Non-reconstructible locally finite graphs

Abstract

Two graphs G and H are hypomorphic if there exists a bijection ϕ :Nash-Williams proved that all locally finite connected graphs with a finite number ě 2 of ends are reconstructible, and asked whether locally finite connected graphs with one end or countably many ends are also reconstructible.In this paper we construct non-reconstructible connected 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.2010 Mathematics Subject Classification. 05C60, 05C63.

A counterexample to the reconstruction conjecture for locally finite trees

“…• p 1 pointing in T n towards the root r(T n ), with L 1 =R n , • p 2 pointing in S n towards the root r(S n ), with L 2 =B n , • p 3 pointing inT n towards the root r(T n+1 ), with L 3 = {r(T n+1 ), y}, • p 4 pointing inS n towards the root r(S n+1 ), with L 4 = {r(S n+1 ), g}. (6) Note that our construction so far has been tailored to provide us with a P -respecting isomorphism…”

“…However, they conjectured that the Reconstruction Conjecture should hold for locally finite trees. In our paper [6], we extend the methods developed in the present paper to construct counterexamples to Problems 1.8 and 1.9.…”

“…In our paper , we extend the methods developed in the present paper to construct counterexamples to Problems and .…”