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I know a nonseparable graph is a connected graph with simply no cut vertices. And that a graph of order at least $3$ is nonseperable if and only if every two vertices lie on a common cycle.

I'm not sure how to advance from here though

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Suppose $v$ is a cut vertex. Consider the smallest component $A$ of $G-v$. Each $a \in A$ has at most $(|A|-1)+1=|A| < \frac{n}{2}$ neighbours in $G$, contradiction.

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  • $\begingroup$ Is the contradiction that $|A| < \frac{n}{2}$ when originally we assumed that $deg$ $v$ $\geq \frac{n}{2}$? $\endgroup$ – atherton Mar 6 '14 at 21:03
  • $\begingroup$ The contradiction is that $\deg a < n/2$ when we assumed that $\deg v \geq n/2$ for every vertex. $\endgroup$ – user133281 Mar 6 '14 at 21:05

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