Domino Crime

You are a detective investigating a crime scene. On the floor lie dominoes--someone arranged them and induced a toppling chain-reaction. The policemen are about to collect the dominoes to identify the suspect's fingerprints. Your sidekick, Dr. Watson, claims,

"We don't have to check all the dominoes. First pick a random domino, and keep looking back at the domino it falls under until one that has no domino topples on. The suspect must have manually pushed that, which then will contain his/her fingerprint."

Here are three configurations of dominoes found at the crime scene. The nodes represent dominoes, and the arrows from \(u\) to \(v\) indicate domino \(u\) toppled on domino \(v\).

Which of these fails Watson's method?

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