Two Kinds of Generalized 3-Connectivities of Alternating Group Networks

• Published in: Frontiers in Algorithmics Vol. 10823; pp. 3 - 14
• Main Authors: Chang, Jou-Ming, Pai, Kung-Jui, Yang, Jinn-Shyong, Wu, Ro-Yu
• Format: Book Chapter
• Language: English
• Published: Switzerland Springer International Publishing AG 2018; Springer International Publishing
• Series: Lecture Notes in Computer Science
• Subjects: Alternating group networks; Connectivity; Generalized connectivity; Interconnection networks
• ISBN: 3319784544; 9783319784540
• ISSN: 0302-9743; 1611-3349
• DOI: 10.1007/978-3-319-78455-7_1

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 $$n\geqslant 3$$ .