Is it subsequence or substring? Don't you want to find the longest common substring, i.e. contiguous block? I always understood subsequence to be non-contiguous.
ABXCDYEFGZ QARBCSTDEFG
You want to find [Q]A[R]B(X)C[ST]D(Y)EFG(Z)
ABCDEFG is the longest common subsequence, QRST are the additions and XYZ are the deletions.