Reconstruction of infinite locally finite connected graphs

Let G be an infinite locally finite connected graph. We study the reconstructibility of G in relation to the structure of its end set E(G). We prove that an infinite locally finite connected graph G is reconstructible if there exists a finite family (Omega(i))(0less than or equal toi<n) (n greater than or equal to 2) of pairwise finitely separable subsets of E(G) such that, for all x, y, x(1), y(1) is an element of V(G) and every isomorphism f of G - {x, y} onto G - {x(1), y(1)} there is a permutation pi of {0,...,n - 1} such that (f) over tilde(Omega(i)) = Omega(pi(i)) for 0 less than or equal to i < n. From this theorem we deduce, as particular consequences, that G is reconstructible if it satisfies one of the following properties: (i) G contains no end-respecting subdivision of the dyadic tree and has at least two ends of maximal order; (ii) the set of thick ends or the one of thin ends of G is finite and of cardinality greater than one. We also prove that if almost all vertices of G are cutvertices, then G is reconstructible, if it contains a free end or if it has at least a vertex which is not a cutvertex. (C) 2003 Elsevier B.V. All rights reserved.