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

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\author{Dagwood Remifa\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 remifa@dis.solatido.edu\\
\and
Forgotten Second Author \qquad  Forgotten Third Author\\
\small School of Hard Knocks\\[-0.8ex]
\small University of Western Nowhere\\[-0.8ex]
\small Nowhere, Australasiaopia\\
\small\tt \{fsa,fta\}@uwn.edu.ao
}

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\date{\dateline{Jan 1, 2012}{Jan 2, 2012}\\
\small Mathematics Subject Classifications: 05C88, 05C89}

\begin{document}

\maketitle

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\begin{abstract}
  The reconstruction conjecture states that the multiset of unlabeled
  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 are optional
  \bigskip\noindent \textbf{Keywords:} graph reconstruction
  conjecture; Broglington manifold; Pipletti's classification
\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 independtly introduced
by Ulam~\cite{Ulam} and Kelly~\cite{Kelly}.  The reconstruction
conjecture is widely studied
\cite{Bollobas,FGH,HHRT,KSU,RM,RR,Stockmeyer} and is very interesting
because...... See \cite{WikipediaReconstruction} for more about the
reconstruction conjecture.

\begin{definition} 
  A graph is \emph{fabulous} if....
\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....
\end{proof}

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\section{Broglington Manifolds}

This section describes background information about Broglington
Manifolds.

\begin{lemma}
  \label{lem:Technical}
  Broglington manifolds are abundant.
\end{lemma}

\begin{proof}

\end{proof}

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\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... Hence
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    |X| &= abcdefghijklmnopqrstuvwxyz \nonumber\\
    &= \alpha\beta\gamma
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  This completes the proof of Theorem~\ref{Thm:FabGraphs}.
\end{proof}

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\subsection*{Acknowledgements}
Thanks to Professor Querty for suggesting the proof of
Lemma~\ref{lem:Technical}.

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\begin{thebibliography}{10}

\bibitem{Bollobas} B{\'e}la Bollob{\'a}s.  \newblock Almost every
  graph has reconstruction number three.  \newblock {\em J. Graph
    Theory}, 14(1):1--4, 1990.

\bibitem{WikipediaReconstruction} Wikipedia contributors.  \newblock
  Reconstruction conjecture.  \newblock {\em Wikipedia, the free
    encyclopedia}, 2011.

\bibitem{FGH} J.~Fisher, R.~L. Graham, and F.~Harary.  \newblock A
  simpler counterexample to the reconstruction conjecture for
  denumerable graphs.  \newblock {\em J. Combinatorial Theory Ser. B},
  12:203--204, 1972.

\bibitem{HHRT} Edith Hemaspaandra, Lane~A. Hemaspaandra,
  Stanis{\l}aw~P. Radziszowski, and Rahul Tripathi.  \newblock
  Complexity results in graph reconstruction.  \newblock {\em Discrete
    Appl. Math.}, 155(2):103--118, 2007.

\bibitem{Kelly} Paul~J. Kelly.  \newblock A congruence theorem for
  trees.  \newblock {\em Pacific J. Math.}, 7:961--968, 1957.

\bibitem{KSU} Masashi Kiyomi, Toshiki Saitoh, and Ryuhei Uehara.
  \newblock Reconstruction of interval graphs.  \newblock In {\em
    Computing and combinatorics}, volume 5609 of {\em Lecture Notes in
    Comput. Sci.}, pages 106--115. Springer, 2009.

\bibitem{RM} S.~Ramachandran and S.~Monikandan.  \newblock Graph
  reconstruction conjecture: reductions using complement, connectivity
  and distance.  \newblock {\em Bull. Inst. Combin. Appl.},
  56:103--108, 2009.

\bibitem{RR} David Rivshin and Stanis{\l}aw~P. Radziszowski.
  \newblock The vertex and edge graph reconstruction numbers of small
  graphs.  \newblock {\em Australas. J. Combin.}, 45:175--188, 2009.

\bibitem{Stockmeyer} Paul~K. Stockmeyer.  \newblock The falsity of the
  reconstruction conjecture for tournaments.  \newblock {\em J. Graph
    Theory}, 1(1):19--25, 1977.

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

\end{thebibliography}

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