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Graph minors. XV: Giant steps. (English) Zbl 0860.05023
This paper continues the series of structural theorems aimed at characterizing the class of graphs not containing a fixed graph as a minor. The main result of this paper reads:
For any surface $$\Sigma$$ with $$\text{bd}(\Sigma)=\varnothing$$, and any integers $$\kappa,\varphi,\mu\geq 0$$ there are integers $$\theta,\lambda,\rho\geq 0$$ such that the following holds. Let $${\mathcal T}^*$$ be a tangle in a graph $$G$$, such that some $$\Sigma$$-span of order $$\geq\theta$$, is $$(\lambda,\mu)$$-flat. Then either: (i) there is a $$\Sigma$$-span of order $$\geq\varphi$$ with $$>\kappa$$ independent eyes, or (ii) there is a $$\Sigma'$$-span of order $$\geq\varphi$$, where $$\Sigma'$$ is a surface obtained by adding a crosscap to $$\Sigma$$, or (iii) there is a $${\mathcal T}^*$$-central segregation of $$G$$ of type $$(\rho,\kappa)$$ with an arrangement in $$\Sigma$$.

##### MSC:
 05C10 Planar graphs; geometric and topological aspects of graph theory
##### Keywords:
drawing; representativity; vortex; minor; surface
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