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

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  • A recent result of Chepoi, Estellon and Vaxes [Disc. Comp. Geom. '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.
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  • Borradaile, G., & Chambers, E. W. (2014). Covering nearly surface-embedded graphs with a fixed number of balls. Discrete & Computational Geometry, 51(4), 979-996. doi:10.1007/s00454-014-9594-5
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