Non-reconstructible locally finite graphs

TL;DR

This paper constructs examples of locally finite graphs with one or countably many ends that are not reconstructible, answering Nash-Williams' questions negatively about the reconstructibility of such graphs.

Contribution

It provides the first known examples of non-reconstructible locally finite graphs with one or countably many ends, extending the understanding of graph reconstruction.

Findings

Constructed non-reconstructible graphs with one end

Constructed non-reconstructible graphs with countably many ends

Answered Nash-Williams' questions negatively

Abstract

Two graphs $G$ and $H$ are \emph{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 \emph{reconstructible} if $H \cong G$ for all $H$ hypomorphic to $G$. Nash-Williams proved that all locally finite graphs with a finite number $\geq 2$ of ends are reconstructible, and asked whether locally finite graphs with one end or countably many ends are also reconstructible. In this paper we construct non-reconstructible graphs of bounded maximum degree with one and countably many ends respectively, answering the two questions of Nash-Williams about the reconstruction of locally finite graphs in the negative.