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Structure connectivity and substructure connectivity of bubble-sort star graph networks

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  • Zhang, Guozhen
  • Wang, Dajin

Abstract

The bubble-sort star graph, denoted BSn, is an interconnection network model for multiprocessor systems, which has attracted considerable interest since its first proposal in 1996 [5]. In this paper, we study the problem of structure/substructure connectivity in bubble-sort star networks. Two basic but important structures, namely path Pi and cycle Ci, are studied. Let T be a connected subgraph of graph G. The T-structure connectivity κ(G; T) of G is the cardinality of a minimum set of subgraphs in G, whose deletion disconnects G and every element in the set is isomorphic to T. The T-substructure connectivity κs(G; T) of G is the cardinality of a minimum set of subgraphs in G, whose deletion disconnects G and every element in the set is isomorphic to a connected subgraph of T. Both T-structure connectivity and T-substructure connectivity are a generalization of the classic notion of node-connectivity. We will prove that for P2k+1, a path on odd nodes (resp. P2k, a path on even nodes), κ(BSn;P2k+1)=κs(BSn;P2k+1)=⌈2n−3k+1⌉ for n ≥ 4 and k+1≤2n−3 (resp. κ(BSn;P2k)=κs(BSn;P2k)=⌈2n−3k⌉ for n ≥ 5 and k≤2n−3). For a cycle on 2k nodes C2k (there are only cycles on even nodes in BSn), κ(BSn;C2k)=κs(BSn;C2k)=⌈2n−3k⌉ for n ≥ 5 and 2≤k≤n−1.

Suggested Citation

  • Zhang, Guozhen & Wang, Dajin, 2019. "Structure connectivity and substructure connectivity of bubble-sort star graph networks," Applied Mathematics and Computation, Elsevier, vol. 363(C), pages 1-1.
  • Handle: RePEc:eee:apmaco:v:363:y:2019:i:c:27
    DOI: 10.1016/j.amc.2019.124632
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    References listed on IDEAS

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    1. Wang, Shiying & Ma, Xiaolei, 2018. "The g-extra connectivity and diagnosability of crossed cubes," Applied Mathematics and Computation, Elsevier, vol. 336(C), pages 60-66.
    2. 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.
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    Cited by:

    1. Li, Xiaowang & Zhou, Shuming & Ren, Xiangyu & Guo, Xia, 2021. "Structure and substructure connectivity of alternating group graphs," Applied Mathematics and Computation, Elsevier, vol. 391(C).
    2. Wang, Na & Meng, Jixiang & Tian, Yingzhi, 2022. "Reliability evaluation of Modified bubble-sort graph networks based on structure fault pattern," Applied Mathematics and Computation, Elsevier, vol. 430(C).
    3. Ba, Lina & Zhang, Heping, 2023. "Structure connectivity of data center networks," Applied Mathematics and Computation, Elsevier, vol. 451(C).
    4. Wang, Yihong & Meng, Jixiang & Fan, Jianxi, 2023. "Reliabilities for two kinds of graphs with smaller diameters," Applied Mathematics and Computation, Elsevier, vol. 446(C).
    5. Wei Feng & Shiying Wang, 2020. "The Diagnosability of Wheel Networks with Missing Edges under the Comparison Model," Mathematics, MDPI, vol. 8(10), pages 1-16, October.
    6. Lei, Yafei & Meng, Jixiang, 2020. "Structure Fault-tolerance of Arrangement Graphs," Applied Mathematics and Computation, Elsevier, vol. 381(C).
    7. Zhang, Guozhen & Wang, Dajin, 2021. "The structure fault tolerance of arrangement graphs," Applied Mathematics and Computation, Elsevier, vol. 400(C).

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