Fault-tolerant path embedding in folded hypercubes with both node and edge faults

Abstract

The folded hypercube FQn is a well-known variation of the hypercube structure. FQn is superior to Qn in many measurements, such as diameter, fault diameter, connectivity, and so on. Let Ṽ(FQn) (resp. Ẽ(FQn)) denote the set of faulty nodes (resp. faulty edges) in FQn. In the case that all nodes in FQn are fault-free, it has been shown that FQn contains a fault-free path of length 2n−1 (resp. 2n−2) between any two nodes of odd (resp. even) distance if |Ẽ(FQn)|≤n−1, where n≥1 is odd; and FQn contains a fault-free path of length 2n−1 between any two nodes if |Ẽ(FQn)|≤n−2, where n≥2 is even. In this paper, we extend the above result to obtain two further properties, which consider both node and edge faults, as follows: 1.FQn contains a fault-free path of length at least 2n−2|Ṽ(FQn)|−1 (resp. 2n−2|Ṽ(FQn)|−2) between any two fault-free nodes of odd (resp. even) distance if |Ṽ(FQn)|+|Ẽ(FQn)|≤n−1, where n≥1 is odd.2.FQn contains a fault-free path of length at least 2n−2|Ṽ(FQn)|−1 between any two fault-free nodes if |Ṽ(FQn)|+|Ẽ(FQn)|≤n−2, where n≥2 is even.