<p>A single vote by a moderator is sufficient for <em>any</em> action: closing, reopening, deletion, undeletion. See <a href="https://meta.stackexchange.com/q/75189">Who are the diamond moderators, and what is their role?</a> </p>

<p>High reputation does not confer this ability. However, having a gold tag badge grants the user a <a href="https://meta.stackexchange.com/a/231212">binding closing vote</a> for <em>duplicate</em> on questions having that tag.</p>

<hr>

<p>Regarding reopening, from <a href="https://meta.stackexchange.com/q/36415">How do you reopen a closed question?</a>: </p>

<blockquote>
  <p>Editing the body of a question within five days of closure will add it to the Reopen Review queue, where people with the the ability to cast reopen votes will assess it. A question can only be added to the Reopen Review queue once per closure via editing.</p>
</blockquote>

<p>So, the edit must change the post body (not just title or tags). The system is unable to judge whether the edit was significant. In my experience, an edit that does nothing but <em>remove</em>  content (distracting off-topic parts, or broken markup that makes the text unreadable) can make a question worth reopening. </p>
