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Fault-Tolerant Path-Embedding of Twisted Hypercube-Like Networks (THLNs)

School of Computer Science and Technology, Dalian University of Technology, Dalian 116024, China
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Mathematics 2019, 7(11), 1066; https://doi.org/10.3390/math7111066
Received: 18 August 2019 / Revised: 15 October 2019 / Accepted: 22 October 2019 / Published: 6 November 2019
(This article belongs to the Section Network Science)
It is known widely that an interconnection network can be denoted by a graph G = ( V , E ) , 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 ( T H L N s ) contain several important hypercube variants. This paper is concerned with the fault-tolerant path-embedding of n-dimensional (n-D) T H L N s . Let G n be an n-D T H L N and F be a subset of V ( G n ) E ( G n ) with | F | n - 2 . We show that for two different arbitrary correct vertices u and v, there is a faultless path P u v of every length l with 2 n - 1 - 1 l 2 n - f v - 1 - α , where α = 0 if vertices u and v form a normal vertex-pair and α = 1 if vertices u and v form a weak vertex-pair in G n - F ( n 5 ).
Keywords: combinatorics; multiprocessor interconnection networks; computer network reliability; network topology; hypercubes; twisted hypercube-like networks THLNs; fault tolerance; path-embedding combinatorics; multiprocessor interconnection networks; computer network reliability; network topology; hypercubes; twisted hypercube-like networks THLNs; fault tolerance; path-embedding
MDPI and ACS Style

Zhang, H.; Xu, X.; Zhang, Q.; Yang, Y. Fault-Tolerant Path-Embedding of Twisted Hypercube-Like Networks (THLNs) . Mathematics 2019, 7, 1066.

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