The disk graph of a handlebody H of genus g ≥ 2 with m ≥ 0 marked points on the boundary is the graph whose vertices are isotopy classes of disks disjoint from the marked points and where two vertices are connected by an edge of length one if they can be realized disjointly. We show that for m=1 the disk graph contains quasi-isometrically embedded copies of R². The same holds true for sphere graphs of the doubled handlebody with one marked points provided that g is even.

Partially supported by ERC Grant "Moduli''.