Covering Nearly Surface-Embedded Graphs with a Fixed Number of Balls

A recent result of Chepoi et al. [Discrete Comput. Geom. 37(2):237–244, 2007] states that any planar graph of diameter at most $$2R$$2R can be covered by a constant number of balls of size $$R$$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 $$g$$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.

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