A path of length n>=4 is a counterexample.
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proxximus
A theorem says that a graph G of order 3+ is connected iff G has two distinct vertices u and v such that G-u and G-v are connected.
Is it true then that Every connected graph G or order 4+ is connected iff G has three distinct vertices u, v, and w such that G-u, G-v, and G-w are connected?