  We prove that a Lefschetz fibration over the disc that, after
compactification, has the same singular fibers as an extremal rational elliptic
surface can be obtained by deleting a singular fiber and a section from the
rational extremal elliptic surface, i.e. such a Lefschetz fibration is
determined up to topological equivalence by its set of singular fibers. We get
a complete clasification of Lefschetz fibrations with 2 $I_1$ fibers as a
byproduct of our results. The proof is inspired by homological mirror symmetry
and Karpov--Nogin's theorem on constructivity of helices on del Pezzo surfaces.
It would be interesting to extend our results to the case of Lefschetz
fibrations that, after compactification, have the same singular fibers as an
extremal elliptic K3 surface.
