On the Mixed Connectivity Conjecture of Beineke and Harary

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.