Fault-Tolerant Path-Embedding of Twisted Hypercube-Like Networks (THLNs) - Huifeng Zhang; Xirong Xu; Qiang Zhang; Yuansheng Yang (Mathematics, 2019)
It is known widely that an interconnection network can be denoted by a graph <semantics> G = ( V , E ) </semantics> , where V denotes the vertex set and E denotes the edge set. Investigating structures of G is necessary to design a suitable topological structure of interconnection network. One of the critical issues in evaluating an interconnection network is graph embedding, which concerns whether a host graph contains a guest graph as its subgraph. Linear arrays (i.e., paths) and rings (i.e., cycles) are two ordinary guest graphs (or basic networks) for parallel and distributed computation. In the process of large-scale interconnection network operation, it is inevitable that various errors may occur at nodes and edges. It is significant to find an embedding of a guest graph into a host graph where all faulty nodes and edges have been removed. This is named as fault-tolerant embedding. The twisted hypercube-like networks ( <semantics> T H L N s </semantics> ) contain several important hypercube variants. This paper is concerned with the fault-tolerant path-embedding of n-dimensional (n-D) <semantics> T H L N s </semantics> . Let <semantics> G n </semantics> be an n-D <semantics> T H L N </semantics> and F be a subset of <semantics> V ( G n ) ∪ E ( G n ) </semantics> with <semantics> | F | ≤ n - 2 </semantics> . We show that for two different arbitrary correct vertices u and v, there is a faultless path <semantics> P u v </semantics> of every length l with <semantics> 2 n - 1 - 1 ≤ l ≤ 2 n - f v - 1 - α </semantics> , where <semantics> α = 0 </semantics> if vertices u and v form a normal vertex-pair and <semantics> α = 1 </semantics> if vertices u and v form a weak vertex-pair in <semantics> G n - F </semantics> ( <semantics> n ≥ 5 </semantics> ).