A counterexample to the reconstruction conjecture for locally finite trees. Issue 4 (26th April 2017) - Bowler, Nathan
Abstract: Two graphs G and H are hypomorphic if there exists a bijection φ : V ( G ) → V ( H ) such that G − v ≅ H − φ ( v ) for each v ∈ V ( G ) . A graph G is reconstructible if H ≅ G for all H hypomorphic to G . It is well known that not all infinite graphs are reconstructible. However, the Harary–Schwenk–Scott Conjecture from 1972 suggests that all locally finite trees are reconstructible. In this paper, we construct a counterexample to the Harary–Schwenk–Scott Conjecture. Our example also answers four other questions of Nash‐Williams, Halin and Andreae on the reconstruction of infinite graphs.