gamma)\setminus\mathfrak{C}_1(\Gamma)\neq\emptyset$ by \eqref{critical_construction_1}, we obtain
$V(\mathcal{C})\subsetneq V(\Gamma\bullet\mathbb{H})$, which contradicts Proposition~\ref{Pcritical_T2n+1} because $\Gamma\bullet\mathbb{H}$ is critical.
It follows that $\mathcal{N}=\emptyset$.
Therefore, \eqref{E1_thm2_Critical_hyper} holds.
\end{proof}