A survey on the generalized connectivity of graphs

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.

💡 Research Summary

The surveyed paper offers a comprehensive overview of the two parallel concepts that have become central in modern graph theory: the generalized k‑connectivity κₖ(G) and its edge‑based counterpart, the generalized k‑edge‑connectivity λₖ(G). Both notions extend the classical vertex‑connectivity κ(G) (which equals κ₂) by requiring that any chosen set S of k vertices be simultaneously spanned by a collection of internally disjoint trees (for κₖ) or edge‑disjoint trees (for λₖ). The authors begin with a concise historical note—κₖ was introduced by Hager in the mid‑1980s, while λₖ was defined only recently by the present authors—then organize the existing literature into nine thematic sections, each of which is summarized below.

1. Generalized (Edge‑)Connectivity of Specific Graph Classes
   Exact values of κₖ and λₖ are known for many highly symmetric families. For the complete graph Kₙ, both parameters equal ⌊n/k⌋, reflecting the fact that the vertex set can be partitioned into ⌊n/k⌋ disjoint k‑subsets, each supporting a spanning tree. For complete bipartite graphs K_{a,b}, the values depend on the smaller part size and on k, with tight formulas given for all ranges. Similar closed‑form results are listed for hypercubes, toroidal grids, regular circulants, and other product graphs. These examples illustrate how density and regularity strongly influence generalized connectivity.

2. Algorithms and Computational Complexity
   Determining κₖ(G) or λₖ(G) for an arbitrary graph is shown to be NP‑hard by reduction from the classic k‑disjoint‑paths problem. Nevertheless, polynomial‑time algorithms exist for restricted families: trees (where the parameters reduce to simple degree counts), graphs of bounded treewidth, and graphs with a fixed maximum degree. The section also surveys approximation schemes: a 2‑approximation for κₖ, an O(log n)‑approximation for λₖ, and fixed‑parameter tractable (FPT) algorithms when k or the solution size is taken as a parameter. Recent advances in matroid‑based packing and in parametric search are highlighted as promising directions.

3. Sharp Bounds for κₖ and λₖ
   A collection of inequalities links the generalized parameters to classical invariants. Degree‑based bounds such as κₖ(G) ≤ δ(G)·(k‑1) and λₖ(G) ≤ Δ(G)·(k‑1) are proved, together with connectivity‑based lower bounds like κₖ(G) ≥ ⌊κ(G)/(k‑1)⌋. More refined results involve the graph’s edge‑density, its arboricity, and its laminarity number. In several cases the authors identify extremal graphs that attain the bounds, demonstrating that the inequalities are best possible.

4. Graphs with Large Generalized (Edge‑)Connectivity
   The authors characterize graphs for which κₖ(G) or λₖ(G) reaches the theoretical maximum n‑k+1. Apart from the complete graph, the only other extremal structures are those obtained by deleting a small, well‑controlled set of edges from Kₙ. This section underscores the relevance of generalized connectivity to network reliability: a graph with κₖ = n‑k+1 guarantees that any k terminals can be simultaneously connected by the maximum possible number of internally disjoint routes.

5. Nordhaus‑Gaddum‑Type Results
   Extending the classic Nordhaus‑Gaddum inequalities (which relate a graph invariant to that of its complement), the paper derives bounds for κₖ(G)+κₖ(Ĝ) and κₖ(G)·κₖ(Ĝ) (and analogously for λₖ). The results show that the sum is at most n‑k+2 and the product is bounded by ⌊(n‑k+2)²/4⌋, with equality cases fully described. These findings reveal a delicate balance: a graph with high generalized connectivity forces its complement to have low generalized connectivity, and vice versa.

6. Effects of Graph Operations
   The behavior of κₖ and λₖ under standard graph operations is systematically examined. For the Cartesian product G□H, a lower bound κₖ(G□H) ≥ min{κₖ(G), κₖ(H)}·k is proved, while for the tensor (direct) product a multiplicative bound is established. The authors also discuss line graphs, graph powers, and the operation of taking the complement, showing which operations preserve or amplify generalized connectivity. These results are valuable for constructing large networks from smaller building blocks while controlling reliability metrics.

7. Extremal Problems
   Given integers n, k, and a target value t, the paper investigates the minimum number of edges a graph must have to satisfy κₖ(G) ≥ t (or λₖ(G) ≥ t). Using Turán‑type extremal constructions, the authors prove that the edge‑count threshold is essentially ⌈t·(n‑1)/(k‑1)⌉, and they identify families of graphs that achieve this bound. The dual problem—maximizing κₖ or λₖ under a fixed edge budget—is also treated, leading to a clear picture of the trade‑off between sparsity and robustness.

8. Random Graphs and Multigraphs
   In the Erdős‑Rényi model G(n,p), the authors determine threshold functions for the emergence of a given generalized connectivity. When p ≫ (log n)/n, with high probability κₖ(G)=λₖ(G)=δ(G). More precisely, the critical probability for κₖ ≥ t is shown to be p_c ≈ (t·log n)/n. For multigraphs, where parallel edges are allowed, the paper extends the definition of λₖ and proves that edge multiplicities can increase λₖ linearly, a fact that has implications for designing resilient communication networks with redundant physical links.

9. Open Problems and Conjectures
   The final section lists several research directions that remain unresolved. Key challenges include: (i) establishing the exact computational complexity of λₖ for fixed k (beyond the known NP‑hardness for variable k); (ii) tightening Nordhaus‑Gaddum bounds and proving their optimality for all k; (iii) determining precise threshold functions for generalized connectivity in other random graph models (e.g., random regular graphs, preferential attachment graphs); (iv) developing unified frameworks that capture the effect of arbitrary graph products on κₖ and λₖ; and (v) constructing efficient, practically implementable routing algorithms that exploit the existence of k‑disjoint trees in real‑world networks. The authors also propose concrete conjectures, such as κₖ(G)·λₖ(G) ≥ k·(n‑k) for all connected G, inviting the community to verify or refute them.

In sum, the survey stitches together a rich tapestry of results spanning pure combinatorial theory, algorithm design, extremal graph theory, and probabilistic methods. By juxtaposing exact formulas for special families, general bounds, algorithmic insights, and a forward‑looking list of open questions, the paper serves both as a reference handbook for seasoned researchers and as a guided roadmap for newcomers eager to explore the fertile ground of generalized connectivity.