\label{thm: main-ucp}
Let $T>0$ and $G\subset [0,T]\times {\mathbb T}$ be a spacetime measurable set. The following statements are equivalent.
\begin{itemize}
\item [(1)] UCP holds on $G$.

\item [(2)] $G$ satisfies \blackhyperref{def:weakGCC}{{\rm (weak GCC)}} and does not obey \blackhyperref{def:OSC}{$({\rm OSC})$} for any non-trivial pair $(A, B)$\footnote{Reminder: for simplicity, when there is no risk of confusion,  sometimes we simply write (OSC) indicating that the set $G$ does not obey the observable symmetry condition for any non-trivial pair $(A, B)$.}.
\end{itemize}

\begin{equation}\label{eq: wave-eq-0}
(\partial_t^2-\partial_x^2)u=0,\quad (u,\partial_tu)\big|_{t=0}=(u_0,u_1)\in \dot H^1(\T)\times L^2(\T),
\end{equation}

\begin{definition}[Symmetric function pair]\label{def:classSc}
We say
$(f,g)$ is a symmetric  function pair if  there exists a decomposition pair $\{(A_k, B_k),\ k = 1,\dots, K\}$, a sequence of different numbers $\{s_k\}_{k= 1}^K$ such that
\begin{equation*}
 f=\sum_{1\leq k\leq K}s_k\chi_{A_k}, \quad g=\sum_{1\leq k\leq K}s_k\chi_{B_k},
\end{equation*}
and
\begin{equation*}
    \int_{\T} (f+ g)(x) \d x= 0.
\end{equation*}
The set of all symmetric function pairs is denoted by $\mathcal{S}^2_c$.
\end{definition}

\begin{equation}\label{eq: wave-ob}
\|u_0\|^2_{\dot H^1(\T)}+\|u_1\|^2_{L^2(\T)}\leq C\int_G|\partial_tu(t,x)|^2\d t\d x.
\end{equation}

Let $T > 0$. Let $G \subset [0,T] \times \mathbb{T}$ be a spacetime measurable set with positive measure.
Consider the {\it observability problem} of the wave equation: whether there exists some $C= C(G)>0$ such that every solution $u$ to
\begin{equation}\label{eq: wave-eq-0}
(\partial_t^2-\partial_x^2)u=0,\quad (u,\partial_tu)\big|_{t=0}=(u_0,u_1)\in \dot H^1({\mathbb T})\times L^2({\mathbb T}),
\end{equation}
satisfies
\begin{equation}\label{eq: wave-ob}
\|u_0\|^2_{\dot H^1({\mathbb T})}+\|u_1\|^2_{L^2({\mathbb T})}\leq C\int_G|\partial_tu(t,x)|^2{\,\rm d} t{\,\rm d} x.
\end{equation}

Thanks to the classical Hilbert uniqueness method and an argument to raise regularity (see Appendix \ref{sec: HUM-app} for details on this reduction), the observability of \eqref{eq: wave-ob} is equivalent to the {\it exact controllability} of
the controlled wave equation on $[0,T]\times{\mathbb T}$ with control force $f\in L^2(G)$:
\begin{equation}\label{eq: wave-eq}
(\partial_t^2-\partial_x^2)u =f\mathbf{1}_G,\quad (u, \partial_t u)\big|_{t=0}=(u_0,u_1)\in \dot H^1({\mathbb T})\times L^2({\mathbb T}).
\end{equation}

Our primary objective is to seek the sufficient and necessary geometric conditions on $G$ such that the unique continuation and observability hold.

\subsection{History and setting}
From the early work of Russell \cite{Russell-1}, the studies of the controllability and observability for wave equations have been central topics in control theory.

\subsubsection{Geometric control condition}
In the pioneering work of Rauch and Taylor \cite{RT}, they first related the observability to a geometric condition on the damped region $\omega$ and the rays of geometric optics in the boundaryless case.
Later, in another pioneering work of Bardos-Lebeau-Rauch \cite{BLR-gcc}, the well-known \textit{Geometric control condition} is introduced: for an open observed region $\omega$, every generalized ray should meet $\omega$ in a finite time.

Since then, it has become one of the most natural assumptions for the controlled waves. In the existing literature, the observation is most often made on cylindrical domains $G=(0,T)\times \omega$, with $\omega$ being an open subset. Under suitable smooth conditions, it is well-known that GCC is sufficient, and depending on the domain, necessary for the observability in the cylindrical domains $(0,T)\times \omega$, see \cite{BLR-gcc,BG-97}. When considering stabilization problems, GCC is also a useful condition for the exponential decay of energy. We refer to \cite{RT,Haraux}. Otherwise, one may have logarithmic type of energy decay results \cite{LR-97, Burq-98}. GCC also plays a role in practical issues, such as sensor designs, tomography techniques used for imaging bodies (see \cite{RouLebeauAnalPDE2017} for example), etc. For a comprehensive reference of the numerical study, we refer to \cite{zuazua-review} and its references therein.

Finally, we give a very brief overview of the boundary control case. There are also fruitful results in this direction. For related GCC, we refer to \cite{Lebeau-boundary}. In particular, for 1D wave equations, one can find many nonlinear results \cite{Li-1,LY-2} and the references therein.

\subsubsection{Unique continuation}
A qualitative version of observability is the unique continuation property.
The easiest way to ensure this property is to apply the analyticity based on Holmgren's theorem. Besides, H\"ormander's pseudo-convexity condition and Carleman estimates are powerful tools dealing with the unique continuation problem. In this direction, there is a large literature such as \cite{RZ-98,Tataru,Hormander-92,Hormander-96} and more recent work includes \cite{Laurent-Leautaud-2019, MS-21,FCL,Shao}. Here we point out that to construct the appropriate weight to apply Carleman estimates, it is crucial to understand the behavior across the suitable spacetime surfaces, which is more delicate than considering a cylindrical region.

\subsubsection{Our setting: spacetime measurable observable region}
Recently, researchers started to focus on
 the spacetime setting \cite{Castro-Cindea-Munch-2014, RouLebeauAnalPDE2017,Shao,PK}. Meanwhile, the study of the case where $G$ is a spacetime region is far from complete, even for open regions.

 In this paper, we focus on the setting that $G\subset[0,T]\times{\mathbb T}$ is measurable of positive measure. The following condition may be viewed as the natural analogue of the standard geometric control condition in this spacetime measurable setting.

\vspace{2mm}
\begin{enumerate}
\item[({\bf GCC})]\label{def:GCC}
Let $T>0$. A measurable set $G \subset [0, T] \times \mathbb{T}$ is said to satisfy
the GCC if there exists a constant
$c_0 > 0$ such that for almost every $x \in \mathbb{T}$,
\begin{equation}\label{eq:GCC-wave:0}
\int_0^T\mathbf{1}_G(s,x\pm s){\,\rm d} s\geq c_0.\footnote{\text{This integral is defined in the sense of \eqref{eq:GCC:measure}.}}
\end{equation}
\end{enumerate}
\vspace{2mm}

From now on, in this paper, \blackhyperref{def:GCC}{({\rm GCC})}  refers to the preceding definition with respect to \eqref{eq:GCC-wave:0}.
It is well-known that under the standard setting, namely the observable region is a cylinder $[0, T]\times \omega$, GCC yields several important properties: weak observability and the necessity of observability. These results can be generalized to the spacetime \blackhyperref{def:GCC}{({\rm GCC})} for a spacetime measurable observable region. We put the former result in Section \ref{sec: weak-ob-sec} and the later in Appendix \ref{sec:app:3}.

\subsection{A new symmetry condition}

Extend the system $2\pi-$periodically to $x\in \mathbb{R}$,  and introduce the null coordinate:
\begin{equation}
    \xi=x+t \; \textrm{ and } \; \eta=x-t. \notag
\end{equation}
Under this new coordinate,
\begin{equation*}
  2  u_{\xi}= u_{x}+ u_t \; \textrm{ and } \;  2  u_{\eta}= u_{x}- u_t
\end{equation*}
and the wave equation \eqref{eq: wave-eq} becomes\footnote{We considered the equation on $U(\xi, \eta):= u(t, x)$ and $F(\xi, \eta)= f(t, x)$. For ease of notation, we still denote $U$ by $u$ and $F$ by $f$.  }
\begin{equation}
    4 \partial_{\xi} \partial_{\eta} u=  f \mathbf{1}_G. \notag
\end{equation}

For any $\xi_0\in\mathbb{R}$, we denote the  line $\{(t,x):x+t=\xi_0\}$ by $L_{\xi=\xi_0}$,  and call it the $\xi$-characteristic. Similarly, we denote the  line $\{(t,x):x- t=\eta_0\}$ by $L_{\eta=\eta_0}$ and  call it $L_{\eta=\eta_0}$ the $\eta$-characteristic. Thus the \blackhyperref{def:GCC}{({\rm GCC})} is equivalent to
\begin{equation}\label{eq:GCC:measure}
      \textrm{ for } a.e. \; x\in {\mathbb T}, \mbox{meas}_\mathbb{R}(G\cap L_{\eta= x})\geq c_0, \;\;
  \mbox{meas}_\mathbb{R}(G\cap L_{\xi=x})\geq c_0.
\end{equation}
We further define measurable cylinders for any measurable sets $A, B\subset [0, 2\pi]$ as
\begin{equation}\label{eq: cylinder}
L_{\xi\in A}:=\bigcup_{\xi_0\in A}L_{\xi=\xi_0} \; \textrm{ and } \; L_{\eta\in B}:=\bigcup_{\eta_0\in B}L_{\eta=\eta_0}.
\end{equation}

Introduce the following symmetry condition:

\vspace{2mm}

\begin{enumerate}
\item[({\bf OSC})]\label{def:OSC}
Let $A, B\subset {\mathbb T}$ be two measurable subsets. $G\in [0, T]\times {\mathbb T}$ is said to obey the {\it observable symmetry condition} for ($A, B$) if\footnote{As usual, $A \Delta B = (A\backslash B)\cup (B\backslash A)$ denotes the symmetric difference between set $A$ and set $B$.

Expression \eqref{wave:nece:2} means that $G\cap L_{\xi\in A} = G\cap L_{\eta\in B}$ modulo zero measure set in $\mathbb{R}^2$.}
\begin{equation}\label{wave:nece:2}
 \mbox{meas}_{\mathbb{R}^2} \Big( [G\cap L_{\xi\in A}] \, \Delta \, [G\cap L_{\eta\in B}]\Big)=0.
\end{equation}
\end{enumerate}

\vspace{2mm}

We call a pair $(A, B)$ trivial, if $|A|= |B|= 0$ or $|A|= |B|= 2\pi$.  In principle, we are interested in {\it non-trivial} pairs $(A, B)$. Since the observable symmetry condition is automatically satisfied for trivial pairs ($A, B$).

The observable symmetry condition is very important in our work, see Fig. \ref{fig:OSC-role}.

\begin{figure}[htp]
    \centering

\tikzset{every picture/.style={line width=0.75pt}}
\begin{tikzpicture}[x=0.75pt,y=0.75pt,yscale=-1,xscale=1]

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\draw    (329,82.5) -- (329,134.5) ;
\draw [shift={(329,136.5)}, rotate = 270] [color={rgb, 255:red, 0; green, 0; blue, 0 }  ][line width=0.75]    (10.93,-3.29) .. controls (6.95,-1.4) and (3.31,-0.3) .. (0,0) .. controls (3.31,0.3) and (6.95,1.4) .. (10.93,3.29)   ;
\draw    (430,82.5) -- (509.44,146.25) ;
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\draw (219,47) node [anchor=north west][inner sep=0.75pt]   [align=left] {Observable symmetry condition};
\draw (81,167.5) node [anchor=north west][inner sep=0.75pt]   [align=left] {Conservation law};
\draw (282,146.5) node [anchor=north west][inner sep=0.75pt]   [align=left] {\begin{minipage}[lt]{66.78pt}\setlength\topsep{0pt}
\begin{center}
Controllability \\Observability \\UCP
\end{center}

\end{minipage}};
\draw (455,167.5) node [anchor=north west][inner sep=0.75pt]   [align=left] {Symetric \ \ \  class };
\draw (515,167.9) node [anchor=north west][inner sep=0.75pt]  [font=\small]  {$\mathcal{S}^{2}$};
\draw (128,106) node [anchor=north west][inner sep=0.75pt]   [align=left] {{\footnotesize Noether}};
\draw (333,106) node [anchor=north west][inner sep=0.75pt]   [align=left] {{\footnotesize necessary}};
\draw (486,106) node [anchor=north west][inner sep=0.75pt]   [align=left] {{\footnotesize generate}};

\end{tikzpicture}

    \caption{The role of (OSC) in this paper}
    \label{fig:OSC-role}
\end{figure}

\vspace{3mm}
The first result is a new conservation law of the forced equation \eqref{eq: wave-eq}.
\begin{proposition}[Conservation law]
\label{prop-sym-conser}
Let $A,B\subset {\mathbb T}$. Assume that $G$ satisfies the ($A, B$)-{\it observable symmetry condition}.
Let $(u,u_t)\in C([0,T];\dot{H}^1({\mathbb T})\times L^2({\mathbb T}))$ be a solution to the forced equation \eqref{eq: wave-eq} and define the energy \footnote{Indeed, under the $(\xi, \eta)-$coordinate this conservation is write as
\begin{equation*}
 \frac{1}{2} I(t)=   \int_{A-t}u_{\xi}(t,x){\,\rm d} x+\int_{B+t}u_{\eta}(t,x){\,\rm d} x.
\end{equation*}
}
$$
I(t)= \int_{A-t}(u_x+u_t)(t,x){\,\rm d} x+\int_{B+t}(u_x-u_t)(t,x){\,\rm d} x.
$$
Then we have
$$
I(t)=I(0), \;  \forall t\in [0, T].
$$
In the case $|A|= |B|= 2\pi$, this conservation law is exactly the condition $\int_{{\mathbb T}} u_x (t, x) {\,\rm d} x= 0$.
\end{proposition}

\begin{remark}
  This proposition can be understood as a variant of Noether’s theorem\footnote{Noether’s theorem: ``Every continuous symmetry of the action (or the equations of motion, in a suitable sense) implies a conserved quantity.''} adapted to a controlled wave equation: a geometric symmetry of the forcing region yields a conserved functional of the solution. Indeed,
the key hypothesis is that the region $G$ satisfies the  {\it observable symmetry condition} for ($A, B$). The conclusion shows that the energy $I(t)$ is invariant in time, regardless of the force term.
\end{remark}

This new type of symmetry provides a {\it necessary condition} for the controllability and observability of \eqref{eq: wave-eq}, and even a {\it necessary condition} for the unique continuation of  \eqref{eq: wave-ob}. See Section \ref{sec:symmetry} for the detailed construction of counterexamples.  Moreover, this construction gives rise to two classes of function pairs satisfying the symmetry condition, $\mathcal{S}^2_c$ and $\mathcal{S}^2$, in Definitions \ref{def:classSc} and \ref{def:classS}.

\begin{figure}[htp]
    \centering
\begin{tikzpicture}[scale=1.0]

  \draw[->] (-0.2,0) -- (6.2,0) node[right] {$x$};
  \draw[->] (0,-0.2) -- (0,6.2) node[above] {$t$};

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  \draw (0,6) node[left] {$2\pi$} -- (0.1,6);

  \draw (0,0) -- (6,0) -- (6,6) -- (0,6) -- cycle;

ll[blue!30] (0,0) coordinate (A1) -- (3,0) coordinate (B1) -- (1.5,1.5) coordinate (C1)-- cycle;

\node at (barycentric cs:A1=1,B1=1,C1=1) {\textcolor{white}{-1}};

ll[red!30] (3,0) coordinate (A2) -- (6,0) coordinate (B2) -- (4.5,1.5) coordinate (C2)-- cycle;

\node at (barycentric cs:A2=1,B2=1,C2=1) {\textcolor{white}{1}};

ll[red!30] (0,3) coordinate (A3) -- (1.5,1.5) coordinate (B3) -- (3,3) coordinate (C3)--(1.5,4.5) coordinate (D3)-- cycle;

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\node at (barycentric cs:A5=1,B5=1,C5=1) {\textcolor{white}{-1}};

ll[red!30] (3,6) coordinate (A6) -- (4.5,4.5) coordinate (B6) -- (6,6) coordinate (C6)-- cycle;

\node at (barycentric cs:A6=1,B6=1,C6=1) {\textcolor{white}{1}};

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\node at (10,4) {$G=G_{0}\cup G_1$};

\node at (10, 3) {$G_{0}=\{(x,t): u_\xi=u_\eta=-1\}$: blue part};

\node at (10, 2) {$G_1=\{(x,t): u_\xi=u_\eta=1\}$: red part};

\end{tikzpicture}
    \caption{Observability fails on $G$, though $G$ satisfies GCC. }
    \label{fig:GCCfail1}
\end{figure}

\vspace{2mm}
\noindent {\bf The counterexample.}
 Let $T= 2\pi$. Let $G\subset [0,2\pi]\times {\mathbb T}$ be given in Fig. \ref{fig:GCCfail1}. Clearly $G$ satisfies \blackhyperref{def:GCC}{({\rm GCC})}.  However, the observable symmetry condition holds for $A=B=(0,\pi)$ or $A=B=(\pi,2\pi)$. Then one can show that the observability inequality \eqref{eq: wave-ob}  fails on $G$, see Section \ref{sec-osc-nece}.

\subsection{Necessary and sufficient unique continuation conditon}
The second result is about the unique continuation property (UCP).  Let $T>0$ and let $G\in [0, T]\times {\mathbb T}$ be a measurable set.  UCP is the qualitative version of the observability and asks:
$$
\mbox{(UCP) \;\;\;\;\;\;\;\;\;\;\;\;\;\; Let $u$ be a solution of \eqref{eq: wave-eq-0} and }   u_t =0 \mbox{ a.e. in } G \Longrightarrow u \equiv 0.  \;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;
$$

Motivated by the geometric features of the system, we formulate the following minimal geometric assumption on the observation region, which is required for unique continuation.

\vspace{2mm}

\begin{enumerate}[label={}, leftmargin=6.5em, labelindent=3em]
\item[({\bf Weak  GCC})]\label{def:weakGCC}
Let $T>0$. A measurable set $G\subset [0, T]\times \mathbb{T}$ is said to satisfy the
 weak GCC if for almost every
$x\in\mathbb{T}$,
\begin{equation}
\int_0^T\mathbf{1}_G(s,x\pm s){\,\rm d} s>0.   \notag
\end{equation}
\end{enumerate}

\vspace{2mm}

This condition is equivalent to
\begin{equation}
      \textrm{ for } a.e. \; x\in {\mathbb T}, \mbox{meas}_\mathbb{R}(G\cap L_{\eta= x})>0, \;\;
  \mbox{meas}_\mathbb{R}(G\cap L_{\xi=x})>0. \notag
\end{equation}
Compared to the standard \blackhyperref{def:GCC}{({\rm GCC})}, it is not required a uniform positive lower bound.

\vspace{2mm}
\begin{enumerate}
\item[({\bf GCC})]\label{def:GCC}
Let $T>0$. A measurable set $G \subset [0, T] \times \mathbb{T}$ is said to satisfy
the GCC if there exists a constant
$c_0 > 0$ such that for almost every $x \in \mathbb{T}$,
\begin{equation}\label{eq:GCC-wave:0}
\int_0^T\mathbf{1}_G(s,x\pm s){\,\rm d} s\geq c_0.\footnote{\text{This integral is defined in the sense of \eqref{eq:GCC:measure}.}}
\end{equation}
\end{enumerate}
\vspace{2mm}

\begin{enumerate}
\item[({\bf OSC})]\label{def:OSC}
Let $A, B\subset {\mathbb T}$ be two measurable subsets. $G\in [0, T]\times {\mathbb T}$ is said to obey the {\it observable symmetry condition} for ($A, B$) if\footnote{As usual, $A \Delta B = (A\backslash B)\cup (B\backslash A)$ denotes the symmetric difference between set $A$ and set $B$.

\vspace{3mm}
The first result is a new conservation law of the forced equation \eqref{eq: wave-eq}.
\begin{proposition}[Conservation law]
\label{prop-sym-conser}
Let $A,B\subset {\mathbb T}$. Assume that $G$ satisfies the ($A, B$)-{\it observable symmetry condition}.
Let $(u,u_t)\in C([0,T];\dot{H}^1({\mathbb T})\times L^2({\mathbb T}))$ be a solution to the forced equation \eqref{eq: wave-eq} and define the energy \footnote{Indeed, under the $(\xi, \eta)-$coordinate this conservation is write as
\begin{equation*}
 \frac{1}{2} I(t)= \int_{A-t}u_{\xi}(t,x){\,\rm d} x+\int_{B+t}u_{\eta}(t,x){\,\rm d} x.
\end{equation*}
}
$$
I(t)= \int_{A-t}(u_x+u_t)(t,x){\,\rm d} x+\int_{B+t}(u_x-u_t)(t,x){\,\rm d} x.
$$
Then we have
$$
I(t)=I(0), \; \forall t\in [0, T].
$$
In the case $|A|= |B|= 2\pi$, this conservation law is exactly the condition $\int_{{\mathbb T}} u_x (t, x) {\,\rm d} x= 0$.
\end{proposition}

\begin{enumerate}[label={}, leftmargin=6.5em, labelindent=3em]
\item[({\bf Weak GCC})]\label{def:weakGCC}
Let $T>0$. A measurable set $G\subset [0, T]\times \mathbb{T}$ is said to satisfy the
 weak GCC if for almost every
$x\in\mathbb{T}$,
\begin{equation}
\int_0^T\mathbf{1}_G(s,x\pm s){\,\rm d} s>0. \notag
\end{equation}
\end{enumerate}

\vspace{2mm}

This condition is equivalent to
\begin{equation}
 \textrm{ for } a.e. \; x\in {\mathbb T}, \mbox{meas}_\mathbb{R}(G\cap L_{\eta= x})>0, \;\;
 \mbox{meas}_\mathbb{R}(G\cap L_{\xi=x})>0. \notag
\end{equation}
Compared to the standard \blackhyperref{def:GCC}{({\rm GCC})}, it is not required a uniform positive lower bound.

\begin{enumerate}
\item[({\bf GCC})]\label{def:GCC}
Let $T>0$. A measurable set $G \subset [0, T] \times \mathbb{T}$ is said to satisfy
the GCC if there exists a constant
$c_0 > 0$ such that for almost every $x \in \mathbb{T}$,
\begin{equation}\label{eq:GCC-wave:0}
\int_0^T\mathbf{1}_G(s,x\pm s)\d s\geq c_0.\footnote{\text{This integral is defined in the sense of \eqref{eq:GCC:measure}.}}
\end{equation}
\end{enumerate}

\begin{enumerate}
\item[({\bf OSC})]\label{def:OSC}
Let $A, B\subset \T$ be two measurable subsets. $G\in [0, T]\times \T$ is said to obey the {\it observable symmetry condition} for ($A, B$) if\footnote{As usual, $A \Delta B = (A\backslash B)\cup (B\backslash A)$ denotes the symmetric difference between set $A$ and set $B$.

Expression \eqref{wave:nece:2} means that $G\cap L_{\xi\in A} = G\cap L_{\eta\in B}$ modulo zero measure set in $\R^2$.}
\begin{equation}\label{wave:nece:2}
 \mbox{meas}_{\R^2} \Big( [G\cap L_{\xi\in A}] \, \Delta \, [G\cap L_{\eta\in B}]\Big)=0.
\end{equation}
\end{enumerate}

\begin{enumerate}[label={}, leftmargin=6.5em, labelindent=3em]
\item[({\bf Weak  GCC})]\label{def:weakGCC}
Let $T>0$. A measurable set $G\subset [0, T]\times \mathbb{T}$ is said to satisfy the
 weak GCC if for almost every
$x\in\mathbb{T}$,
\begin{equation}
\int_0^T\mathbf{1}_G(s,x\pm s)\d s>0.   \notag
\end{equation}
\end{enumerate}

\begin{equation}\label{eq: wave-eq}
(\partial_t^2-\partial_x^2)u =f\mathbf{1}_G,\quad (u, \partial_t u)\big|_{t=0}=(u_0,u_1)\in \dot H^1(\T)\times L^2(\T).
\end{equation}

\begin{equation}\label{eq:GCC:measure}
      \textrm{ for } a.e. \; x\in \T, \mbox{meas}_\R(G\cap L_{\eta= x})\geq c_0, \;\;
  \mbox{meas}_\R(G\cap L_{\xi=x})\geq c_0.
\end{equation}

\begin{figure}[htp]
 \centering

\subsection{Necessary and sufficient observability/controllability condition}
Finally, we obtain the following sharp and complete geometric characterization of observable regions.
\begin{theorem}\label{thm: main-ob}
Let $T>0$ and let $G\subset [0, T]\times \T$ be a measurable set. Then the following two statements are equivalent.
\begin{itemize}
 \item[(1)] The observability inequality \eqref{eq: wave-ob} holds on $G$;
 \item [(2)] $G$ satisfies \blackhyperref{def:GCC}{{\rm (GCC)}} and does not obey \blackhyperref{def:OSC}{$({\rm OSC})$} for any non-tirvial pair ($A, B$).
\end{itemize}
\end{theorem}

\begin{proposition}[Unique continuation implies weak GCC]
Let $T>0$ and $G\in [0, T]\times \T$. Assume that the following unique continuation property holds
$$
\mbox{ Let u be a solution of \eqref{eq: wave-eq-0} and } u_t =0 \mbox{ a.e. in } G \Longrightarrow u \equiv 0.
$$
Then $G$ satisfies the \blackhyperref{def:weakGCC}{{\rm (weak GCC)}}.
\end{proposition}
\begin{proof}
We argue by contradiction. Suppose that $G$ does not satisfy the \blackhyperref{def:weakGCC}{({\rm weak GCC})}, there exists a subset $A\subset \T$ such that $\mbox{meas}_\R(A)>0$ and
\begin{align*}
\mbox{meas}_\R(G\cap L_{\eta\in A})=0 \quad \mbox{ or } \quad
 \mbox{meas}_\R(G\cap L_{\xi\in A})=0.
\end{align*}
Without loss of generality, we only consider the case
\begin{align}\label{equ-ucp-wgcc-1}
\mbox{meas}_\R(G\cap L_{\eta\in A})=0.
\end{align}
Since $A$ has positive measure, one can find a non-zero function $\phi$ such that
\begin{align}\label{equ-ucp-wgcc-2}
 \int_A \phi(x)\d x=0, \quad \int_A|\phi| \d x=\mbox{meas}_{\R}(A)>0.
\end{align}
Set
$$
u_{0x}=-u_1=\chi_A \phi.
$$
This is possible since $\int_\T u_{0,x}\d x =\int_A\phi \d x =0$. Then we have
$$
\partial_\xi u|_{t=0}=u_{0x}+u_1=0, \quad \partial_\eta u|_{t=0}=u_{0x}-u_1=2\chi_A\phi.
$$
It follows that
$$
\partial_\xi u=0 \quad \mbox{ on } [0,T]\times \T
$$
and
$$
\partial_\eta u= 0 \quad \mbox{ on } L_{\eta\in \T\backslash A}.
$$
Thus we have
$$
u_t=\frac{1}{2}(\partial_\xi u-\partial_\eta u)=0 \quad \mbox{ on } L_{\eta\in \T\backslash A}.
$$
Thanks to \eqref{equ-ucp-wgcc-1}, we see $G\subset L_{\eta\in \T\backslash A}$, and thus $u_t=0$ on $G$. By the UCP, we must have $\partial_\eta u|_{t=0}=2\chi_A\phi\equiv 0$, which leads a contradiction with \eqref{equ-ucp-wgcc-2}.
\end{proof}

\subsection{OSC and weak GCC imply unique continuation }\label{sec: OSC+WGCC>UCP}
In this sequel, we generalize the result in Proposition \ref{prop-ucp-S2}, which forms the following proposition.
\begin{proposition}[Weak GCC implies UCP up to $\mathcal{S}^2$]\label{prop-weakGCC-ucp-S2}
 Let $T>0$. Let $G\in [0, T]\times \T$ satisfy the \blackhyperref{def:weakGCC}{{\rm (weak GCC)}}.
 If $u$ solves the wave equation \eqref{eq: wave-eq-0} and $u_t = 0$ on $G$, then its initial state $(\partial_\xi u, \partial_\eta u)|_{t=0}$ belongs to the symmetry function pair class $\mathcal{S}^2$.
\end{proposition}
\begin{proof}
We use the same notation in the proof of Proposition \ref{prop-ucp-S2}. Recall that we have the relation \eqref{equ-xi-eta-exchange}, namely
\begin{equation*} \partial_{\eta}u|_{\{t= 0\}\cap L_{\eta= \eta_0}}=\partial_{\eta}u(\xi_0,\eta_0)= \partial_{\xi}u|_{\{t= 0\}\cap L_{\xi\in \mathcal{U}_{\eta= \eta_0}}}
\end{equation*}
up to sets with zero measure.
Namely, the value of $\partial_{\eta}u(\xi_0,\eta_0)$ determines almost every value of $\partial_{\xi}u|_{\{t= 0\}\cap L_{\xi\in \mathcal{U}_{\eta= \eta_0}}}$. Since $G$ satisfies the \blackhyperref{def:weakGCC}{({\rm weak GCC})}, we know
$$
\mbox{meas}_\R(\{t= 0\}\cap L_{\xi\in \mathcal{U}_{\eta= \eta_0}})>0.
$$
In other words, $\partial_{\xi}u|_{\{t= 0\}}$ equals to $\partial_{\eta}u(\xi_0,\eta_0)$ at least on a positive measure subset set of $\T$. Thus the set
$$
\mathcal{U}_{(\xi_0,\eta_0)}:=\{x_0\in \T: \partial_{\xi}u|_{\{t= 0,x=x_0\}}=\partial_{\eta}u(\xi_0,\eta_0)\}
$$
has positive measure. Repeating this process for other point $(\xi_0',\eta_0')\in [0,T]\times \T$, we shall find that, there exists a family sets $\mathcal{U}_\ell (\ell\in I)$, $I$ is a index set, such that
\begin{align}\label{equ-Ul-129}
 \mathcal{U}_\ell\cap \mathcal{U}_{\ell'}=\emptyset \mbox{ for } \ell\neq \ell', \quad \mbox{meas}_\R(\mathcal{U}_\ell)>0 \mbox{ for } \ell\in I , \quad \T=\cup_{\ell\in I}\mathcal{U}_\ell
\end{align}
and $\partial_{\xi}u|_{\{t= 0\}}$ is a constant on each $\mathcal{U}_\ell$.
One can show that, under the restrictions \eqref{equ-Ul-129}, the index set $I$ is at most countable, thus, after relabeling if necessary, $\{\mathcal{U}_\ell, \ell\ge 1\}$ is a weak decomposition of $\T$. Moreover, there exists a sequence of different numbers $\{s_\ell\}$ such that
$$
\partial_{\xi}u|_{\{t= 0\}}=s_\ell \mbox{ on } \mathcal{U}_\ell.
$$
Note that $\partial_{\xi}u|_{\{t= 0\}}\in L^2(\T)$, and $\mathcal{U}_\ell$ are disjoint for different $\ell$, we find
$$
\|\partial_{\xi}u|_{\{t= 0\}}\|_{L^2(\T)}^2=\sum_\ell |s_\ell|^2\mbox{meas}_\R(\mathcal{U}_\ell)<\infty.
$$
Thus we obtain
$$
\partial_{\xi}u|_{\{t= 0\}} = \sum_{\ell\geq 1} s_\ell \chi_{\mathcal{U}_\ell} \quad \mbox{ in } L^2(\T).
$$
Similarly, we can find a weak decomposition $\mathcal{V}_{\ell}(\ell\geq 1)$ such that
$$
\partial_{\eta}u|_{\{t= 0\}} = \sum_{\ell\geq 1} s_\ell \chi_{\mathcal{V}_\ell} \quad \mbox{ in } L^2(\T).
$$
In summary, we find that $(\partial_{\xi}u|_{\{t= 0\}}, \partial_{\eta}u|_{\{t= 0\}})\in \mathcal{S}^2$.
\end{proof}

The following theorem gives a sufficient and necessary condition of the observable set for the transport equation. We only state the result for \eqref{eq: transport-eq}, the reader easily figures out the the necessary modifications for \eqref{eq: transport-eq-2}.
\begin{theorem}\label{thm-tran-suff-nece}
Let $T>0$ and $G$ be a measurable subset of $[0,T]\times \T$ with positive measure. Then the observability inequality
\begin{align}\label{equ-tran-suff-necc-1}
 \|u_0\|^2_{L^2(\T)}\leq C\int_G|u(t,x)|^2\d x \d t
\end{align}
holds with a positive constant $C>0$ for all solutions to the transport equation \eqref{eq: transport-eq} if and only if there exists a constant $c_0>0$ such that
\begin{equation}\label{equ-tran-suff-necc-2}
\int_0^T\mathbf{1}_G(s,x- s)\d s\geq c_0 \quad \mbox{ for a.e. } x\in\T
\end{equation}
\end{theorem}
\begin{proof}
The direction $\eqref{equ-tran-suff-necc-2}\Longrightarrow \eqref{equ-tran-suff-necc-1}$ follows clearly from Proposition \ref{prop: tran-eq-ob}. To show the inverse direction, namely $\eqref{equ-tran-suff-necc-1}\Longrightarrow \eqref{equ-tran-suff-necc-2}$, we use the contradiction argument. Suppose that \eqref{equ-tran-suff-necc-2} is not true, then for any $\varepsilon>0$, there exists a set $E_\varepsilon\subset \T$ with positive measure $|E_\varepsilon|>0$ such that
\begin{align}\label{equ-tran-suff-necc-3}
\int_0^T\mathbf{1}_G(s,x- s)\d s<\varepsilon \quad \mbox{ for a.e. } x\in E_\varepsilon.
\end{align}
Let $u_{0\varepsilon}(x)=1_{E_\varepsilon}(x)$. Then $u_{0\varepsilon}\in L^2(\T)$ and its support is contained in $E_\varepsilon$. Then, similar to the proof in Proposition \ref{prop: tran-eq-ob}, we have
\begin{align*}
\int_G|u(t,x)|^2\d x \d t =\int_{\T}|u_{0\varepsilon}(x)|^2\int_{\T}\mathbf{1}_G(t,x-t)\d t\d x\leq \varepsilon \|u_{0\varepsilon}\|^2_{L^2(\T)}.
\end{align*}
Taking $\varepsilon>0$ small enough, say $\varepsilon=C/2$, we obtain a contradiction with \eqref{equ-tran-suff-necc-1}.
\end{proof}
\subsection{GCC is necessary for the wave observability}\label{sec:app:3}
\begin{corollary}\label{wave-GCC-nece}
If the observability \eqref{eq: wave-ob} holds on some observation set $G\subset [0,T]\times \T$, then $G$ needs to satisfies \blackhyperref{def:GCC}{{\rm (GCC)}}.
\end{corollary}
\begin{proof}

\begin{remark}
    The essence of this result lies in the role of the observable symmetry condition. Under the minimal \blackhyperref{def:weakGCC}{{\rm (weak GCC)}} assumption, unique continuation can be established only up to a symmetric function pair belonging to the class $\mathcal{S}^2$, which is generated precisely by the observable symmetry condition (see Definitions~\ref{def:classS}). In other words, the \blackhyperref{def:weakGCC}{{\rm (weak GCC)}} eliminates all obstructions except those arising from the intrinsic symmetry group, and full uniqueness is recovered once solutions are taken modulo this symmetry.
\end{remark}

\begin{figure}[htp]
    \centering

\tikzset{every picture/.style={line width=0.75pt}}
\begin{tikzpicture}[x=0.75pt,y=0.75pt,yscale=-1,xscale=1]

\draw   (141,130) -- (292,130) -- (292,170) -- (141,170) -- cycle ;
\draw   (425,130) -- (539,130) -- (539,170) -- (425,170) -- cycle ;
\draw    (303,146.5) -- (415,146.5) ;
\draw [shift={(417,146.5)}, rotate = 180] [color={rgb, 255:red, 0; green, 0; blue, 0 }  ][line width=0.75]    (10.93,-3.29) .. controls (6.95,-1.4) and (3.31,-0.3) .. (0,0) .. controls (3.31,0.3) and (6.95,1.4) .. (10.93,3.29)   ;
\draw    (416,154.5) -- (305,153.52) ;
\draw [shift={(303,153.5)}, rotate = 0.51] [color={rgb, 255:red, 0; green, 0; blue, 0 }  ][line width=0.75]    (10.93,-3.29) .. controls (6.95,-1.4) and (3.31,-0.3) .. (0,0) .. controls (3.31,0.3) and (6.95,1.4) .. (10.93,3.29)   ;

\draw (149,143) node [anchor=north west][inner sep=0.75pt]   [align=left] {\blackhyperref{def:OSC}{${\rm OSC}$} \ + \ \blackhyperref{def:weakGCC}{{\rm weak GCC}}};
\draw (462,142) node [anchor=north west][inner sep=0.75pt]   [align=left] {UCP};
\draw (337,130) node [anchor=north west][inner sep=0.75pt]   [align=left] {{\footnotesize Sec \ref{sec: OSC+WGCC>UCP}}};
\draw (320,156) node [anchor=north west][inner sep=0.75pt]   [align=left] {{\footnotesize Sec \ref{sec:neosctoucp}, Sec \ref{sec: ucp-wgcc}}};

\end{tikzpicture}

    \caption{Outline of the proof to the equivalence.}
    \label{fig:OSC+WGCC=UCP}
\end{figure}

\subsection{Necessary and sufficient observability/controllability condition}
Finally, we obtain the following sharp and complete geometric characterization of observable regions.
\begin{theorem}\label{thm: main-ob}
Let $T>0$ and let $G\subset [0, T]\times {\mathbb T}$ be a measurable set. Then the following two statements are equivalent.
\begin{itemize}
    \item[(1)] The observability inequality \eqref{eq: wave-ob}  holds on $G$;
    \item [(2)] $G$ satisfies \blackhyperref{def:GCC}{{\rm (GCC)}} and does not obey \blackhyperref{def:OSC}{$({\rm OSC})$} for any non-tirvial pair ($A, B$).
\end{itemize}
\end{theorem}

To the best of our knowledge, this work provides the first necessary and sufficient geometric condition for observability of the wave equation on spacetime region. The observable symmetry condition plays a central role: it completes the classical GCC by capturing the additional geometric structure. This viewpoint also leads to necessary and sufficient conditions in a variety of other important geometric configurations, such as $[0, T]\times \omega$,  measurable Cartesian products  $E_t\times F_{x}$, and general spacetime open sets.  A detailed comparison is presented in Section~\ref{sec:example}.

\begin{figure}[htp]
    \centering

\tikzset{every picture/.style={line width=0.75pt}}
\begin{tikzpicture}[x=0.75pt,y=0.75pt,yscale=-1,xscale=1]

\draw   (178,130) -- (292,130) -- (292,170) -- (178,170) -- cycle ;
\draw   (425,130) -- (539,130) -- (539,170) -- (425,170) -- cycle ;
\draw    (303,146.5) -- (415,146.5) ;
\draw [shift={(417,146.5)}, rotate = 180] [color={rgb, 255:red, 0; green, 0; blue, 0 }  ][line width=0.75]    (10.93,-3.29) .. controls (6.95,-1.4) and (3.31,-0.3) .. (0,0) .. controls (3.31,0.3) and (6.95,1.4) .. (10.93,3.29)   ;
\draw    (416,154.5) -- (305,153.52) ;
\draw [shift={(303,153.5)}, rotate = 0.51] [color={rgb, 255:red, 0; green, 0; blue, 0 }  ][line width=0.75]    (10.93,-3.29) .. controls (6.95,-1.4) and (3.31,-0.3) .. (0,0) .. controls (3.31,0.3) and (6.95,1.4) .. (10.93,3.29)   ;

\draw (188,143) node [anchor=north west][inner sep=0.75pt]   [align=left] {\blackhyperref{def:OSC}{${\rm OSC}$} \ + \ \blackhyperref{def:GCC}{{\rm GCC}}};
\draw (439,142) node [anchor=north west][inner sep=0.75pt]   [align=left] {Observability};
\draw (343,130) node [anchor=north west][inner sep=0.75pt]   [align=left] {{\footnotesize Sec \ref{sec:sharp:obser}}};
\draw (320,156) node [anchor=north west][inner sep=0.75pt]   [align=left] {{\footnotesize Sec \ref{sec-osc-nece}, Sec \ref{sec:app:3}}};

\end{tikzpicture}
    \caption{Outline of the proof to the equivalence.}
    \label{fig:OSC+GCC=OB}
\end{figure}

\begin{remark}
The main theorem reveals two structural features of observability that distinguish the wave equation from other important models, for example, the heat equation,  \cite{AEWZ,WWZZ}. First, the family of observable regions $\mathcal{O}(T)$ depends intrinsically on $T$; non-observable sets arise from geometric obstructions that vary with $T$, leading to a genuinely time-dependent structure for $\{\mathcal{O}(T)\}_{T>0}$.
Second, our characterization also suggests that no waiting time is needed for observability, indicating that, for wave equations, the spacetime geometry of the observation region may play a more decisive role than the duration of control. See Section~\ref{sec:furthercomment}.
\end{remark}

Moreover, we believe that the symmetry mechanism uncovered in this work may extend to other instances of wave equations and to a broader range of models, including coupled systems, semilinear equations, and geometric wave equations. These ideas may also provide insight for higher-dimensional problems, where geometric propagation and microlocal structures are considerably more intricate. In addition, the geometric perspectives here may prove useful in questions of sensor placement, optimization of observation regions, and numerical implementations of control and observability.

\subsection*{Acknowledgements}
The authors would like to thank Nicolas Burq for valuable and useful discussions during the preparation of this manuscript.  Shengquan Xiang is partially supported by the NSFC under grants 12571474 and 12301562.   Ming Wang was partially supported by the NSFC under grants 12571260, 12171442 and 12171178.

A proof sketch for Theorem~\ref{thm: main-ucp} is indicated as follows.

- The key mechanism is the \blackhyperref{def:OSC}{(OSC)} (\emph{observable symmetry condition}): “Under the minimal \blackhyperref{def:weakGCC}{{\rm (weak GCC)}} assumption, unique continuation can be established only up to a symmetric function pair belonging to the class $\mathcal{S}^2$, which is generated precisely by the observable symmetry condition (see Definitions~\ref{def:classS}).”

- Conceptual structure: “the \blackhyperref{def:weakGCC}{{\rm (weak GCC)}} eliminates all obstructions except those arising from the intrinsic symmetry group, and full uniqueness is recovered once solutions are taken modulo this symmetry.”

- The introduction also provides an explicit “Outline of the proof to the equivalence” (Figure~\ref{fig:OSC+WGCC=UCP}), depicting the equivalence between “\blackhyperref{def:OSC}{{\rm OSC}} + \blackhyperref{def:weakGCC}{{\rm weak GCC}}” and “UCP,” with the relevant parts located in “Sec \ref{sec: OSC+WGCC>UCP}” and “Sec \ref{sec:neosctoucp}, Sec \ref{sec: ucp-wgcc}.”

A proof sketch for the main theorem is indicated as follows.

- The key mechanism is the following observable symmetry condition:
\begin{enumerate}
\item[({\bf OSC})]\label{def:OSC}
Let $A, B\subset \T$ be two measurable subsets. $G\in [0, T]\times \T$ is said to obey the {\it observable symmetry condition} for ($A, B$) if\footnote{As usual, $A \Delta B = (A\backslash B)\cup (B\backslash A)$ denotes the symmetric difference between set $A$ and set $B$.

Expression \eqref{wave:nece:2} means that $G\cap L_{\xi\in A} = G\cap L_{\eta\in B}$ modulo zero measure set in $\R^2$.}
\begin{equation}\label{wave:nece:2}
 \mbox{meas}_{\R^2} \Big( [G\cap L_{\xi\in A}] \, \Delta \, [G\cap L_{\eta\in B}]\Big)=0.
\end{equation}
\end{enumerate}
Under the minimal assumption
\begin{enumerate}[label={}, leftmargin=6.5em, labelindent=3em]
\item[({\bf Weak  GCC})]\label{def:weakGCC}
Let $T>0$. A measurable set $G\subset [0, T]\times \mathbb{T}$ is said to satisfy the
 weak GCC if for almost every
$x\in\mathbb{T}$,
\begin{equation}
\int_0^T\mathbf{1}_G(s,x\pm s)\d s>0.   \notag
\end{equation}
\end{enumerate}
unique continuation can be established only up to a symmetric function pair belonging to the class $\mathcal{S}^2$, where
\begin{definition}[Weak symmetric function pair]\label{def:classS}
We say
$(f,g)$ is a weak symmetric  function pair if  there exists a weak decomposition pair $\{(A_k, B_k), k\in I\}$, a sequence of different  numbers $\{s_k\}_{k\in I}$ such that
\begin{equation*}
 f=\sum_{k\in I}s_k\chi_{A_k}, \quad g=\sum_{k\in I}s_k\chi_{B_k} \quad \mbox{ in } L^2(\T)
\end{equation*}
and
\begin{equation*}
    \int_{\T} (f+ g)(x) \d x= 0.
\end{equation*}
The set of all weak symmetric function pairs is denoted by $\mathcal{S}^2$.
\end{definition}
In particular, this class is generated precisely by the observable symmetry condition above.

- Conceptual structure: the weak GCC condition stated above eliminates all obstructions except those arising from the intrinsic symmetry group, and full uniqueness is recovered once solutions are taken modulo this symmetry.

- The introduction also provides an explicit “Outline of the proof to the equivalence” (Figure~\ref{fig:OSC+WGCC=UCP}), depicting the equivalence between “the observable symmetry condition above + the weak GCC condition above” and “UCP,” with the relevant parts located later and in \ref{sec: OSC+WGCC>UCP} and \ref{sec:neosctoucp}, \ref{sec: ucp-wgcc}.

\label{thm: main-ucp}
Let $T>0$ and $G\subset [0,T]\times {\mathbb T}$ be a spacetime measurable set. The following statements are equivalent.
\begin{itemize}
\item [(1)] UCP holds on $G$.

\item [(2)] $G$ satisfies
\begin{enumerate}[label={}, leftmargin=6.5em, labelindent=3em]
\item[({\bf Weak  GCC})]\label{def:weakGCC}
Let $T>0$. A measurable set $G\subset [0, T]\times \mathbb{T}$ is said to satisfy the
 weak GCC if for almost every
$x\in\mathbb{T}$,
\begin{equation}
\int_0^T\mathbf{1}_G(s,x\pm s)\d s>0.   \notag
\end{equation}
\end{enumerate}
and does not obey
\begin{enumerate}
\item[({\bf OSC})]\label{def:OSC}
Let $A, B\subset \T$ be two measurable subsets. $G\in [0, T]\times \T$ is said to obey the {\it observable symmetry condition} for ($A, B$) if\footnote{As usual, $A \Delta B = (A\backslash B)\cup (B\backslash A)$ denotes the symmetric difference between set $A$ and set $B$.

Expression \eqref{wave:nece:2} means that $G\cap L_{\xi\in A} = G\cap L_{\eta\in B}$ modulo zero measure set in $\R^2$.}
\begin{equation}\label{wave:nece:2}
 \mbox{meas}_{\R^2} \Big( [G\cap L_{\xi\in A}] \, \Delta \, [G\cap L_{\eta\in B}]\Big)=0.
\end{equation}
\end{enumerate}
for any non-trivial pair $(A, B)$\footnote{Reminder: for simplicity, when there is no risk of confusion,  sometimes we simply write (OSC) indicating that the set $G$ does not obey the observable symmetry condition for any non-trivial pair $(A, B)$.}.
\end{itemize}

Biconditional or Equivalence
Classification or Bijection

Question: Let $T>0$ and let $G\subset [0,T]\times \mathbb T$ be a measurable spacetime set, where $\mathbb T$ is the one-dimensional torus. Which explicit geometric condition on $G$ is equivalent to the unique continuation property (UCP) on $G$ for the wave setting?

$G$ satisfies the weak geometric control condition, meaning that for almost every $x\in\mathbb T$ one has both $\int_0^T \mathbf 1_G(s,x+s)\,ds>0$ and $\int_0^T \mathbf 1_G(s,x-s)\,ds>0$, and moreover $G$ does not obey the observable symmetry condition for any non-trivial pair $(A,B)$ of measurable subsets of $\mathbb T$; equivalently, there is no non-trivial pair $(A,B)$ such that $$\operatorname{meas}_{\mathbb R^2}\Big(\,[G\cap\{(t,x):x+t\in A\}]\,\Delta\,[G\cap\{(t,x):x-t\in B\}]\,\Big)=0,$$ where $\Delta$ denotes symmetric difference.

Choices:
B. $G$ satisfies the geometric control condition in the strong sense that there exists a constant $c_0>0$ such that for almost every $x\in\mathbb T$ one has both $\int_0^T \mathbf 1_G(s,x+s)\,ds\ge c_0$ and $\int_0^T \mathbf 1_G(s,x-s)\,ds\ge c_0$, and moreover $G$ does not obey the observable symmetry condition for any non-trivial pair $(A,B)$ of measurable subsets of $\mathbb T$; equivalently, there is no non-trivial pair $(A,B)$ such that $$\operatorname{meas}_{\mathbb R^2}\Big(\,[G\cap\{(t,x):x+t\in A\}]\,\Delta\,[G\cap\{(t,x):x-t\in B\}]\,\Big)=0.$$
C. $G$ satisfies the weak geometric control condition, meaning that for almost every $x\in\mathbb T$ one has both $\int_0^T \mathbf 1_G(s,x+s)\,ds>0$ and $\int_0^T \mathbf 1_G(s,x-s)\,ds>0$.
D. $G$ satisfies the weak geometric control condition, meaning that for almost every $x\in\mathbb T$ one has both $\int_0^T \mathbf 1_G(s,x+s)\,ds>0$ and $\int_0^T \mathbf 1_G(s,x-s)\,ds>0$, and moreover $G$ does not obey the observable symmetry condition for any pair $(A,B)$ of measurable subsets of $\mathbb T$, including the trivial pair; equivalently, for every measurable pair $(A,B)$ one has $$\operatorname{meas}_{\mathbb R^2}\Big(\,[G\cap\{(t,x):x+t\in A\}]\,\Delta\,[G\cap\{(t,x):x-t\in B\}]\,\Big)>0.$$
E. $G$ satisfies the weak geometric control condition, meaning that for almost every $x\in\mathbb T$ at least one of the two integrals $\int_0^T \mathbf 1_G(s,x+s)\,ds$ or $\int_0^T \mathbf 1_G(s,x-s)\,ds$ is strictly positive, and moreover $G$ does not obey the observable symmetry condition for any non-trivial pair $(A,B)$ of measurable subsets of $\mathbb T$; equivalently, there is no non-trivial pair $(A,B)$ such that $$\operatorname{meas}_{\mathbb R^2}\Big(\,[G\cap\{(t,x):x+t\in A\}]\,\Delta\,[G\cap\{(t,x):x-t\in B\}]\,\Big)=0.$$

restriction to non-trivial symmetry pairs

both characteristic directions required in weak GCC


Question: Let \(T>0\) and let \(G\subset [0,T]\times \mathbb T\) be a measurable set, where \(\mathbb T\) is the one-dimensional torus. Which of the following statements is equivalent to the unique continuation property (UCP) holding on \(G\)?

\(G\) satisfies the weak geometric control condition, i.e. for almost every \(x\in\mathbb T\), both \(\int_0^T \mathbf 1_G(s,x+s)\,ds>0\) and \(\int_0^T \mathbf 1_G(s,x-s)\,ds>0\), and \(G\) does not obey the observable symmetry condition for any non-trivial pair \((A,B)\) of measurable subsets of \(\mathbb T\); that is, there is no non-trivial pair \((A,B)\) such that \(\operatorname{meas}_{\mathbb R^2}(([G\cap L_{\xi\in A}]\,\Delta\,[G\cap L_{\eta\in B}]))=0\), where \(\xi=x+t\), \(\eta=x-t\), \(L_{\xi\in A}=\{(t,x):x+t\in A\}\), and \(L_{\eta\in B}=\{(t,x):x-t\in B\}\).

Choices:
B. \(G\) satisfies the geometric control condition in the strong sense: there exists a constant \(c_0>0\) such that for almost every \(x\in\mathbb T\), both \(\int_0^T \mathbf 1_G(s,x+s)\,ds\ge c_0\) and \(\int_0^T \mathbf 1_G(s,x-s)\,ds\ge c_0\), and \(G\) does not obey the observable symmetry condition for any non-trivial pair \((A,B)\) of measurable subsets of \(\mathbb T\); that is, there is no non-trivial pair \((A,B)\) such that \(\operatorname{meas}_{\mathbb R^2}(([G\cap L_{\xi\in A}]\,\Delta\,[G\cap L_{\eta\in B}]))=0\), where \(\xi=x+t\), \(\eta=x-t\), \(L_{\xi\in A}=\{(t,x):x+t\in A\}\), and \(L_{\eta\in B}=\{(t,x):x-t\in B\}\).
C. \(G\) satisfies the weak geometric control condition, i.e. for almost every \(x\in\mathbb T\), both \(\int_0^T \mathbf 1_G(s,x+s)\,ds>0\) and \(\int_0^T \mathbf 1_G(s,x-s)\,ds>0\).
D. \(G\) satisfies the weak geometric control condition, i.e. for almost every \(x\in\mathbb T\), both \(\int_0^T \mathbf 1_G(s,x+s)\,ds>0\) and \(\int_0^T \mathbf 1_G(s,x-s)\,ds>0\), and \(G\) does not obey the observable symmetry condition for any pair \((A,B)\) of measurable subsets of \(\mathbb T\), trivial or non-trivial; that is, for every measurable \((A,B)\), one has \(\operatorname{meas}_{\mathbb R^2}(([G\cap L_{\xi\in A}]\,\Delta\,[G\cap L_{\eta\in B}]))>0\), where \(\xi=x+t\), \(\eta=x-t\), \(L_{\xi\in A}=\{(t,x):x+t\in A\}\), and \(L_{\eta\in B}=\{(t,x):x-t\in B\}\).
E. \(G\) satisfies the weak geometric control condition, i.e. for almost every \(x\in\mathbb T\), at least one of the two integrals \(\int_0^T \mathbf 1_G(s,x+s)\,ds\) or \(\int_0^T \mathbf 1_G(s,x-s)\,ds\) is strictly positive, and \(G\) does not obey the observable symmetry condition for any non-trivial pair \((A,B)\) of measurable subsets of \(\mathbb T\); that is, there is no non-trivial pair \((A,B)\) such that \(\operatorname{meas}_{\mathbb R^2}(([G\cap L_{\xi\in A}]\,\Delta\,[G\cap L_{\eta\in B}]))=0\), where \(\xi=x+t\), \(\eta=x-t\), \(L_{\xi\in A}=\{(t,x):x+t\in A\}\), and \(L_{\eta\in B}=\{(t,x):x-t\in B\}\).

uniform positive lower bound in weak GCC

restriction to non-trivial pairs in OSC exclusion

both characteristic families must be seen in weak GCC


\begin{theorem}\label{thm:clique_classification}
For any clique $\mathcal K=\{P_1,\dots,P_m\}$ with $m\ge 1$ in $\B_k(G)$, exactly one of the following holds:
\begin{enumerate}
    \item $\mathcal K=\{P_1,P_2,P_3\}$ is a cyclic $T$-triangle;
    \item $\mathcal K=\{P_1,P_2,P_3\}$  is a radial $T$-triangle;
    \item $\mathcal K=\{P_1,P_2,P_3,P_4\}$  is a split $T$-tetrahedron;
    \item $\mathcal K=\{P_1,P_2,P_3,P_4\}$ is a fused $T$-tetrahedron; or
    \item $\mathcal K=\{P_1,\dots,P_m\}$ is an $S$-clique.
\end{enumerate}
\end{theorem}

\begin{tikzpicture}[
    vertex/.style={circle, draw, fill=white, minimum size=8mm, inner sep=0pt, font=\small},
    edge label/.style={midway, fill=white, font=\footnotesize, inner sep=1.5pt},
    part label/.style={font=\scriptsize, color=black, align=center},
    x={(-0.6cm, -0.4cm)}, y={(1cm, 0cm)}, z={(0cm, 1cm)},
    scale=1.2
]

\coordinate (P1) at (0, -2, 0);
\coordinate (P2) at (2, 1, 0);
\coordinate (P3) at (-2, 1, 0);
\coordinate (P4) at (0, 0, 3);

\draw (P1) -- (P2) node[edge label, pos=0.53] {$w$};
\draw (P1) -- (P3) node[edge label, pos=0.55] {$w$};
\draw (P2) -- (P3) node[edge label] {$w$};
\draw (P4) -- (P1) node[edge label, pos=0.5] {$w, v_1$};
\draw (P4) -- (P2) node[edge label, pos=0.4] {$w, v_2$};
\draw (P4) -- (P3) node[edge label, pos=0.6] {$w, v_3$};

\node[vertex] at (P1) {$P_1$};
\node[part label, below left=0.3cm of P1] {$\{\{w, v_1\}, \{v_2\}, \{v_3\}, \emptyset\}$};

\node[vertex] at (P2) {$P_2$};
\node[part label, below=0.4cm of P2] {$\{\{w, v_2\}, \{v_1\}, \{v_3\}, \emptyset\}$};

\node[vertex] at (P3) {$P_3$};
\node[part label, below right=0.3cm of P3] {$\{\{w, v_3\}, \{v_1\}, \{v_2\}, \emptyset \}$};

\node[vertex] at (P4) {$P_4$};
\node[part label, above=0.4cm of P4] {$\{\{w\}, \{v_1\}, \{v_2\}, \{v_3\}\}$};

\end{tikzpicture}
\caption{Illustration of $\mathcal{B}_4(K_{3}\sqcup K_1)\cong K_4$. The graph $K_3\sqcup K_1$ has vertex set $\{v_1, v_2, v_3, w\}$ with edges $v_1v_2, v_2v_3, v_3v_1$. Edge labels indicate the vertices responsible for the adjacency.}
\label{fig:Bell3-K3-plus-K1}
\end{figure}

Our work is motivated in part by the question of which graphs can arise as Bell coloring graphs. Every coloring graph is a Bell coloring graph because $\B_k(G\sqcup K_k) \cong \mathcal{C}_k(G)$ for any graph $G$ and integer $k\ge 1$ \cite{FM14}*{Proposition 2.5}. However, the class of graphs realizable as standard coloring graphs is quite limited. For example, $K_1$ and $K_2$ are the \emph{only} trees that are realizable as coloring graphs, and $C_3$, $C_4$, and $C_6$ are the \emph{only} cycles that are realizable as coloring graphs \cite{BFHRS16}*{Theorems 7 and 23}. In contrast, we prove the following:

\begin{theoremA}[Theorem~\ref{thm:trees-cycles}]
All trees and all cycles are realizable as Bell coloring graphs.
\end{theoremA}

We prove Theorem~\ref{thm:trees-cycles} by showing that a more general class of graphs, namely certain reconfiguration graphs of matchings, are Bell coloring graphs.

The papers \cites{BFHRS16, ABFR17} examine forbidden subgraphs in standard coloring graphs. To treat the analogous question for Bell coloring graphs, we first obtain a structural description of their cliques (see Theorem~\ref{thm:clique_classification}), showing that every clique belongs to one of two explicit families. This classification allows us to construct an infinite family of forbidden induced subgraphs of Bell coloring graphs.

\begin{theoremA}[Theorem~\ref{thm:forbidden}]
The graph $K_{6}-e$ is not an induced subgraph of any Bell coloring graph. Hence, the set of graphs $K_n-e$ for $n\ge 6$ is an infinite family of forbidden induced subgraphs.
\end{theoremA}

Another natural question is whether the structure of the reconfiguration graph uniquely determines the base graph. This is known for standard coloring graphs; in particular, if $T$ is a tree, $\mathcal{C}_3(T)$ uniquely determines $T$ (\cite{AKL25}*{Theorem 1.1} or \cite{BBHvdHHP25}*{Theorem 1.2}). Having lost the color labels, $\mathcal{B}_3(T)$ has less symmetry than the $3$-coloring graph $\mathcal{C}_3(T)$. Despite this compression, our next result guarantees that a tree $T$ can be reconstructed uniquely from its Bell $3$-coloring graph. Let $\mathsf{Trees}$ and $\mathsf{Graphs}$ denote the sets of isomorphism classes of all (finite) trees and graphs, respectively.

\begin{theoremA}[Theorem~\ref{thm:tree-reconstruction}]
The map $\mathsf{Trees} \to \mathsf{Graphs}$ given by $T\mapsto \mathcal{B}_3(T)$ is injective.
\end{theoremA}

We also find it useful to work with a multigraph variant, the \emph{Bell coloring multigraph}, denoted $\BellMulti{k}{G}$, which uses multiple edges to encode how many vertices are responsible for each adjacency. If $P$ and $P'$ are two stable partitions of $G$, then in the Bell coloring graph, $P$ and $P'$ are adjacent whenever there exists a vertex $v\in V(G)$ with $P-v = P'-v$. Thus, multiple vertices may witness the same adjacency, but $\B_k(G)$ still records only a single edge between $P$ and $P'$. In the multigraph version, we instead add a separate parallel edge between $P$ and $P'$ for each distinct vertex $v$ such that $P-v = P'-v$.

The additional information means that Bell coloring multigraphs serve as a more refined graph invariant. For example, Figures~\ref{fig:Bell3-K13}~and~\ref{fig:Bell3-K3-plus-K1} show that $\B_3(K_{1,3})\cong \B_4(K_3\sqcup K_1)$. However, each edge incident to the partition $P_4$ in the latter graph is witnessed by two vertices, so
$\BellMulti{3}{K_{1,3}}\not\cong\BellMulti{4}{K_3\sqcup K_{1}}$.

Yet even a Bell coloring multigraph cannot serve as a complete graph invariant. For instance, $\BellMulti{3}{K_{1,3}}\cong \BellMulti{2}{\overline{K}_3}$. This ambiguity arises because the center vertex in $K_{1,3}$ is adjacent to every other vertex. When a vertex $v$ is adjacent to every vertex in $G-v$, we call $v$ a \emph{universal vertex}.
If $G$ has a universal vertex $w$, then $\B_{k+1}(G) \cong \B_k(G-w)$ for every $k\in\mathbb{N}$ due to the natural bijection between stable $k$-partitions of $G-w$ and stable $(k+1)$-partitions of $G$ in which $\{w\}$ forms an additional singleton part.

To handle this, let $U(G)$ denote the set of universal vertices of $G$. We define the \emph{core} of $G$, denoted $G^{\circ}$, as the induced subgraph $G[V(G)\setminus U(G)]$. We define the equivalence relation $\sim_{\text{uni}}$ on $\mathsf{Graphs}$ as follows: $G\sim_{\text{uni}} G'$ if their cores are isomorphic, $G^{\circ} \cong (G')^{\circ}$.
The equivalence classes of graphs under this relation form the set $\mathsf{Graphs}^{\circ} = \mathsf{Graphs}/\!\sim_{\text{uni}}$.

Our final result shows that a base graph of order $n$ can be reconstructed from its Bell $n$-coloring multigraph up to universal vertices:

\begin{theoremA}[Theorem~\ref{thm:general-reconstruction-multigraph:intro}]
The map $\mathsf{Graphs}^{\circ} \to \mathsf{Multigraphs}$ given by $G\mapsto \BellMultiIntro{|V(G)|}{G}$  is injective.
\end{theoremA}

\textbf{Outline of the paper.} Section~\ref{sec:preliminaries} introduces Bell $k$-coloring graphs, gives small examples, and develops basic tools for understanding adjacency, including a description of edges that are realized by two vertices. In Section~\ref{sec:cliques}, we classify cliques in Bell coloring graphs and apply this to show that $K_4-e$ is not a Bell coloring graph and that $K_n-e$ is not an induced subgraph of any Bell coloring graph for $n\ge 6$. Section~\ref{sec:matchings} introduces a matching reconfiguration graph and shows that, for triangle-free graphs, it coincides with a Bell coloring graph; this yields realizations of all trees and all cycles as Bell coloring graphs. Section~\ref{sec:tree-reconstruction} proves that trees are reconstructible from their Bell
$3$-coloring graphs. Section~\ref{sec:reconstruction-multigraph} proves the multigraph reconstruction theorem: the Bell $n$-coloring multigraph determines $G$ up to universal vertices.

\section{Preliminaries}\label{sec:preliminaries}

\subsection{Notation and conventions} Let $\mathbb{N}=\{1, 2, 3, \dots\}$ denote the set of positive integers. For sets $X$ and $Y$, we use the notation $X\setminus Y=\{x\in X \ | \ x\notin Y\}$ for the set difference and $X\cup Y$ for set union; in the special case when $Y=\{v\}$ is a singleton, we also denote $X\setminus\{v\}$ by $X-v$ and $X\cup\{v\}$ by $X+v$. Given a graph $G$, we let $\overline{G}$ denote its complement. We write $G\sqcup H$ for the disjoint union of graphs $G$ and $H$.

\begin{defn}\label{def:stable-partition}
A \emph{stable $k$-partition} of a graph $G$ is a multiset $P=\{V_1,\dots,V_k\}$ of independent sets that partition $V(G)$, where we allow some of the parts $V_i$ to be empty.
\end{defn}

Let $\cP_k(G)$ denote the set of all stable $k$-partitions of $G$. For $v\in V(G)$, we write
\[
P-v \colonequals \{V_i-v : 1\leq i\leq k\},
\]
viewed as a multiset of $k$ subsets.

A Bell $k$-coloring graph $\B_k(G)$ is the graph whose vertex set is $\cP_k(G)$. Two vertices $P_1, P_2 \in \cP_k(G)$ are adjacent if and only if $P_1-v=P_2-v$ for some $v\in V(G)$.

Throughout, partitions $P, Q, R \in \cP_k(G)$ are treated as \emph{unordered multisets} of parts. We write $P \sim Q$ to denote adjacency in $\B_k(G)$. To specify the vertex responsible for the adjacency, we use the following notation:
\[
P \sim_v Q \iff P \sim Q \text{ and } P-v = Q-v.
\]
We say that a vertex $v\in V(G)$ is \emph{responsible} for an edge $PQ$ in $\B_k(G)$ if $P\sim_v Q$. Equivalently, the edge $PQ$ is \emph{realized} by $v$. The condition $P-v=Q-v$ implies that $P$ and $Q$ agree on the structure of the partition when restricted to $V(G)\setminus\{v\}$.

We record the following useful fact, which we use several times in the paper: if $P_1\sim_v P_2$ and $P_2\sim_v P_3$, then $P_1$ and $P_3$ agree on $V(G)\setminus\{v\}$, so $P_1\sim_v P_3$.

\subsection{Example Bell coloring graphs} We begin with a few illustrative examples.

\begin{ex} Consider $\B_2(\overline{K}_2)$ where $V(\overline{K}_2)=\{1,2\}$. The stable 2-partitions are $R_1 = \{\{1,2\}, \emptyset\}$ and $R_2 = \{\{1\}, \{2\}\}$.
We have $R_1-1 = \{\{2\}, \emptyset\}$ and $R_2-1 = \{\emptyset, \{2\}\}$, so $R_1 \sim_1 R_2$. Similarly $R_1 \sim_2 R_2$. Thus $\B_2(\overline{K}_2) \cong K_2$.
\end{ex}

\begin{ex}\label{ex:K3}
Consider $G=K_3 \sqcup K_1$ on vertices $\{1,2,3,4\}$, where $\{1,2,3\}$ form the $K_3$ and the vertex $4$ is isolated. We examine $\B_3(G)$. Since $\{1,2,3\}$ is a clique, any stable 3-partition must place them in different parts. The vertex $4$ can join any part.
\begin{align*}
P_1 &= \{\{1,4\}, \{2\}, \{3\}\}, \\
P_2 &= \{\{1\}, \{2,4\}, \{3\}\}, \\
P_3 &= \{\{1\}, \{2\}, \{3,4\}\}.
\end{align*}
These are the only stable 3-partitions of $G$. We check adjacencies: $P_1-4 = P_2-4 = P_3-4 = \{\{1\}, \{2\}, \{3\}\}$. Thus, we have $P_1 \sim_4 P_2$, $P_2 \sim_4 P_3$, and $P_3 \sim_4 P_1$, so $\B_3(G) \cong K_3$.
\end{ex}

\begin{ex}\label{ex:P4}
Consider the path graph $G$ on vertices $\{1,2,3,4\}$ with edges 12, 23, 34. The stable $3$-partitions are:
\[
\begin{aligned}
R_1 &= \{\{1,3\}, \{2\}, \{4\}\}, & R_2 &= \{\{1,4\}, \{2\}, \{3\}\},\\
R_3 &= \{\{1\}, \{2,4\}, \{3\}\}, & R_4 &= \{\{1,3\}, \{2,4\}, \emptyset\}.
\end{aligned}
\]
A direct check shows that $R_1 \sim_1 R_2$, $R_2 \sim_4 R_3$, $R_3 \sim_1 R_4$, and $R_4 \sim_4 R_1$. The graph $\B_3(P_4)$ is the cycle $C_4$; see Figure~\ref{fig:B3_P4}.
\end{ex}

\begin{figure}[h]
    \centering
    \begin{tikzpicture}[
        partition/.style={rectangle, draw, rounded corners, thick, inner sep=5pt, align=center, fill=white},
        edge label/.style={midway, fill=white, font=\small, inner sep=1.5pt},
        scale=1.1
    ]

    \node[partition] (R1) at (0, 2) {$\{1,3\}, \{2\}, \{4\}$};

    \node[partition] (R2) at (5, 2) {$\{1,4\}, \{2\}, \{3\}$};

    \node[partition] (R3) at (5, 0) {$\{1\}, \{2,4\}, \{3\}$};

    \node[partition] (R4) at (0, 0) {$\{1,3\}, \{2,4\}, \emptyset$};

    \draw[thick] (R1) -- (R2) node[edge label] {$1$};

    \draw[thick] (R2) -- (R3) node[edge label] {$4$};

    \draw[thick] (R3) -- (R4) node[edge label] {$1, 3$};

    \draw[thick] (R4) -- (R1) node[edge label] {$2, 4$};

    \end{tikzpicture}

\begin{theorem}\label{thm:general-reconstruction-multigraph:intro}
Suppose $G_1$ and $G_2$ are two graphs of orders $n_1$ and $n_2$, respectively. If $\BellMulti{n_1}{G_1} \cong \BellMulti{n_2}{G_2}$, then $G_1$ and $G_2$ have isomorphic cores.
\end{theorem}

\begin{theorem}\label{thm:forbidden}
The graph $K_{6}-e$ is not an induced subgraph of any Bell coloring graph. Hence, the set of graphs $K_n-e$ for $n\ge 6$ is an infinite family of forbidden induced subgraphs.
\end{theorem}

\label{sec:intro}

Graph coloring is a vibrant area of research that bridges theoretical questions and practical applications. Several excellent references survey both the classic landscape and modern frontiers; see, for example, the monograph \cite{JT95}, the themed collection \cite{BW15}, and the recent text \cite{Cra24}, which focuses on contemporary techniques. One research theme that has gained momentum is the study of not only the colorings themselves but also the \emph{reconfiguration graphs} that connect them. In a standard $k$-coloring graph, vertices represent proper $k$-colorings, and edges correspond to changing the color of a single vertex. These graphs retain much of the structure of the underlying graph. Indeed, recent work has shown that coloring graphs can serve as complete graph invariants \cites{HSTT24, BBHvdHHP25, AKL25}.

In this paper, we study a compressed version of a coloring graph obtained by treating the color classes as indistinguishable. In this model, we retain the partition of vertices into independent sets but `forget' the specific color labels assigned to each set. Given a base graph $G$ and $k\in\mathbb{N}$, we define the \emph{Bell $k$-coloring graph} $\mathcal{B}_k(G)$ as follows.
\begin{itemize}
    \item The vertices of $\mathcal{B}_k(G)$ are partitions of the vertex set $V(G)$ into $k$ independent sets (some possibly empty); we call such partitions \emph{stable} $k$-partitions.
    \item Two distinct partitions $P_1$ and $P_2$ are adjacent if they differ only by the placement of a single vertex $v\in V(G)$, formalized as $P_1-v = P_2-v$.
\end{itemize}
The notation $P-v$ means the partition $P$ with the vertex $v$ removed from its part; that is, $P-v$ is the restriction of $P$ to $G-v$. Figures~\ref{fig:Bell3-K13}~and~\ref{fig:Bell3-K3-plus-K1} depict examples of Bell coloring graphs with vertices labeled to indicate the corresponding partitions and edges labeled to indicate the vertex or vertices responsible for each edge.

The name `Bell coloring graph' is motivated by the Bell numbers, which count the total number of set partitions. This object, also studied by Haas \cite{Haa12} as the `isomorphic color graph', interpolates between enumerative invariants (graphical Bell numbers \cite{DP09}) and reconfiguration structure. In contrast to standard coloring graphs, only a handful of papers (e.g., \cite{Haa12}, \cite{FM14}) address Bell coloring graphs.

\begin{figure}[h]
\centering
\begin{tikzpicture}[
    vertex/.style={circle, draw, fill=white, minimum size=8mm, inner sep=0pt, font=\small},
    edge label/.style={midway, fill=white, font=\footnotesize, inner sep=1.5pt},
    part label/.style={font=\scriptsize, color=black, align=center},
    x={(-0.6cm, -0.4cm)}, y={(1cm, 0cm)}, z={(0cm, 1cm)},
    scale=1.2
]

\coordinate (P1) at (0, -2, 0);
\coordinate (P2) at (2, 1, 0);
\coordinate (P3) at (-2, 1, 0);
\coordinate (P4) at (0, 0, 3);

\draw (P1) -- (P2) node[edge label, pos=0.53] {$v_3$};
\draw (P1) -- (P3) node[edge label, pos=0.55] {$v_2$};
\draw (P2) -- (P3) node[edge label] {$v_1$};
\draw (P4) -- (P1) node[edge label, pos=0.5] {$v_1$};
\draw (P4) -- (P2) node[edge label, pos=0.4] {$v_2$};
\draw (P4) -- (P3) node[edge label, pos=0.6] {$v_3$};

\node[vertex] at (P1) {$P_1$};
\node[part label, below left=0.3cm of P1] {$\{\{u\}, \{v_2, v_3\}, \{v_1\} \}$};

\node[vertex] at (P2) {$P_2$};
\node[part label, below=0.4cm of P2] {$\{\{u\}, \{v_1, v_3\}, \{v_2\} \}$};

\node[vertex] at (P3) {$P_3$};
\node[part label, below right=0.3cm of P3] {$\{\{u\}, \{v_1, v_2\}, \{v_3\} \}$};

\node[vertex] at (P4) {$P_4$};
\node[part label, above=0.4cm of P4] {$\{\{u\}, \{v_1, v_2, v_3\}, \emptyset\}$};

\end{tikzpicture}
\caption{Illustration of $\mathcal{B}_3(K_{1,3})\cong K_4$. The claw graph $K_{1, 3}$ has vertex set $\{u, v_1, v_2, v_3\}$ and three edges $uv_i$ for $i=1,2,3$. Edge labels indicate the vertices responsible for the adjacency.}
\label{fig:Bell3-K13}
\end{figure}

\begin{figure}[h]
\centering
\begin{tikzpicture}[
    vertex/.style={circle, draw, fill=white, minimum size=8mm, inner sep=0pt, font=\small},
    edge label/.style={midway, fill=white, font=\footnotesize, inner sep=1.5pt},
    part label/.style={font=\scriptsize, color=black, align=center},
    x={(-0.6cm, -0.4cm)}, y={(1cm, 0cm)}, z={(0cm, 1cm)},
    scale=1.2
]

\coordinate (P1) at (0, -2, 0);
\coordinate (P2) at (2, 1, 0);
\coordinate (P3) at (-2, 1, 0);
\coordinate (P4) at (0, 0, 3);

\draw (P1) -- (P2) node[edge label, pos=0.53] {$w$};
\draw (P1) -- (P3) node[edge label, pos=0.55] {$w$};
\draw (P2) -- (P3) node[edge label] {$w$};
\draw (P4) -- (P1) node[edge label, pos=0.5] {$w, v_1$};
\draw (P4) -- (P2) node[edge label, pos=0.4] {$w, v_2$};
\draw (P4) -- (P3) node[edge label, pos=0.6] {$w, v_3$};

\node[vertex] at (P1) {$P_1$};
\node[part label, below left=0.3cm of P1] {$\{\{w, v_1\}, \{v_2\}, \{v_3\}, \emptyset\}$};

\node[vertex] at (P2) {$P_2$};
\node[part label, below=0.4cm of P2] {$\{\{w, v_2\}, \{v_1\}, \{v_3\}, \emptyset\}$};

\node[vertex] at (P3) {$P_3$};
\node[part label, below right=0.3cm of P3] {$\{\{w, v_3\}, \{v_1\}, \{v_2\}, \emptyset \}$};

\node[vertex] at (P4) {$P_4$};
\node[part label, above=0.4cm of P4] {$\{\{w\}, \{v_1\}, \{v_2\}, \{v_3\}\}$};

\end{tikzpicture}
\caption{Illustration of $\mathcal{B}_4(K_{3}\sqcup K_1)\cong K_4$. The graph $K_3\sqcup K_1$ has vertex set $\{v_1, v_2, v_3, w\}$ with edges $v_1v_2, v_2v_3, v_3v_1$. Edge labels indicate the vertices responsible for the adjacency.}
\label{fig:Bell3-K3-plus-K1}
\end{figure}

Our work is motivated in part by the question of which graphs can arise as Bell coloring graphs. Every coloring graph is a Bell coloring graph because $\B_k(G\sqcup K_k) \cong \mathcal{C}_k(G)$ for any graph $G$ and integer $k\ge 1$ \cite{FM14}*{Proposition 2.5}. However, the class of graphs realizable as standard coloring graphs is quite limited. For example, $K_1$ and $K_2$ are the \emph{only} trees that are realizable as coloring graphs, and $C_3$, $C_4$, and $C_6$ are the \emph{only} cycles that are realizable as coloring graphs \cite{BFHRS16}*{Theorems 7 and 23}. In contrast, we prove the following:

Graph coloring is a vibrant area of research that bridges theoretical questions and practical applications. Several excellent references survey both the classic landscape and modern frontiers; see, for example, the monograph \cite{JT95}, the themed collection \cite{BW15}, and the recent text \cite{Cra24}, which focuses on contemporary techniques. One research theme that has gained momentum is the study of not only the colorings themselves but also the \emph{reconfiguration graphs} that connect them. In a standard $k$-coloring graph, vertices represent proper $k$-colorings, and edges correspond to changing the color of a single vertex. These graphs retain much of the structure of the underlying graph. Indeed, recent work has shown that coloring graphs can serve as complete graph invariants \cites{HSTT24, BBHvdHHP25, AKL25}.

In this paper, we study a compressed version of a coloring graph obtained by treating the color classes as indistinguishable. In this model, we retain the partition of vertices into independent sets but `forget' the specific color labels assigned to each set. Given a base graph $G$ and $k\in\mathbb{N}$, we define the \emph{Bell $k$-coloring graph} $\mathcal{B}_k(G)$ as follows.
\begin{itemize}
 \item The vertices of $\mathcal{B}_k(G)$ are partitions of the vertex set $V(G)$ into $k$ independent sets (some possibly empty); we call such partitions \emph{stable} $k$-partitions.
 \item Two distinct partitions $P_1$ and $P_2$ are adjacent if they differ only by the placement of a single vertex $v\in V(G)$, formalized as $P_1-v = P_2-v$.
\end{itemize}
The notation $P-v$ means the partition $P$ with the vertex $v$ removed from its part; that is, $P-v$ is the restriction of $P$ to $G-v$. Figures~\ref{fig:Bell3-K13}~and~\ref{fig:Bell3-K3-plus-K1} depict examples of Bell coloring graphs with vertices labeled to indicate the corresponding partitions and edges labeled to indicate the vertex or vertices responsible for each edge.

The name `Bell coloring graph' is motivated by the Bell numbers, which count the total number of set partitions. This object, also studied by Haas \cite{Haa12} as the `isomorphic color graph', interpolates between enumerative invariants (graphical Bell numbers \cite{DP09}) and reconfiguration structure. In contrast to standard coloring graphs, only a handful of papers (e.g., \cite{Haa12}, \cite{FM14}) address Bell coloring graphs.

\end{tikzpicture}
\caption{Illustration of $\mathcal{B}_3(K_{1,3})\cong K_4$. The claw graph $K_{1, 3}$ has vertex set $\{u, v_1, v_2, v_3\}$ and three edges $uv_i$ for $i=1,2,3$. Edge labels indicate the vertices responsible for the adjacency.}
\label{fig:Bell3-K13}
\end{figure}

\end{tikzpicture}
\caption{Illustration of $\mathcal{B}_4(K_{3}\sqcup K_1)\cong K_4$. The graph $K_3\sqcup K_1$ has vertex set $\{v_1, v_2, v_3, w\}$ with edges $v_1v_2, v_2v_3, v_3v_1$. Edge labels indicate the vertices responsible for the adjacency.}
\label{fig:Bell3-K3-plus-K1}
\end{figure}

Our work is motivated in part by the question of which graphs can arise as Bell coloring graphs. Every coloring graph is a Bell coloring graph because $\B_k(G\sqcup K_k) \cong \mathcal{C}_k(G)$ for any graph $G$ and integer $k\ge 1$ \cite{FM14}*{Proposition 2.5}. However, the class of graphs realizable as standard coloring graphs is quite limited. For example, $K_1$ and $K_2$ are the \emph{only} trees that are realizable as coloring graphs, and $C_3$, $C_4$, and $C_6$ are the \emph{only} cycles that are realizable as coloring graphs \cite{BFHRS16}*{Theorems 7 and 23}. In contrast, we prove the following:

Our work is motivated in part by the question of which graphs can arise as Bell coloring graphs. Every coloring graph is a Bell coloring graph because $\B_k(G\sqcup K_k) \cong \mathcal{C}_k(G)$ for any graph $G$ and integer $k\ge 1$ \cite{FM14}*{Proposition 2.5}. However, the class of graphs realizable as standard coloring graphs is quite limited. For example, $K_1$ and $K_2$ are the \emph{only} trees that are realizable as coloring graphs, and $C_3$, $C_4$, and $C_6$ are the \emph{only} cycles that are realizable as coloring graphs \cite{BFHRS16}*{Theorems 7 and 23}. In contrast, we prove the following:

We prove Theorem~\ref{thm:trees-cycles} by showing that a more general class of graphs, namely certain reconfiguration graphs of matchings, are Bell coloring graphs.

\textbf{Outline of the paper.} Section~\ref{sec:preliminaries} introduces Bell $k$-coloring graphs, gives small examples, and develops basic tools for understanding adjacency, including a description of edges that are realized by two vertices. In Section~\ref{sec:cliques}, we classify cliques in Bell coloring graphs and apply this to show that $K_4-e$ is not a Bell coloring graph and that $K_n-e$ is not an induced subgraph of any Bell coloring graph for $n\ge 6$. Section~\ref{sec:matchings} introduces a matching reconfiguration graph and shows that, for triangle-free graphs, it coincides with a Bell coloring graph; this yields realizations of all trees and all cycles as Bell coloring graphs. Section~\ref{sec:tree-reconstruction} proves that trees are reconstructible from their Bell
$3$-coloring graphs. Section~\ref{sec:reconstruction-multigraph} proves the multigraph reconstruction theorem: the Bell $n$-coloring multigraph determines $G$ up to universal vertices.

In this section, we introduce the matching reconfiguration graph and establish its connection to Bell coloring graphs. As a consequence, we obtain Theorem~\ref{thm:trees-cycles}, stating that all trees and cycle graphs are realizable as Bell coloring graphs.

The isomorphism in Proposition~\ref{prop:matchings} enables us to realize all cycles and trees as Bell coloring graphs. We focus on the case where $n=|V(G)|=2k+1$. More formally, if $n=2k+1$ then we call $\mathcal{M}_k(G)$ the \emph{near-perfect matching graph} of $G$. The following lemma gives three equivalent conditions for two near-perfect matchings to be adjacent in a matching graph, which helps us reason about matching graphs that are cycles. In particular, it shows that near-perfect matchings are adjacent if and only if the unmatched vertices $u$ and $v$ have a common neighbor $w$ such that the symmetric difference of the matching is $\{uw,uv\}$.

Having shown that all cycles are realizable as near-perfect matching graphs and hence Bell coloring graphs, we turn to the task of realizing all trees as Bell coloring graphs. To do this, we first describe a procedure for joining near-perfect matching graphs to create larger near-perfect matching graphs. This joining construction relies on a particular property of odd-order graphs. For a graph $G$ on $n=2k+1$ vertices, a vertex $v$ is \emph{uniquely unmatched} if the graph $G\setminus v$ has a unique perfect matching. For two such graphs $H_1$ and $H_2$, Figure~\ref{fig:npm} depicts a near-perfect matching of these graphs joined on their uniquely unmatched vertices. We show that the near-perfect matching graph of this joined graph is exactly the joined near-perfect matching graphs of $H_1$ and $H_2$.

\begin{theorem}\label{thm:trees-cycles}
 All trees and all cycle graphs are realizable as Bell coloring graphs.
\end{theorem}

This section proves Theorem~\ref{thm:tree-reconstruction} through a sequence of lemmas that yield an algorithmic procedure for reconstructing a tree given its Bell 3-coloring graph. First, we prove that three categories of trees can be differentiated based on the degree sequence of their Bell coloring graphs, and then we show that within each category, other Bell coloring graph features determine the specific tree. As we can manually verify the result for all trees of order at most $5$, we assume $n\geq 6$ throughout this section.

\begin{theorem}\label{thm:tree-reconstruction}
For trees $T_1$ and $T_2$, we have $\B_3(T_1)\cong \B_3(T_2)$ if and only if $T_1\cong T_2$.
\end{theorem}

\begin{theorem}\label{thm:trees-cycles}
    All trees and all cycle graphs are realizable as Bell coloring graphs.
\end{theorem}

We prove Theorem~\ref{thm:trees-cycles} by showing that a more general class of graphs, namely certain reconfiguration graphs of matchings, are Bell coloring graphs. In particular, Section~\ref{sec:matchings} introduces a matching reconfiguration graph and shows that, for triangle-free graphs, it coincides with a Bell coloring graph; this yields realizations of all trees and all cycles as Bell coloring graphs.

\begin{theorem}\label{thm:trees-cycles}
    All trees and all cycle graphs are realizable as Bell coloring graphs.
\end{theorem}
All trees and all cycles are realizable as Bell coloring graphs.

Question: Let \(H\) be a graph that is either a tree or a cycle graph. A graph is said to be realizable as a Bell coloring graph if there exist a graph \(G\) and an integer \(k\ge 1\) such that \(H\cong \mathcal{B}_k(G)\), where \(\mathcal{B}_k(G)\) is the graph whose vertices are the stable \(k\)-partitions of \(V(G)\) (that is, partitions of \(V(G)\) into \(k\) independent sets, some possibly empty), and where two such partitions \(P_1\) and \(P_2\) are adjacent when they differ only in the placement of a single vertex, equivalently \(P_1-v=P_2-v\) for some vertex \(v\in V(G)\). Which statement holds for every such \(H\)?

Every tree and every cycle graph is realizable as a Bell coloring graph; equivalently, for each such graph \(H\), there exist a graph \(G\) and an integer \(k\ge 1\) with \(H\cong \mathcal{B}_k(G)\).

Choices:
B. Every tree is realizable as a Bell coloring graph, but among cycle graphs only the even cycles are realizable; equivalently, for each tree or even cycle graph \(H\), there exist a graph \(G\) and an integer \(k\ge 1\) with \(H\cong \mathcal{B}_k(G)\).
C. Every tree is realizable as a Bell coloring graph; equivalently, for each tree \(H\), there exist a graph \(G\) and an integer \(k\ge 1\) with \(H\cong \mathcal{B}_k(G)\).
D. For every graph \(H\) that is either a tree or a cycle graph, there exists an integer \(k\ge 1\) such that for every such \(H\) one can find a graph \(G\) with \(H\cong \mathcal{B}_k(G)\).
E. Every tree and every cycle graph is realizable as a Bell coloring graph; moreover, for each such graph \(H\), one may always choose \(G=H\) and some integer \(k\ge 1\) so that \(H\cong \mathcal{B}_k(H)\).

existence of some realizing graph G replaced by G=H