I don't think you are doing what you intend to. 
my @third_fragments = grep !/$rsite3/, $second_fragments[$i];

[download]
only checks the i'th element of the @second_fragments array, and $i is associated with @first_fragments. Perhaps you meant 
my @third_fragments = grep !/$rsite3/, @second_fragments;

[download]

Also, you should be aware that the vast majority of your scalar invocations are unnecessary. <, - and most other operators enforce scalar context already. And you can count from the end of an array using negative indices, thus you could have written e.g. 

$first_fragments[-1] = $rsite1.$first_fragments[-1];

[download]

#11929 First ask yourself `How would I do this without a computer?' Then have the computer do it the same way.

In reply to Re: Bioinformatics- Use of uninitialized value by kennethk
in thread Bioinformatics- Use of uninitialized value by mlsmit10

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