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Every connected graph $G$ of order $4$ or more contains three distinct vertices $u$, $v$ and $w$ such that $G-u$, $G-v$ and $G-w$ are connected.

Why is this statement false? Is there a counterexample or something contradictory about this statement?

I know that if we let $G$ be a graph of order $3$ or more. Then $G$ is connected if and only if $G$ contains two distinct vertices $u$ and $v$ such that $G-u$ and $G-v$ are connected.

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What is the simplest connected graph on 4 vertices?

Does this satisfy the statement?

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