Further hardness results on the generalized connectivity of graphs

TL;DR

This paper investigates the computational complexity of generalized connectivity and edge-connectivity in graphs, proving two conjectures and extending understanding of these graph invariants.

Contribution

It determines the complexity status of generalized connectivity measures and confirms two conjectures, advancing theoretical knowledge in graph connectivity.

Findings

Complexity of generalized connectivity is established.

Complexity of generalized edge-connectivity is established.

Two conjectures related to these concepts are proved.

Abstract

The generalized $k$-connectivity $\kappa_k(G)$ of a graph $G$ was introduced by Chartrand et al. in 1984, which is a nice generalization of the classical connectivity. Recently, as a natural counterpart, Li et al. proposed the concept of generalized edge-connectivity for a graph. In this paper, we determine the computational complexity of the generalized connectivity and generalized edge-connectivity of a graph. Two conjectures are also proved to be true.