The fiber-sum construction gives
us many interesting examples of Lefschetz fibrations. Which
Lefschetz fibrations can be decomposed as fiber-sums? Stipsicz
obtained some results on the fiber-sum decomposition, which
state about the relationship between the minimality and the
fiber-sum decomposability of Lefschetz fibrations. He proved
that every Lefschetz fibration with section of self-intersection
number $-1$ cannot be decomposed as any nontrivial fiber-sum.
In this paper, we show that the reverse of this theorem does
not hold and we characterize genus-2 decomposable Lefschetz
fibrations with $b_{2}^{+} = 1$.