Abstract
We present new results on the fault tolerability of k-ary n-cube (denoted Q n k ) networks. Q n k is a topological model for interconnection networks that has been extensively studied since proposed, and this paper is concerned with the structure/substructure connectivity of Q n k networks, for paths and cycles, two basic yet important network structures. Let G be a connected graph and T a connected subgraph of G. The T-structure connectivity κ(G; T) of G is the cardinality of a minimum set of subgraphs in G, such that each subgraph is isomorphic to T, and the set's removal disconnects G. The T -substructure connectivity κ s (G; T) of G is the cardinality of a minimum set of subgraphs in G, such that each subgraph is isomorphic to a connected subgraph of T, and the set's removal disconnects G. In this paper, we study κ(Q n k ; T) and κ s (Q n k ; T) for T = P i , a path on i nodes (resp. T = C i , a cycle on i nodes). Lv et al. determined κ(Q n k ; T) and κ s (Q n k ; T) for T ∈ {P 1 , P 2 , P 3 }. Our results generalize the preceding results by determining κ(Q n k ; P i ) and κ s (Q n k ; P i ). In addition, we have also established κ(Q n k ; C i ) and κs(Q n k ; C i ).
Highlights
Interconnection networks play an important role in largescale multiprocessor systems
The g-extra connectivity κg(G) of a connected graph G is the minimum cardinality of a set of nodes in G, if such a set exists, whose deletion disconnects
The T -substructure connectivity κs(G; T ) of G is the cardinality of a minimum set of subgraphs F = {H1, H2, . . . , Hm}, such that every Hi ∈ F is isomorphic to a connected subgraph of T, and F’s deletion disconnects G
Summary
Interconnection networks play an important role in largescale multiprocessor systems. Lin et al [17] considered the fault status of a certain structure, rather than individual nodes, and proposed structure connectivity and substructure connectivity. Wang: Structure Connectivity and Substructure Connectivity of k-Ary n-Cube Networks. There are already many works in the literature studying path/cycle-structure fault tolerance for some well-known networks. We determine the path- and cycle-structure/substructure connectivity for k-ary n-cubes. Our result of κ(Qkn; C2l+1) ≤ 2n − 2 provides an upperbound on the structure connectivity for odd-cycles. This ‘‘half-solved’’ κ(Qkn; C2l+1) and the unknown κ(Q3n; C3l+1) are the two missing pieces for a complete solution to Qkn’s structure/substructure connectivity for paths and cycles.
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