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Structure and Substructure Connectivity of Balanced Hypercubes

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Abstract

The connectivity of a network directly signifies its reliability and fault tolerance. Structure connectivity and substructure connectivity are two novel generalizations of the connectivity. Let H be a subgraph of a connected graph G. The structure connectivity (resp. substructure connectivity) of G, denoted by \(\kappa (G;H)\) [resp. \(\kappa ^s(G;H)\)], is defined to be the minimum cardinality of a set F of connected subgraphs in G, if exists, whose removal disconnects G and each element of F is isomorphic to H (resp. a subgraph of H). In this paper, we present the upper bounds for \(\kappa (BH_n;K_{1,M})\) and \(\kappa ^s(BH_n;K_{1,M})\) and show that the upper bounds are tight for \(1\le M\le 5\). Furthermore, we establish both \(\kappa (BH_n;H)\) and \(\kappa ^s(BH_n;H)\) for \(H\in \{K_1,C_4\}\).

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Acknowledgements

The authors are grateful to the anonymous referees for their kind suggestions and comments that greatly improved the original manuscript.

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Authors and Affiliations

  1. School of Mathematical Sciences, University of Electronic Science and Technology of China, Chengdu, 610054, Sichuan, People’s Republic of China

    Huazhong Lü

  2. School of Mathematics and Statistics, Qinghai Nationalities University, Xining, 810007, Qinghai, People’s Republic of China

    Tingzeng Wu

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  1. Huazhong Lü
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Correspondence to Huazhong Lü.

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Sanming Zhou.

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This research was partially supported by the National Natural Science Foundation of China (Nos. 11801061 and 11761056), the Chunhui Project of Ministry of Education (No. Z2017047) and the Fundamental Research Funds for the Central Universities (No. ZYGX2018J083).

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Lü, H., Wu, T. Structure and Substructure Connectivity of Balanced Hypercubes. Bull. Malays. Math. Sci. Soc. 43, 2659–2672 (2020). https://doi.org/10.1007/s40840-019-00827-4

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  • DOI: https://doi.org/10.1007/s40840-019-00827-4

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