A survey on the generalized connectivity of graphs

TL;DR

This survey reviews the current state of research on generalized connectivity and edge-connectivity in graphs, covering theoretical bounds, algorithms, special graph classes, and open problems in the field.

Contribution

It compiles and summarizes recent results, conjectures, and open problems on generalized (edge-)connectivity, providing a comprehensive overview of this research area.

Findings

Summarized bounds for generalized connectivity and edge-connectivity.

Identified classes of graphs with large generalized (edge-)connectivity.

Presented open problems and conjectures for future research.

Abstract

The generalized $k$-connectivity $\kappa_k(G)$ of a graph $G$ was introduced by Hager before 1985. As its a natural counterpart, we introduced the concept of generalized edge-connectivity $\lambda_k(G)$, recently. In this paper we summarize the known results on the generalized connectivity and generalized edge-connectivity. After an introductory section, the paper is then divided into nine sections: the generalized (edge-)connectivity of some graph classes, algorithms and computational complexity, sharp bounds of $\kappa_k(G)$ and $\lambda_k(G)$, graphs with large generalized (edge-)connectivity, Nordhaus-Gaddum-type results, graph operations, extremal problems, and some results for random graphs and multigraphs. It also contains some conjectures and open problems for further studies.