Sounds like you need Dijkstra's Algorithm for graph searching; the order in which it visits those nodes should be the order you're looking for.

True. The problem is SPT_Dijkstra in Graph is sometimes wrong.

#!/usr/bin/perl
use warnings;
use strict;
use feature qw{ say };
use Syntax::Construct qw{ // };

use Graph;

my %graph;
my $first;

# Handle the sample data where there are sometimes less than 4 neighbo
+urs.
while (<DATA>) {
    my ($node, @neighbours) = split;
    $first //= $node;
    my $distance = 1;
    for my $neighbour (@neighbours) {
        if (not exists $graph{$node}{$neighbour}
            or $graph{$node}{$neighbour} < $distance) {
            $_ = $distance for $graph{$node}{$neighbour},
                               $graph{$neighbour}{$node};
        }
        $distance++;
    }
}


my $g = 'Graph::Undirected'->new;
for my $u (keys %graph) {
    for my $v (keys %{ $graph{$u} }) {
        $g->add_weighted_edge($u, $v, $graph{$u}{$v});
    }
}

my $dij = $g->SPT_Dijkstra($first);
my %weights = ($first => 0);
$weights{$_} //= $dij->get_vertex_attribute($_, 'weight') for keys %gr
+aph;
say join ' ', sort { $weights{$a} <=> $weights{$b} }
              keys %weights;

__DATA__
O1 O3 O5 O11 O73
O2 O72 O54 O12 O7
O3 O1 O6 O5 O12
O5 O1 O3
O6 O3
O7 O2
O11 O1
O12 O2 O3
O54 O2
O72 O2
O73 O1
[download]

لսႽ† ᥲᥒ⚪⟊Ⴙᘓᖇ Ꮅᘓᖇ⎱ Ⴙᥲ𝇋ƙᘓᖇ

Smells like the Travelling Salesman, see some discussion here Travelling problem

Basically, you are given a graph by its set of edges. I changed the input to

O1 O3 O5 O11 O73
O2 O72 O54 O12 O7
O3 O1 O6 O5 O12
O5 O1 O3
O6 O3
O7 O2
O11 O1
O12 O2 O3
O54 O2
O72 O2
O73 O1
[download]

Now, you want to turn in into a weighted graph. The original edges will have weight 1. Any combination of two edges (e.g. 01 -- 072) will have weight 2, and so on. Then, you are just searching for the lightest path that visits all the nodes. You did not specify in what node to start.