Graphs without vertices of degree n-1 are uniquely determined by their Bell colouring graphs, which encode partitions into independent sets.
Then for eachi∈ {1,2}we haveB ≥ki(Gi) ∼= B(Gi), and so the claim follows by Theorem 1.4
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Reconstructing a graph from its Bell colouring graph
Graphs without vertices of degree n-1 are uniquely determined by their Bell colouring graphs, which encode partitions into independent sets.