In this paper, we propose two algorithms, B-N2N and B-N2S, that solve the node-to-node and node-to-set disjoint paths problems in the bicube, respectively. We prove their correctness and that the time complexities of the B-N2N and B-N2S algorithms are O(n2) and O(n2 log n), respectively, if they are applied in an n-dimensional bicube with n ≥ 5. Also, we prove that the maximum lengths of the paths generated by B-N2N and B-N2S are both n + 2. Furthermore, we have shown that the algorithms can be applied in the locally twisted cube, too, with the same performance.

In a multiprocessor system, processors are connected based on various types of network topologies. A network topology is usually represented by a graph. Let G be a graph and u, v be any two distinct vertices of G. We say that G is pancyclic if G has a cycle C of every length l(C) satisfying 3≤l(C)≤|V(G)|, where |V(G)| denotes the total number of vertices in G. Moreover, G is panpositionably pancyclic from r if for any integer m satisfying $r leq m leq rac{|V(G)|}{2}$, G has a cycle C containing u and v such that dC(u,v)=m and 2m≤l(C)≤|V(G)|, where dC(u,v) denotes the distance of u and v in C. In this paper, we investigate the panpositionable pancyclicity problem with respect to the n-dimensional locally twisted cube LTQn, which is a popular topology derived from the hypercube. Let D(LTQn) denote the diameter of LTQn. We show that for n≥4 and for any integer m satisfying $D(LTQ_n) + 2 leq m leq rac{|V(LTQ_n)|}{2}$, there exists a cycle C of LTQn such that dC(u,v)=m, where (i) 2m+1≤l(C)≤|V(LTQn)| if m=D(LTQn)+2 and n is odd, and (ii) 2m≤l(C)≤|V(LTQn)| otherwise. This improves on the recent result that u and v can be positioned with a given distance on C only under the condition that l(C)=|V(LTQn)|. In parallel and distributed computing, if cycles of different lengths can be embedded, we can adjust the number of simulated processors and increase the flexibility of demand. This paper demonstrates that in LTQn, the cycle embedding containing any two distinct vertices with a feasible distance is extremely flexible.

In this paper, a new family of interconnection networks which we call the Incrementally Extensible Twisted Cube (IETQ) is proposed. The topology of this network is a novel generalization of the twisted cube. It inherits all the merits but without the limitations owned by a twisted cube. First, this proposed IETQ is incrementally extensible and can be adapted for use in any number of nodes; therefore, this network is particularly well suited for the design of a distributed communication network with an arbitrary number of nodes. Second, the vertex connectivity of IETQ is n. Measured by this vertex connectivity, we demonstrate that this network is optimally fault-tolerant . And it is almost regular, because the difference between the maximum and minimum degree of any node in an IETQ is at most one. A shortestpath routing algorithm for IETQ is proposed to generate path for any given pair of vertices in the network. Third, comparing with most of the other competitors, the diameter of this IETQ network is only half in size. This low diameter helps to reduce the internode communication delay. Moreover, IETQ also possesses the property of a pancyclic network. This attractive property would enable us to map rings of any length into the proposed network.