Reducible mapping class of the canonical Heegaard splitting in a mapping torus

Let S be an orientable closed surface with genus at least two. From S×I, for a given orientation-reversing homeomorphism f from S×{1} to S×{0}, there is an orientable closed 3-manifold M _f =S×I/f which is called a mapping torus. It is known that M _f admits a canonical Heegaard splitting H ₁ ∪ _Σ H ₂ . By the construction of Namazi [H.Namazi, Topology Appl. 154 (2007), no. 16, 2939-2949], the mapping class group of this Heegaard splitting, denoted by Mod(Σ;H ₁ ,H ₂ ), contains a reducible mapping class which has infinitely order. So it is interesting to know that for a given element in Mod(Σ;H ₁ ,H ₂ ), whether it is reducible or not. Using the translation length of f in the curve complex, we prove that if f is the identity map or its translation length is at least 8, then each element of Mod(Σ;H ₁ ,H ₂ ) is reducible.