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arXiv · 1207.5357

On (2k,k)-connected graphs

Olivier Durand de Gevigney·Zoltán Szigeti
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Abstract

A graph G is called (2k, k)-connected if G is 2k-edge-connected and G-v is k-edge-connected for every vertex v. The study of (2k, k)-connected graphs is motivated by a conjecture of Frank which states that a graph has a 2-vertex-connected orientation if and only if it is (4, 2)-connected. In this paper, we provide a construction of the family of (2k, k)-connected graphs for k even which generalizes the construction given by Jord\'an for k = 2. We also solve the corresponding connectivity augmentation problem: given a graph G and an integer k \geq 2, what is the minimum number of edges to be added to make G (2k, k)-connected. Both these results are based on a new splitting-off theorem for (2k, k)-connected graphs.

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Olivier Durand de Gevigney, Zoltán Szigeti. 2012-07-23. On (2k,k)-connected graphs. https://arxiv.org/abs/1207.5357

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