A survey on the generalized connectivity of graphs

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.

51 pages. arXiv admin note: text overlap with arXiv:1303.3881 by other authors

References in corpus (11)

• On the generalized (edge-)connectivity of graphs
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• The generalized 3-connectivity of Cartesian product graphs
• The minimal size of a graph with given generalized 3-edge-connectivity
• The generalized 3-connectivity of random graphs
• The minimal size of a graph with generalized connectivity ${\kappa }_3=2$
• Note on packing of edge-disjoint spanning trees in sparse random graphs
• Graphs with large generalized (edge-)connectivity
• Lower bounds for the spanning tree numbers of two graph products
• On extremal graphs with at most $\ell$ internally disjoint Steiner trees connecting any n-1 vertices
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Cited by in corpus (3)

• The 4-Component Connectivity of Alternating Group Networks
• The generalized 3-connectivity of random graphs
• Strong subgraph $k$-connectivity bounds