$start = {
           'lineitem' => {
                           'invoice' => 1
                         },
           'market_x_branch' => {
                                  'insurer' => 1
                                },
           'invoice' => {
                          'lineitem' => 1
                        },
           'insurer' => {
                          'market_x_branch' => 1
                        }
};

##</code><code>##

$end = [
          [
            'lineitem',
            'invoice'
          ],
          [
            'market_x_branch',
            'insurer'
          ]
        ];

##</code><code>##

foreach my $x (keys %$start)
{
    next unless exists $start->{$x};

    my @values = ($x, keys %{$start->{$x}});
    push @$end, \@values;
    delete $start->{$_} for @values;
}

##</code><code>##

$start = {
          'lineitem' => {
                          'invoice' => 1
                        },
          'invoice' => {
                         'lineitem' => 1,
                         'claim' => 1
                       },
          'insurer' => {
                         'claim' => 1
                       },
          'claim' => {
                       'invoice' => 1,
                       'insurer' => 1
                     }
        };

$end = [
    [
        'invoice',
        'claim',
        'insurer',
        'lineitem',
     ],
];