for my $node ($graph->nodes) {
  my @paths = $graph->find_shortest_paths_starting_from($node);

  # Now, suppose @paths is an array containing arrays of
  # nodes that are the actual nodes on the shortest paths,
  # with the first being $node and the last being the end of
  # the path. For instance, [ $node, $other_node, ..., $end_node ] 
  # for the shortest path from $node to $end_node.

  # Next, for each endpoint, try to find a path back, but
  # remove the nodes from consideration that are on the
  # known shortest path. This is to prevent the algorithm
  # from re-discovering the shortest path, and will ensure
  # that the two possible found paths are mutually exclusive,
  # i.e. do not contain same nodes. (Otherwise you would have
  # an even shorter cycle... could this be used for our benefit?)
  for my $path (@paths) {
    my $new_graph = $graph->copy;
    # Slice away the first and last nodes, as they are still of
    # interest to us.
    $new_graph->remove_nodes(@$path[1 .. $#{@$path}-1]);
    my @paths_back = $new_graph->find_shortest_paths_starting_from($path->[-1]);

    for my $path_back (@paths_back) {
        if ($path_back->[-1] eq $node) {
          print "We have a cycle involving $node!\n";
        }
    }
  }
}