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.

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It is a well-known conjecture of S. M. Ulam that any finite graph of order at least three can be reconstructed from its maximal vertex-deleted subgraphs. Formally (writing G υ for Gv) Ulam's Conjecture states: if G and H are finite graphs of order at least three such that there is a bijection σ: V(G)-> V(H) with the property (1) G Ό s* H aM for all v e V(G) , then G = H. This conjecture has not been proved in general, although it was shown by P. J. Kelly to be true for disconnected graphs and trees and has also been verified for several other classes of graphs. The purpose of this paper is to examine Ulam's Conjecture for infinite graphs. (It is trivial to determine, from any G v , whether or not a graph G is infinite.) Results are obtained which can loosely be viewed as extensions of Kelly's work on disconnected graphs and trees. In §2 it is shown that infinite graphs G and H satisfying (1) must have the same finite components, occurring with the same multiplicity. Corollaries of this are that if G either has only finite components, or has some finitely occurring finite component, then G ~ H. In § 3 the conjecture is proved for m-coherent locally finite trees, where m is finite and greater than one. This furnishes a partial solution to the reconstruction problem for infinite trees, raised by C. St. J. A. Nash-Williams. We have used the language of reconstruction in our proofs. However, it should be noted that the results are existential in nature and not algorithmic. Throughout the paper G and H will denote infinite graphs satisfying condition (1) of Ulam's Conjecture. Any notation and terminology not defined can be found in Harary [3]. 2* Disconnected graphs* We denote by c(G) the number of components of G, and by c(G; K) the number of components of G that are isomorphic to K. A finite connected graph J is called a K-producer if c(J v ; K) > 0 for all ve V(J). (Since J has a non-cutvertex, J must be regular and of order one more than the order of K; hence K determines / up to isomorphism.) An endvertex of G is a vertex of degree one. LEMMA 2.1. If L is infinite and connected and K is finite, then there is an infinite set S £ V(L) such that c(L v ; K) = 0 for all ve S.

We show that if G is a 3-connected graph of order at least seven, then every longest path between distinct vertices in G contains at least two contractible edges. An immediate corollary is that longest cycles in such graphs contain at least three contractible edges.We consider only finite undirected simple graphs. An edge of a 3-connected graph is called contractible if its contraction results in a 3-connected graph; otherwise it is called noncontractible. Note that in a 3-connected graph of order at least five, an edge uu is noncontractible if and only if there exists a 3-cutset containing both u and u. We call such a 3-cutset one associated with the edge uu. Other terminology is as in [2].It easily follows from results of Tutte [5] that every 3-connected graph on at least five vertices contains a contractible edge. Thomassen [4] gave a simpler proof of this (and then used it in his elegant induction proof of Kuratowski's Theorem). Using methods similar to Thomassen's, Dean, Hemminger, and Toft [3] improved on this by showing that, in the same setting, longest (x,y)-paths contain at least one contractible edge if xy is an edge. In this paper, we use different methods to get the best result of this type; namely, with two small excep-