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<p><it>Abstract</it>—In this paper, we first reviewed a 1-fault-tolerant (1-ft) hypercube model with degree 2<it>r</it>, the ring-connected network (RCN), which has the lowest degree among all 1-ft, one spare node, <it>r</it>-dimensional hypercube architecture yet discovered. Then we proposed a constant-time reconfiguration algorithm via an add-and-modulo automorphism. Furthermore, by introducing the equivalence from hypercubes to cube-connected cycles (CCCs) and to butterflies (BFs), we find there is also a corresponding equivalence from RCNs to cubical ring connected cycles (CRCCs) and to dynamic redundancy networks (DRNs). From this fact, we find out that once a symmetric fault-tolerant structure has been discovered for one of the three models, then it can apply directly to the other hypercubic networks. Applying the technique, we find a degree 6, 1-ft Benes network. Another point is we think that the strong relationship between hypercubes, CCCs and BFs should be paid more attention, and finally from this equivalence relationship we propose three new bounded-degree <it>k</it>-ft models: <it>k</it>-ft CCCs, <it>k</it>-ft BFs, and <it>k</it>-ft Benes networks.</p>
Butterfly, circulant graph, cube-connected cycle, graph automorphism, hypercubic networks, interconnection network, multiple-fault tolerant, ring-connected network.

I. Y. Lee and S. Wang, "Ring-Connected Networks and Their Relationship to Cubical Ring Connected Cycles and Dynamic Redundancy Networks," in IEEE Transactions on Parallel & Distributed Systems, vol. 6, no. , pp. 988-996, 1995.
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