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
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print "strict graph {\n";
while (<DATA>) {
    my ($node, @neighbours) = split;
    print "    $node -- $_\n" for @neighbours;
}
print "}\n";
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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.