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Two Kinds of Generalized 3-Connectivities of Alternating Group Networks

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Frontiers in Algorithmics (FAW 2018)

Part of the book series: Lecture Notes in Computer Science ((LNTCS,volume 10823))

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Abstract

To strengthen the classical connectivity of graphs, two kinds of generalized k-connectivities of a graph G, denoted by \(\kappa _k'(G)\) and \(\kappa _k(G)\), were introduced by Chartrand et al. [1] and Hager [7], respectively. The former is the so-called cut-version definition of connectivity, whereas the latter is the path-version definition of connectivity (a synonym was also called the tree-connectivity by Okamoto and Zhang [32]). Since the underlying topologies of interconnection networks are usually modeled as undirected simple graphs, as applications of these two kinds of generalized connectivities, one can be used to assess the vulnerability of the corresponding network, and the other can serve to measure the capability of connection for a set of k nodes in the network. So far the exact values of these two types of generalized connectivities are known only for small classes of graphs. In this paper, we study the two kinds of generalized 3-connectivities in the n-dimensional alternating group networks \(AN_n\). Consequently, we determine the exact values: \(\kappa '_3(AN_n)=2n-3\) for \(n\geqslant 4\) and \(\kappa _3(AN_n)=n-2\) for \(n\geqslant 3\).

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Acknowledgments

This research was partially supported by MOST grants 104-2221-E-141-002-MY3 (Jou-Ming Chang),105-2221-E-131-027 (Kung-Jui Pai), 106-2221-E-141-001 (Jinn-Shyong Yang) and 104-2221-E-262-005 (Ro-Yu Wu) from the Ministry of Science and Technology, Taiwan.

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Authors and Affiliations

  1. Institute of Information and Decision Sciences, National Taipei University of Business, Taipei, Taiwan

    Jou-Ming Chang

  2. Department of Industrial Engineering and Management, Ming Chi University of Technology, New Taipei City, Taiwan

    Kung-Jui Pai

  3. Department of Information Management, National Taipei University of Business, Taipei, Taiwan

    Jinn-Shyong Yang

  4. Department of Industrial Management, Lunghwa University of Science and Technology, Taoyuan, Taiwan

    Ro-Yu Wu

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  1. Jou-Ming Chang
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  2. Kung-Jui Pai
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  3. Jinn-Shyong Yang
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  4. Ro-Yu Wu
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Correspondence to Jou-Ming Chang .

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Editors and Affiliations

  1. Texas A&M University, College Station, Texas, USA

    Jianer Chen

  2. Shanghai University of Finance and Economics, Shanghai, China

    Pinyan Lu

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Chang, JM., Pai, KJ., Yang, JS., Wu, RY. (2018). Two Kinds of Generalized 3-Connectivities of Alternating Group Networks. In: Chen, J., Lu, P. (eds) Frontiers in Algorithmics. FAW 2018. Lecture Notes in Computer Science(), vol 10823. Springer, Cham. https://doi.org/10.1007/978-3-319-78455-7_1

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  • DOI: https://doi.org/10.1007/978-3-319-78455-7_1

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  • Online ISBN: 978-3-319-78455-7

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