Sebastian S. Johann, Sven O. Krumke, Manuel Streicher

• The conjecture of Beineke and Harary states that for any two vertices which can be separated by k vertices and l edges for \(l\ge 1\) 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 prove this conjecture for \(l=2\) and every \(k\in \mathbb {N}\) . We utilize this result to prove that the conjecture holds for all graphs of treewidth at most 3 and all k and l.