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\title{An elementary proof\\ of the reconstruction conjecture}

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\author[1]{First Author\thanks{Supported by NASA grant ABC123.}}
\author[2]{Some Second Author}
\author[3]{Some Third Author}

\affil[1]{Department of Inconsequential Studies, Solatido College, North Kentucky, U.S.A.\newline
\email{fa@solatido.edu}%
}

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\affil[2,3]{School of Hard Knocks, University of Western Nowhere, Somewhere, Australia\newline
\email{ssa@uwn.edu.au}, \email{sta@uwn.edu.au}%
}



\begin{document}
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\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.
% KEYWORDS (optional)
\keywords{Broglington manifolds}
\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
    % Use \includegraphics to import figures; for example 
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  \caption{Here is an informative figure.\label{fig:InformativeFigure}}
\end{figure}

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\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 
%    include everything in this file.  Don't use natbib please.
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\begin{thebibliography}{1}

\bibitem{Bollobas}
B\'{e}la Bollob\'{a}s, \emph{Almost every graph has reconstruction number
  three}, J. Graph Theory \textbf{14} (1990), no.~1, 1--4. \MR{1037416}

\bibitem{BH}
J.~A. Bondy and R.~L. Hemminger, \emph{Graph reconstruction---a survey}, J.
  Graph Theory \textbf{1} (1977), no.~3, 227--268. \MR{480189}

\bibitem{FGH}
J.~Fisher, R.~L. Graham, and F.~Harary, \emph{A simpler counterexample to the
  reconstruction conjecture for denumerable graphs}, J. Combinatorial Theory
  Ser. B \textbf{12} (1972), 203--204. \MR{295946}

\bibitem{HHRT}
Edith Hemaspaandra, Lane~A. Hemaspaandra, Stanis\l aw~P. Radziszowski, and
  Rahul Tripathi, \emph{Complexity results in graph reconstruction}, Discrete
  Appl. Math. \textbf{155} (2007), no.~2, 103--118. \MR{2287413}

\bibitem{Kelly}
Paul~J. Kelly, \emph{A congruence theorem for trees}, Pacific J. Math.
  \textbf{7} (1957), 961--968. \MR{87949}

\bibitem{KSU}
Masashi Kiyomi, Toshiki Saitoh, and Ryuhei Uehara, \emph{Reconstruction of
  interval graphs}, Computing and combinatorics, Lecture Notes in Comput. Sci.,
  vol. 5609, Springer, Berlin, 2009, pp.~106--115. \MR{2545774}

\bibitem{WS}
Hannah Spinoza and Douglas~B. West, \emph{Reconstruction from the deck of
  {$k$}-vertex induced subgraphs}, J. Graph Theory \textbf{90} (2019), no.~4,
  497--522. \MR{3915185}

\bibitem{Stockmeyer}
Paul~K. Stockmeyer, \emph{The falsity of the reconstruction conjecture for
  tournaments}, J. Graph Theory \textbf{1} (1977), no.~1, 19--25. \MR{453584}

\bibitem{Ulam}
S.~M. Ulam, \emph{A collection of mathematical problems}, Interscience Tracts
  in Pure and Applied Mathematics, no. 8, Interscience Publishers, New
  York-London, 1960. \MR{0120127}

\end{thebibliography}

\end{document}



