A counterexample to the reconstruction conjecture for locally finite trees

TL;DR

This paper presents a counterexample to a long-standing conjecture claiming all locally finite trees are reconstructible, thereby disproving the conjecture and answering several related open questions in infinite graph theory.

Contribution

The paper constructs the first known counterexample to the Harary-Schwenk-Scott Conjecture for locally finite trees, challenging previous assumptions in graph reconstruction theory.

Findings

Counterexample disproves the conjecture

Answers open questions by Nash-Williams, Halin, and Andreae

Shows not all locally finite trees are reconstructible

Abstract

Two graphs $G$ and $H$ are hypomorphic if there exists a bijection $\varphi \colon V(G) \rightarrow V(H)$ such that $G - v \cong H - \varphi(v)$ for each $v \in V(G)$. A graph $G$ is reconstructible if $H \cong G$ 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.