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Note on the hardness of generalized connectivity

Author

Listed:
  • Shasha Li

    (Nankai University)

  • Xueliang Li

    (Nankai University)

Abstract

Let G be a nontrivial connected graph of order n and let k be an integer with 2≤k≤n. For a set S of k vertices of G, let κ(S) denote the maximum number ℓ of edge-disjoint trees T 1,T 2,…,T ℓ in G such that V(T i )∩V(T j )=S for every pair i,j of distinct integers with 1≤i,j≤ℓ. Chartrand et al. generalized the concept of connectivity as follows: The k-connectivity, denoted by κ k (G), of G is defined by κ k (G)=min{κ(S)}, where the minimum is taken over all k-subsets S of V(G). Thus κ 2(G)=κ(G), where κ(G) is the connectivity of G, for which there are polynomial-time algorithms to solve it. This paper mainly focus on the complexity of determining the generalized connectivity of a graph. At first, we obtain that for two fixed positive integers k 1 and k 2, given a graph G and a k 1-subset S of V(G), the problem of deciding whether G contains k 2 internally disjoint trees connecting S can be solved by a polynomial-time algorithm. Then, we show that when k 1 is a fixed integer of at least 4, but k 2 is not a fixed integer, the problem turns out to be NP-complete. On the other hand, when k 2 is a fixed integer of at least 2, but k 1 is not a fixed integer, we show that the problem also becomes NP-complete.

Suggested Citation

  • Shasha Li & Xueliang Li, 2012. "Note on the hardness of generalized connectivity," Journal of Combinatorial Optimization, Springer, vol. 24(3), pages 389-396, October.
  • Handle: RePEc:spr:jcomop:v:24:y:2012:i:3:d:10.1007_s10878-011-9399-x
    DOI: 10.1007/s10878-011-9399-x
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    Citations

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    Cited by:

    1. Li, Hengzhe & Ma, Yingbin & Yang, Weihua & Wang, Yifei, 2017. "The generalized 3-connectivity of graph products," Applied Mathematics and Computation, Elsevier, vol. 295(C), pages 77-83.
    2. Li, Hengzhe & Wang, Jiajia, 2018. "The λ3-connectivity and κ3-connectivity of recursive circulants," Applied Mathematics and Computation, Elsevier, vol. 339(C), pages 750-757.
    3. Gao, Hui & Lv, Benjian & Wang, Kaishun, 2018. "Two lower bounds for generalized 3-connectivity of Cartesian product graphs," Applied Mathematics and Computation, Elsevier, vol. 338(C), pages 305-313.
    4. Li, Shasha & Zhao, Yan & Li, Fengwei & Gu, Ruijuan, 2019. "The generalized 3-connectivity of the Mycielskian of a graph," Applied Mathematics and Computation, Elsevier, vol. 347(C), pages 882-890.
    5. Li, Shasha & Tu, Jianhua & Yu, Chenyan, 2016. "The generalized 3-connectivity of star graphs and bubble-sort graphs," Applied Mathematics and Computation, Elsevier, vol. 274(C), pages 41-46.
    6. 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.
    7. Zhao, Shu-Li & Hao, Rong-Xia & Wei, Chao, 2022. "Internally disjoint trees in the line graph and total graph of the complete bipartite graph," Applied Mathematics and Computation, Elsevier, vol. 422(C).
    8. Zhao, Shu-Li & Hao, Rong-Xia, 2019. "The generalized 4-connectivity of exchanged hypercubes," Applied Mathematics and Computation, Elsevier, vol. 347(C), pages 342-353.
    9. Hongyu Liang & Tiancheng Lou & Haisheng Tan & Yuexuan Wang & Dongxiao Yu, 2015. "On the complexity of connectivity in cognitive radio networks through spectrum assignment," Journal of Combinatorial Optimization, Springer, vol. 29(2), pages 472-487, February.
    10. Shasha Li & Wei Li & Yongtang Shi & Haina Sun, 2017. "On minimally 2-connected graphs with generalized connectivity $$\kappa _{3}=2$$ κ 3 = 2," Journal of Combinatorial Optimization, Springer, vol. 34(1), pages 141-164, July.

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