One-to-one communication in twisted cubes under restricted connectivity

The dimensions of twisted cubes are only limited to odd integers. In this paper, we first extend the dimensions of twisted cubes to all positive integers. Then, we introduce the concept of the restricted faulty set into twisted cubes. We further prove that under the condition that each node of the n-dimensional twisted cube TQ _n has at least one fault-free neighbor, its restricted connectivity is 2 n 2, which is almost twice as that of TQ _n under the condition of arbitrary faulty nodes, the same as that of the n -dimensional hypercube. Moreover, we provide an O ( N log N ) fault-free unicast algorithm and simulations result of the expected length of the fault-free path obtained by our algorithm, where N denotes the node number of TQ _n . Finally, we propose a polynomial algorithm to check whether the faulty node set satisfies the condition that each node of the n -dimensional twisted cube TQ _n has at least one fault-free neighbor.