(untitled) (Communications in Mathematical Research, 2021)
For a handlebody H with ∂H = S, let F ⊂ S be an essential connected subsurface of S. Let C(S) be the curve complex of S, AC(F) be the arc and curve complex of F, D(H) ⊂ C(S) be the disk complex of H and π_F (D(H)) ⊂ AC(F) be the image of D(H) in AC(F). We introduce the definition of subsurface 1-distance between the 1-simplices of AC(F) and show that under some hypothesis, π_F (D(H)) comes within subsurface 1-distance at most 4 of every 1-simplex of AC(F).