On $H^{\prime }$-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., $M=X{\cup }_{F}Y$), then we say that $F$ is an $H^{\prime }$-splitting surface for $M$ and call $X{\cup }_{F}Y$ an $H^{\prime }$-splitting for $M$. When the $H^{\prime }$-splitting surface $F$ is incompressible in a handlebody $H$, a characteristic of an $H^{\prime }$-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 $g\ge 1$, $X{\cup }_{F}Y$ an $H^{\prime }$-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 $H^{\prime }$-splitting $H_1{\cup }_F{H}_2$ to denote a handlebody.