Switching reconstruction conjecture

Importance: Medium ✭✭
Author(s): Stanley, Richard P.
Subject: Graph Theory
Keywords: reconstruction
Recomm. for undergrads: no
Posted by: fhavet
on: March 7th, 2013

\begin{conjecture} Every simple graph on five or more vertices is switching-reconstructible. \end{conjecture}

To \emph{switch} a vertex of a simple graph is to exchange its sets of neighbours and non-neighbours. The graph so obtained is called a \emph{switching} of the graph. The collection of switchings of a graph G is called the \emph{switching deck} of $G$. A graph is \emph{switching-reconstructible} if every graph with the same deck as $G$ is isomorphic to $G$.

There are four pairs of non-isomorphic graphs of order $4$ with the same switching deck. One of them consists of the empty graph and the $4$-cycle.

Stanley [S] proved that a graph on $n$ vertices is switching-reconstructible if $n \not\equiv 0 (\mod 4)$.

An \OPrefnum[analogous problem]{46952} was posed for digraphs. Instead of complementing the edges at a vertex, one reverses each of its incident arc.

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*[S] R. P. Stanley Reconstruction from vertex-switching. J. Combin. Theory Ser. B, 38 (1985), 132--138.

% Example: %*[B] Claude Berge, Farbung von Graphen, deren samtliche bzw. deren ungerade Kreise starr sind, Wiss. Z. Martin-Luther-Univ. Halle-Wittenberg Math.-Natur. Reihe 10 (1961), 114. % %[CRS] Maria Chudnovsky, Neil Robertson, Paul Seymour, Robin Thomas: \arxiv[The strong perfect graph theorem]{math.CO/0212070}, % Ann. of Math. (2) 164 (2006), no. 1, 51--229. \MRhref{MR2233847} % % (Put an empty line between individual entries)

* indicates original appearance(s) of problem.