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\title{An elementary proof\\ of the reconstruction conjecture}
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\author{Jo Firstauthor\authornote{1}
\and
Jim Secondauthor\authornote{2}
\and
Janet Thirdauthor\authornote{1,2}
}
\authortext{1}{Department of Mathematics, University of Sutherland, Reay, Scotland
(\email{j.firstauthor@uos.ac.uk}, \email{j.thirdauthor@uos.ac.uk}).}
\authortext{2}{Stochastics Department, Boyle County College, Danville,
Kentucky, USA (\email{jsecond@bcc.edu}).}
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\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 Paul Kelly (1957) and Stanis\l aw Ulam (1960). 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 Kelly~\cite{Kelly} and Ulam~\cite{Ulam}. The reconstruction
conjecture is widely studied
\cite{Bollobas,FGH,HHRT,KSU,Stockmeyer,WS} and is very interesting.
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 by well-ordering there is a smallest planar graph that is
not fabulous. It is not the trivial graph, and we can easily see
that the property of being not fabulous is preserved by edge contraction.
This gives 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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%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
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\subsection*{Acknowledgements}
Thanks to Professor Qwerty for suggesting the proof of
Lemma~\ref{lem:Technical}.
Jim Secondauthor received support from NASA grant MOON1969.
%BIBLIOGRAPHY
% You do not have to use the same format for your references, but
% include everything in this file.
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\begin{thebibliography}{99}
\bibitem{Bollobas} B. Bollob{\'a}s. Almost every
graph has reconstruction number three. \emph{J. Graph Theory},
14(1): 1--4, 1990.
\bibitem{BH} J.\,A. Bondy and R. Hemminger,
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.
A simpler counterexample to the reconstruction conjecture for
denumerable graphs. \emph{J. Combinatorial Theory, Ser. B},
12: 203--204, 1972.
\bibitem{HHRT} E. Hemaspaandra, L.\,A. Hemaspaandra,
S.~P. Radziszowski, and R. Tripathi.
Complexity results in graph reconstruction. \emph{Discrete
Appl. Math.}, 155(2): 103--118, 2007.
\bibitem{Kelly} P.\,J. Kelly. A congruence theorem for
trees. \emph{Pacific J. Math.}, 7: 961--968, 1957.
\bibitem{KSU} M. Kiyomi, T. Saitoh, and R. Uehara.
Reconstruction of interval graphs. In
\emph{Computing and combinatorics}, volume 5609 of
\emph{Lecture Notes in Comput. Sci.}, pages 106--115. Springer, 2009.
\bibitem{Stockmeyer} P.~K. Stockmeyer. The falsity of the
reconstruction conjecture for tournaments. \emph{J. Graph
Theory}, 1(1):19--25, 1977.
\bibitem{Ulam} S.\,M. Ulam. A collection of
mathematical problems. Interscience Tracts in Pure and
Applied Mathematics, no. 8. Interscience Publishers, New
York-London, 1960.
\bibitem{WS} D.~B. West and H. Spinoza.
Reconstruction from $k$-decks for graphs with maximum degree~2.
\arxiv{1609.00284vi}, 2016.
\end{thebibliography}
\end{document}