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Computation and algorithm for the minimum k-edge-connectivity of graphs

Author

Listed:
  • Yuefang Sun

    (Ningbo University
    Shaoxing University)

  • Chenchen Wu

    (Tianjin University of Technology)

  • Xiaoyan Zhang

    (Nanjing Normal University)

  • Zhao Zhang

    (Zhejiang Normal University)

Abstract

Boesch and Chen (SIAM J Appl Math 34:657–665, 1978) introduced the cut-version of the generalized edge-connectivity, named k-edge-connectivity. For any integer k with $$2\le k\le n$$ 2 ≤ k ≤ n , the k-edge-connectivity of a graph G, denoted by $$\lambda _k(G)$$ λ k ( G ) , is defined as the smallest number of edges whose removal from G produces a graph with at least k components. In this paper, we first compute some exact values and sharp bounds for $$\lambda _k(G)$$ λ k ( G ) in terms of n and k. We then discuss the relationships between $$\lambda _k(G)$$ λ k ( G ) and other generalized connectivities. An algorithm in $$\mathcal {O}(n^2)$$ O ( n 2 ) time will be provided such that we can compute a sharp upper bound in terms of the maximum degree. Among our results, we also compute some exact values and sharp bounds for the function f(n, k, t) which is defined as the minimum size of a connected graph G with order n and $$\lambda _k(G)=t$$ λ k ( G ) = t .

Suggested Citation

  • Yuefang Sun & Chenchen Wu & Xiaoyan Zhang & Zhao Zhang, 2022. "Computation and algorithm for the minimum k-edge-connectivity of graphs," Journal of Combinatorial Optimization, Springer, vol. 44(3), pages 1741-1752, October.
    • Handle: RePEc:spr:jcomop:v:44:y:2022:i:3:d:10.1007_s10878-020-00541-z
    DOI: 10.1007/s10878-020-00541-z
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    References listed on IDEAS

    as
    1. Lily Chen & Xueliang Li & Mengmeng Liu & Yaping Mao, 2017. "A solution to a conjecture on the generalized connectivity of graphs," Journal of Combinatorial Optimization, Springer, vol. 33(1), pages 275-282, January.
    2. Olivier Goldschmidt & Dorit S. Hochbaum, 1994. "A Polynomial Algorithm for the k-cut Problem for Fixed k," Mathematics of Operations Research, INFORMS, vol. 19(1), pages 24-37, February.
    3. Yuefang Sun & Xueliang Li, 2017. "On the difference of two generalized connectivities of a graph," Journal of Combinatorial Optimization, Springer, vol. 33(1), pages 283-291, January.
    Full references (including those not matched with items on IDEAS)

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