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while (my ($k,$v) = first %hash)
{
  delete $hash{$_} for $k, @$v;
}

use strict;
use warnings;
use feature 'say';
...
  say "Starting from $k -> [ @$v ]";
  rec_delete %hash => $k;
}

Input hash:{ 1 => [2 .. 7], 2 => [1], 3 => [1], 4 => [1], 5 => [1], 6 
+=> [1], 7 => [1], 8 => [9], 9 => [8] }

Try first then each twice in a row to check that the iteration is rest
+arted
...
     Removing 4 -> [ 1 ]
     Removing 6 -> [ 1 ]
     Removing 7 -> [ 1 ]

While the hash isn't empty
  Take one key (at random) from the hash
    Delete that key and all connected keys