  A recent result of Chepoi, Estellon and Vaxes [DCG '07] states that any
planar graph of diameter at most 2R can be covered by a constant number of
balls of size R; put another way, there are a constant-sized subset of vertices
within which every other vertex is distance half the diameter. We generalize
this result to graphs embedded on surfaces of fixed genus with a fixed number
of apices, making progress toward the conjecture that graphs excluding a fixed
minor can also be covered by a constant number of balls. To do so, we develop
two tools which may be of independent interest. The first gives a bound on the
density of graphs drawn on a surface of genus $g$ having a limit on the number
of pairwise-crossing edges. The second bounds the size of a non-contractible
cycle in terms of the Euclidean norm of the degree sequence of a graph embedded
on surface.
