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\title{An elementary proof\\ of the reconstruction conjecture}
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\author{First Author\thanks{Supported by NASA grant ABC123.}\\
\small Department of Inconsequential Studies\\[-0.8ex]
\small Solatido College\\[-0.8ex]
\small North Kentucky, U.S.A.\\
\small\tt first.author@dis.solatido.edu\\
\and
Some Second Author \qquad Some Third Author\\
\small School of Hard Knocks\\[-0.8ex]
\small University of Western Nowhere\\[-0.8ex]
\small Somewhere, Australia\\
\small\tt \{ssa,sta\}@uwn.edu.au}
\begin{document}
\maketitle
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\begin{abstract}
The reconstruction conjecture states that the multiset of
vertex-deleted subgraphs of a graph determines the graph, provided
it has at least 3 vertices. This problem was independently introduced
by Stanis\l aw Ulam (1960) and Paul Kelly (1957). In this paper,
we prove the conjecture by elementary methods.
It is only necessary
to integrate the Lenkle potential of the Broglington manifold over
the quantum supervacillatory measure in order to reduce the set of
possible counterexamples to a small number (less than a trillion).
A simple computer program that implements Pipletti's classification
theorem for torsion-free Aramaic groups with simplectic socles can
then finish the remaining cases.
\end{abstract}
\section{Introduction}
The reconstruction conjecture states that the multiset of unlabeled
vertex-deleted subgraphs of a graph determines the graph, provided it
has at least three vertices. This problem was independently introduced
by Ulam~\cite{Ulam} and Kelly~\cite{Kelly}. The reconstruction
conjecture is widely studied
\cite{Bollobas,FGH,HHRT,KSU,Stockmeyer,WS} and is very interesting
because it is. See \cite{BH} for more about the
reconstruction conjecture.
\begin{definition}
A graph is \emph{fabulous} if \emph{rest of definition here}.
\end{definition}
\begin{theorem}\label{Thm:FabGraphs}
All planar graphs are fabulous.
\end{theorem}
\begin{proof}
Suppose on the contrary that some planar graph is not fabulous.
Then we have a contradiction.
\end{proof}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\section{Broglington Manifolds}
This section describes background information about Broglington
Manifolds.
\begin{lemma}\label{lem:Technical}
Broglington manifolds are abundant.
\end{lemma}
\begin{proof}
A proof is given here.
\end{proof}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\section{Proof of Theorem~\ref{Thm:FabGraphs}}
In this section we complete the proof of Theorem~\ref{Thm:FabGraphs}.
\begin{proof}[Proof of Theorem~\ref{Thm:FabGraphs}]
Let $G$ be a graph. We have
% The align environment for multi-line equations is defined in the amsmath package.
\begin{align}
|X| &= a+b+c \nonumber\\
&= \alpha\beta\gamma.
\end{align}
This completes the proof of Theorem~\ref{Thm:FabGraphs}.
\end{proof}
\begin{figure}[ht]
\centering
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\caption{Here is an informative figure.\label{fig:InformativeFigure}}
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%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\subsection*{Acknowledgements}
Thanks to Professor Qwerty for suggesting the proof of
Lemma~\ref{lem:Technical}.
%BIBLIOGRAPHY
% You do not have to use the same format for your references, but
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% If you use BibTeX to create a bibliography, copy the .bbl file into here.
% \newblock is optional (it adds a little space)
\begin{thebibliography}{99}
\bibitem{Bollobas} B. Bollob{\'a}s. \newblock Almost every
graph has reconstruction number three. \newblock \emph{J. Graph Theory},
14(1):1--4, 1990.
\bibitem{BH} J.~A. Bondy and R. Hemminger,
\newblock Graph reconstruction---a survey.
\emph{J. Graph Theory}, 1:227--268, 1977. \doi{10.1002/jgt.3190010306}.
\bibitem{FGH} J.~Fisher, R.~L. Graham, and F.~Harary. \newblock A
simpler counterexample to the reconstruction conjecture for
denumerable graphs. \newblock \emph{J. Combinatorial Theory, Ser. B},
12:203--204, 1972.
\bibitem{HHRT} E. Hemaspaandra, L.~A. Hemaspaandra,
S.~P. Radziszowski, and R. Tripathi. \newblock
Complexity results in graph reconstruction. \newblock \emph{Discrete
Appl. Math.}, 155(2):103--118, 2007.
\bibitem{Kelly} P.~J. Kelly. \newblock A congruence theorem for
trees. \newblock \emph{Pacific J. Math.}, 7:961--968, 1957.
\bibitem{KSU} M. Kiyomi, T. Saitoh, and R. Uehara.
\newblock Reconstruction of interval graphs. \newblock In
\emph{Computing and combinatorics}, volume 5609 of
\emph{Lecture Notes in Comput. Sci.}, pages 106--115. Springer, 2009.
\bibitem{Stockmeyer} P.~K. Stockmeyer. \newblock The falsity of the
reconstruction conjecture for tournaments. \newblock \emph{J. Graph
Theory}, 1(1):19--25, 1977.
\bibitem{Ulam} S.~M. Ulam. \newblock \newblock {A collection of
mathematical problems}. \newblock Interscience Tracts in Pure and
Applied Mathematics, no. 8. Interscience Publishers, New
York-London, 1960.
\bibitem{WS} D.~B. West and H. Spinoza.
\newblock Reconstruction from $k$-decks for graphs with maximum degree~2.
\newblock \arxiv{1609.00284vi}, 2016.
\end{thebibliography}
\end{document}