On the Mixed Connectivity Conjecture of Beineke and Harary

TL;DR

This paper investigates a specific case of the Mixed Connectivity Conjecture, proving it for graphs with treewidth at most 3 and establishing NP-completeness for certain separation problems.

Contribution

It proves the conjecture for l=2 and all k, and for graphs with treewidth at most 3, while also showing the problem's NP-completeness.

Findings

Conjecture holds for l=2 and all k.

Valid for graphs with treewidth ≤ 3.

Deciding separation by k vertices and l edges is NP-complete.

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.