Let D be a graph or a digraph. If δ (D) is the minimum degree, λ (D) the edge-connectivity and κ (D) the vertex-connectivity, then κ (D) ≤ λ (D) ≤ δ (D) is a well-known basic relationship between these parameters. The graph or digraph D is called maximally edge-connected if λ (D) = δ (D) and maximally vertex-connected if κ (D) = δ (D). In this survey we mainly present sufficient conditions for graphs and digraphs to be maximally edge-connected as well as maximally vertex-connected. We also discuss the concept of conditional or restricted edge-connectivity and vertex-connectivity, respectively. © 2007 Elsevier B.V. All rights reserved.
Hellwig, A., & Volkmann, L. (2008). Maximally edge-connected and vertex-connected graphs and digraphs: A survey. Discrete Mathematics, 308(15), 3265–3296. https://doi.org/10.1016/j.disc.2007.06.035