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arXiv · 1908.11621

On the Mixed Connectivity Conjecture of Beineke and Harary

Sebastian S. Johann·Sven O. Krumke·Manuel Streicher
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Abstract

The conjecture of Beineke and Harary states that for any two vertices which can be separated by $k$ vertices and $l$ edges for $l\geq 1$ but neither by $k$ vertices and $l-1$ edges nor $k-1$ vertices and $l$ edges there are $k+l$ edge-disjoint paths connecting these two vertices of which $k+1$ are internally disjoint. In this paper we consider this conjecture for $l=2$ and any $k\in \mathbb{N}$. Afterwards, we utilize this result to prove that the conjecture holds for all graphs of treewidth at most $3$ and all $k$ and $l$. We also show that it is NP-complete to decide whether two vertices can be separated by $k$ vertices and $l$ edges.

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Sebastian S. Johann, Sven O. Krumke, Manuel Streicher. 2019-08-30. On the Mixed Connectivity Conjecture of Beineke and Harary. https://arxiv.org/abs/1908.11621

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