2. An Exercise in Graph Theory

An Exercise in Graph Theory
Elmar Guseinov
May 2021
The following exercise concerns the reconstruction problem [1]. Before reading the solution, try to do it by yourself.
Problem. Find a pair of non-isomorphic graphs G and H, such that:
1. V (G) = {g 1 , ..., g n }, V (H) = {h 1 , ..., h n }, n ≥ 3
2. deg(g 1 ) = deg(h 1 ), ..., deg(g n ) = deg(h n )
3. There are isomorphisms between vertex-deleted subgraphs G − g 1
∼ = H − h1 , G − g 2
∼ = H − h2
Solution
It’s natural to try to define some infinite class of componentwise non-isomorphic pairs (G, H) satisfying conditions 1-3.
Yet another generalization is to define an infinite sequence of pairs (G i , H i ), such that:
1. V (G i ) = {g 1 , ..., g n }, V (H i ) = {h 1 , ..., h n }, n ≥ max{i + 1, 3}
2. deg(g 1 ) = deg(h 1 ), ..., deg(g n ) = deg(h n )
3. There are isomorphisms between vertex-deleted subgraphs G − g 1
∼ = H − h1 , ..., G − g i+1
∼ = H − hi+1 ”.
Given this sequence, if the condition n = i + 1 holds for some i ∈ N, the reconstruction conjecture fails.
References
1. https://en.wikipedia.org/wiki/Reconstruction conjecture