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arXiv · 1811.06411

Characterising $k$-connected sets in infinite graphs

J. Pascal Gollin·Karl Heuer
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Abstract

A $k$-connected set in an infinite graph, where $k > 0$ is an integer, is a set of vertices such that any two of its subsets of the same size $\ell \leq k$ can be connected by $\ell$ disjoint paths in the whole graph. We characterise the existence of $k$-connected sets of arbitrary but fixed infinite cardinality via the existence of certain minors and topological minors. We also prove a duality theorem for the existence of such $k$-connected sets: if a graph contains no such $k$-connected set, then it has a tree-decomposition which, whenever it exists, precludes the existence of such a $k$-connected set.

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J. Pascal Gollin, Karl Heuer. 2018-11-15. Characterising $k$-connected sets in infinite graphs. https://arxiv.org/abs/1811.06411

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