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 2n - 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(NlogN) 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.