On $ H' $-splittings of a handlebody

Let $ M $ be a compact connected orientable 3-manifold and $ F $ be a compact connected orientable surface properly embedded in $ M $. If $ F $ cuts $ M $ into two handlebodies $ X $ and $ Y $ (i.e., $ M = X\cup_FY $), then we say that $ F $ is an $ H' $-splitting surface for $ M $ and call $ X\cup_FY $ an $ H' $-splitting for $ M $. When the $ H' $-splitting surface $ F $ is incompressible in a handlebody $ H $, a characteristic of an $ H' $-splitting $ H_1\cup_F H_2 $ to denote $ H $ is already known. In the present paper, we generalize the above result as follows: Let $ H $ be a handlebody of genus $ g\geq 1 $, $ X\cup_F Y $ an $ H' $-splitting for $ H $. Then, either $ X\cup_F Y $ is stabilized, or there exists a reducing system $ \mathcal{J}_1\cup\mathcal{K}_1 $ of $ F $, such that $ \mathcal{J}_1 $ is quasi-primitive in $ Y $ and $ \mathcal{K}_1 $ is quasi-primitive in $ X $. Combining the result with the known result, we obtain a characteristic of an $ H' $-splitting $ H_1\cup_F H_2 $ to denote a handlebody.