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.

[1]  E. Helly.  Über Mengen konvexer Körper mit gemeinschaftlichen Punkte. , 1923 .

[2]  Jirí Matousek,et al.  Bounded VC-Dimension Implies a Fractional Helly Theorem , 2004, Discret. Comput. Geom..

[3]  R. Ho.  Algebraic Topology , 2022 .

[4]  Erik D. Demaine,et al.  Subexponential parameterized algorithms on graphs of bounded-genus and H-minor-free graphs , 2004, SODA '04.

[5]  Tom Jenkyns,et al.  The toroidal crossing number of the complete graph , 1968 .

[6]  Bartosz Walczak,et al.  New bounds on the maximum number of edges in k-quasi-planar graphs , 2015, Comput. Geom..

[7]  Victor Chepoi,et al.  Covering Planar Graphs with a Fixed Number of Balls , 2007, Discret. Comput. Geom..

[8]  Erik D. Demaine,et al.  Subexponential parameterized algorithms on bounded-genus graphs and H-minor-free graphs , 2005, JACM.

[9]  János Pach,et al.  The Number of Edges in k-Quasi-planar Graphs , 2011, SIAM J. Discret. Math..

[10]  Vladimir Vapnik,et al.  Chervonenkis: On the uniform convergence of relative frequencies of events to their probabilities , 1971 .

[11]  Andrew Suk.  $k$-quasi Planar Graphs , 2011, Graph Drawing.

[12]  George Pólya.  A note of welcome , 1977, J. Graph Theory.

[13]  Gary L. Miller,et al.  Planar Separators and the Euclidean Norm , 1990, SIGAL International Symposium on Algorithms.

[14]  Carsten Thomassen,et al.  Graphs on Surfaces , 2001, Johns Hopkins series in the mathematical sciences.

[15]  Krzysztof Diks,et al.  Edge Separators of Planar and Outerplanar Graphs with Applications , 1993, J. Algorithms.

[16]  André Raspaud,et al.  Small k-Dominating Sets in Planar Graphs with Applications , 2001, WG.

[17]  G. Ringel,et al.  Solution of the heawood map-coloring problem. , 1968, Proceedings of the National Academy of Sciences of the United States of America.

[18]  Paul D. Seymour,et al.  Graph Minors. XVI. Excluding a non-planar graph , 2003, J. Comb. Theory, Ser. B.

[19]  Micha Sharir,et al.  Quasi-planar graphs have a linear number of edges , 1995, GD.

[20]  Farhad Shahrokhi,et al.  Applications of the crossing number , 1994, SCG '94.

[21]  Gary L. Miller,et al.  Finding small simple cycle separators for 2-connected planar graphs. , 1984, STOC '84.

[22]  F. Shahrokhi,et al.  The Crossing Number of a Graph on a Compact 2-Manifold , 1996 .

[23]  Farhad Shahrokhi,et al.  Improving Bounds for the Crossing Numbers on Surfaces of Genus g , 1993, WG.