A unifying approach to splitting-off


We study the behaviour of splitting-off algorithms when applied to the problem of covering a symmetric skew-supermodular set function by a graph. This hard problem is a natural generalization of many solved connectivity augmentation problems, such as local edge-connectivity augmentation of graphs, global arc-connectivity augmentation of mixed graphs with undirected edges, or the node-to-area connectivity augmentation problem in graphs. Using a simple lemma we characterize the situation when a splitting-off algorithm can get stuck. This characterization enables us to give very simple proofs for the results mentioned above. Finally we apply our observations on generalizations of the above problems: we consider three connectivity augmentation problems in hypergraphs where the objective is to use hyperedges of minimum total size without increasing the rank. The first is local edge-connectivity augmentation of undirected hypergraphs. The second is global arc-connectivity augmentation of mixed hypergraphs with undirected hyperedges. The third is a hypergraphic generalization of the node-to-area connectivity augmentation problem. We show that a greedy approach (almost) works in these cases.

This is a preview of subscription content, access via your institution.


  1. [1]

    J. Bang-Jensen, A. Frank and Bill Jackson: Preserving and increasing local edge-connectivity in mixed graphs, SIAM J. Discrete Math. 8 (1995), 155–178.

    MathSciNet  MATH  Article  Google Scholar 

  2. [2]

    A. A. Benczúr and A. Frank: Covering symmetric supermodular functions by graphs, Math. Program. 84 Ser. B, (1999), 483–503, Connectivity augmentation of networks: structures and algorithms (Budapest, 1994).

    MathSciNet  MATH  Article  Google Scholar 

  3. [3]

    A. Bernáth and T. Király: A new approach to splitting-off, Tech. Report TR-2008-02, Egerváry Research Group, Budapest, 2008, www.cs.elte.hu/egres.

  4. [4]

    Y. H. Chan, W. S. Fung, L. C. Lau and C. K. Yung: Degree bounded network design with metric costs, Proceedings of the 2008 49th Annual IEEE Symposium on Foundations of Computer Science (Washington, DC, USA), IEEE Computer Society, 2008, 125–134.

  5. [5]

    B. Cosh: Vertex splitting and connectivity augmentation in hypergraphs, PhDThesis, University of London, (2000).

  6. [6]

    B. Cosh, B. Jackson, and Z. Király: Local edge-connectivity augmentation in hypergraphs is NP-complete, Discrete Applied Mathematics 158 723–727, 2010.

    MathSciNet  MATH  Article  Google Scholar 

  7. [7]

    A. Frank: Augmenting graphs to meet edge-connectivity requirements, SIAM J. Discrete Math. 5 (1992), 25–53.

    MathSciNet  MATH  Article  Google Scholar 

  8. [8]

    A. Frank: On a theorem of Mader, Discrete Math. 101 (1992), 49–57; Special volume to mark the centennial of Julius Petersen’s “Die Theorie der regulären Graphs”, Part II.

    MathSciNet  MATH  Article  Google Scholar 

  9. [9]

    R. Grappe and Z. Szigeti: Note: Covering symmetric semi-monotone functions, Discrete Appl. Math. 156 (2008), 138–144.

    MathSciNet  MATH  Article  Google Scholar 

  10. [10]

    T. Ishii and M. Hagiwara: Minimum augmentation of local edge-connectivity between vertices and vertex subsets in undirected graphs, Discrete Appl. Math. 154 (2006), 2307–2329.

    MathSciNet  MATH  Article  Google Scholar 

  11. [11]

    W. Mader: A reduction method for edge-connectivity in graphs, Ann. Discrete Math. 3 (1978), 145–164; Advances in graph theory (Cambridge Combinatorial Conf., Trinity College, Cambridge, 1977).

    MathSciNet  MATH  Article  Google Scholar 

  12. [12]

    H. Miwa and H. Ito: NA-edge-connectivity augmentation problems by adding edges, J. Oper. Res. Soc. Japan 47 (2004), 224–243.

    MathSciNet  MATH  Google Scholar 

  13. [13]

    Z. Nutov: Approximating connectivity augmentation problems, SODA’ 05: Proceedings of the sixteenth annual ACM-SIAM symposium on Discrete algorithms (Philadelphia, PA, USA), Society for Industrial and Applied Mathematics, 2005, 176–185.

  14. [14]

    Z. Szigeti: Hypergraph connectivity augmentation, Math. Program. Ser. B 84 (1999), 519–527; Connectivity augmentation of networks: structures and algorithms (Budapest, 1994).

    MathSciNet  MATH  Article  Google Scholar 

Download references

Author information



Corresponding author

Correspondence to Attila Bernáth.

Additional information

An extended abstract version of this paper has appeared in the proceedings of IPCO 2008, pages 401–415

Supported by OTKA grants K60802 and TS 049788.

Supported by grants OTKA K60802 and OMFB-01608/2006.

Rights and permissions

Reprints and Permissions

About this article

Cite this article

Bernáth, A., Király, T. A unifying approach to splitting-off. Combinatorica 32, 373–401 (2012). https://doi.org/10.1007/s00493-012-2548-8

Download citation

Mathematics Subject Classification (2000)

  • 90C27. 05C40
  • 05C85