346 research outputs found
A single exponential bound for the redundant vertex Theorem on surfaces
Let s1, t1,. . . sk, tk be vertices in a graph G embedded on a surface \sigma
of genus g. A vertex v of G is "redundant" if there exist k vertex disjoint
paths linking si and ti (1 \lequal i \lequal k) in G if and only if such paths
also exist in G - v. Robertson and Seymour proved in Graph Minors VII that if v
is "far" from the vertices si and tj and v is surrounded in a planar part of
\sigma by l(g, k) disjoint cycles, then v is redundant. Unfortunately, their
proof of the existence of l(g, k) is not constructive. In this paper, we give
an explicit single exponential bound in g and k
The -genus of Kuratowski minors
A drawing of a graph on a surface is independently even if every pair of
nonadjacent edges in the drawing crosses an even number of times. The
-genus of a graph is the minimum such that has an
independently even drawing on the orientable surface of genus . An
unpublished result by Robertson and Seymour implies that for every , every
graph of sufficiently large genus contains as a minor a projective
grid or one of the following so-called -Kuratowski graphs: , or
copies of or sharing at most common vertices. We show that
the -genus of graphs in these families is unbounded in ; in
fact, equal to their genus. Together, this implies that the genus of a graph is
bounded from above by a function of its -genus, solving a problem
posed by Schaefer and \v{S}tefankovi\v{c}, and giving an approximate version of
the Hanani-Tutte theorem on orientable surfaces. We also obtain an analogous
result for Euler genus and Euler -genus of graphs.Comment: 23 pages, 7 figures; a few references added and correcte
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