Fault-free Path Of Length Research Articles

Conditional fault-tolerant routing of (n, k)-star graphs

As a generalization of the star graph, the -star graph is one of the most important interconnection networks for parallel and distributed computing system. In this paper, we investigate the fault-tolerant routing problem of -star graphs under the conditional fault assumption (CFA), where all the neighbours of any fault-free node cannot be faulty simultaneously. We show that, under the CFA, the-star graph can tolerate up to faulty nodes, and there is a fault-free path of length at most the diameter of the -star graph plus 6 between any two fault-free nodes.

Embedding cycles of various lengths into star graphs with both edge and vertex faults

The n-dimensional star graph S n is an attractive alternative to the hypercube graph and is a bipartite graph with two partite sets of equal size. Let F v and F e be the sets of faulty vertices and faulty edges of S n , respectively. We prove that S n − F v − F e contains a fault-free cycle of every even length from 6 to n! − 2∣ F v ∣ with ∣ F v ∣ + ∣ F e ∣ ⩽ n − 3 for every n ⩾ 4. We also show that S n − F v − F e contains a fault-free path of length n! − 2∣ F v ∣ − 1 (respectively, n! − 2∣ F v ∣ − 2) between two arbitrary vertices of S n in different partite sets (respectively, the same partite set) with ∣ F v ∣ + ∣ F e ∣ ⩽ n − 3 for every n ⩾ 4.

Longest paths and cycles in faulty star graphs

In this paper, we investigate the star graph S n with faulty vertices and/or edges from the graph theoretic point of view. We show that between every pair of vertices with different colors in a bicoloring of S n , n ⩾ 4 , there is a fault-free path of length at least n ! - 2 f v - 1 , and there is a path of length at least n ! - 2 f v - 2 joining a pair of vertices with the same color, when the number of faulty elements is n - 3 or less. Here, f v is the number of faulty vertices. S n , n ⩾ 4 , with at most n - 2 faulty elements has a fault-free cycle of length at least n ! - 2 f v unless the number of faulty elements are n - 2 and all the faulty elements are edges incident to a common vertex. It is also shown that S n , n ⩾ 4 , is strongly hamiltonian-laceable if the number of faulty elements is n - 3 or less and the number of faulty vertices is one or less.

Longest fault-free paths in star graphs with vertex faults

The star graph Sn has been recognized as an attractive alternative to the hypercube. Since S1,S2, and S3 have trivial structures, we focus our attention on Sn with n⩾4 in this paper. Let Fv denote the set of faulty vertices in Sn. We show that when |Fv|⩽n−5,Sn with n⩾6 contains a fault-free path of length n!−2|Fv|−2(n!−2|Fv|−1) between arbitrary two vertices of even (odd) distance. Since Sn is bipartite with two partite sets of equal size, the path is longest for the worst-case scenario. The situation of n⩾4 and |Fv|>n−5 is also discussed.