1. Fill pot with water
2. Carry pot to stove
3. Light stove
4. Wait for pot to boil

Then they are asked for an algorithm for boiling a pot which already contains water.

The engineer simply replies with steps 2-4 above.

The mathematician replies:  "Step one = pour out the water.  Now the problem is reduced to the same as the first!"

My simplistic answer would be to take your array and convert it to pairs (using a hash along the way):

my @T1 = ( 'a|b|c', 'c|d', 'j|k|l|m', 'm|n|o', 'o|p|q|r|s' );
my %newT1;
foreach (@T1) {
    my @points = split /\|/;
    for (my $i = 1; $i < @points; $i++) {
        my $link = $points[$i-1] . '|' . $points[$i];
        $newT1{$link}++;
    }
}
my @newT1 = keys %newT1;

# Verify that the new array contains the desired points
printf "\@newT1 = %s\n", Dumper(\@newT1);
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Now the problem's been reduced to the first one.