Subsurface 1-distance of the Handleb o dy

For a handlebody $H$ with $?H = S$, let $F ? S$ be an essential connected subsurface of $S$. Let$\mathcal{C}(S)$ be the curve complex of $S$,$\mathcal{AC}(F)$ be the arc and curve complex of $F$,$\mathcal{D}(H) ?\mathcal{C}(S)$ be the disk complex of $H$ and $π_F (\mathcal{D}(H)) ?\mathcal{AC}(F)$ be the image of$\mathcal{D}(H)$ in$\mathcal{AC}(F)$. We introduce the definition of subsurface 1-distance between the 1-simplices of $\mathcal{AC}(F)$ and show that under some hypothesis, $π_F (\mathcal{D}(H))$ comes within subsurface 1-distance at most 4 of every 1-simplex of $\mathcal{AC}(F)$.