... my best guess is that the server has not permissions to open graph.png for output ...

It's also a good idea to let yourself know about such problems as they develop. Something along these lines might be informative:

my $filename = 'graph.png';
open my $fh, '>', $filename or die "opening $filename: $!";
binmode $fh;
print $fh $mygraph->plot(\@data)->png or die "writing $filename: $!";
close $fh or die "closing $filename: $!";
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