Which of the graphs are isomorphic (GRE math subject test)

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Could you guys take a look at the following two screenshots? I believe I, II, and III are all isomorphic to each other, while the answer only gives I and III.

For II, consider mapping ABCD to EGFH, through which, I believe, it gives both a bijection of vertices while preserving their adjacencies.

Can someone confirm me that the answer is incorrect, or can someone tell me where I am wrong?

Thanks

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