A counterexample to the reconstruction conjecture for locally finite trees

Abstract

Abstract. Two graphs G and H are hypomorphic if there exists a bijection ϕ :

Non-reconstructible locally finite graphs

“…Informally, we think of ℓ P L i as the 'promise' that if we extend G to a graph H Ą G, we will do so in such a way that Hp q ℓ q -Hp p i q. In [4], given a promise structure P "´G, P , L¯, it is shown how to construct a graph clpGq Ą G which has the following properties. .…”

“…In order to glue F n`1˝N onto G 1 n`1 and H 1 n`1 we will need to show that that analogues of (:14) and (:15) hold for G 1 n`1 , H 1 n`1 and F n`1 . Our next lemma is essentially [4,Claim 4.13], and is an analogue of (:14). We briefly remind the reader of the details, as we need to know the nature of our extensions in our later claims.…”

“…However, very recently, the present authors have constructed a counterexample to the conjecture of Harary, Schwenk and Scott. Theorem 1.1 (Bowler, Erde, Heinig, Lehner, Pitz [4]). There exists a non-reconstructible tree of maximum degree three.…”

“…Theorem 1.1 (Bowler, Erde, Heinig, Lehner, Pitz [4]). There exists a non-reconstructible tree of maximum degree three.…”

“…We note that this is a slightly different definition of an bare extension to that in[4] 4. If the statement involves an object indexed by n´1 we only require that it holds for n ě 1.…”