Well, I'd do it by leaving in the undef statement. Either that, or replace it with something commented:

  {
    ...
    $give_id->($start_node);
    $give_id = 0;  # Break circular structure and avoid memory leak
  }
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That is, just make sure to set $give_id to some value that doesn't depend on $give_id before $give_id goes out of scope.