Graph reconstruction—a survey

Abstract

The Reconstruction Conjecture asserts that every finite simple undirected graph on three or more vertices is determined, up to isomorphism, by its collection of vertex-deleted subgraphs. This article reviews the progress made on the conjecture since it was first formulated in 1941 and discusses a number of related questions.The Reconstruction Conjecture is generally regarded as one of the foremost unsol;ed problems in graph theory. Indeed, Harary (1969) has even classified it as a "graphical disease" because of its contagious nature. According to reliable sources, it was discovered in Wisconsin in 1941 by Kelly and Ulam, and claimed its first victim (P. J. Kelly) in 1942.* There are now more than sixty recorded cases, and relapses occur frequently (this article being a case in point). Our purpose here is to describe and analyse the current status of the disease, identify its more interesting variants, and suggest possible remedies.We shall, €or the most part, use the terminology and notation of Bondy and Murty;? so a graph G has vertex set V(G), edge set E ( G ) , v ( G ) vertices and E ( G ) edges. A subgraph of G obtained by deleting a vertex v together with its incident edges will be referred to as a vertex-deleted subgraph and denoted by G, (rather than G -v ) . Figure 1 exhibits the vertex-deleted subgraphs of a graph.

Identification of Resistors in Electrical Networks

“…In the graph theory, problems involving graph identification with a partial information have been among the most important and famous open problems ( [6]). Most of the work on this subject has concentrated on spectral graph theory (see [4], [8], [12], [9], [11], [10], [13], [14] and [16]).…”

On the complexity of graph reconstruction

“…The reconstruction conjecture states: Any graph with at least three vertices can be reconstructed from the collection of its one-vertex-deleted subgraphs, It is widely viewed as one of the most interesting and challenging open problems in graph theory, and has generated many excellent surveys [15,19,5,33]. As noted above, the first result in this field was Kelly's proof that the conjecture is true when restricted to trees; i.e., trees are reconstructible [23].…”

HEREDITARY HEMIMORPHY OF {-Κ}-Hemimorphic TOURNAMENTS FOR ≥ 5

“…Say then that a tournament is strongly connected when it admits at most a single strongly connected component. For example, the tournament C 3 = ({1, 2, 3}, { (1,2), (2,3), (3, 1)}) is strongly connected. However, the two tournaments D 1 = ({1, 2, 3, 4}, { (1,2), (2,3), (3,1), (1,4), (2,4), (3,4)}) and D 2 = D * 1 , are non-strongly connected and admit two strongly connected components: {1, 2, 3} and {4}.…”