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Zadeh has shown that any self-dual network, fixed or linear time varying, is a constant resistance network. To date, the only known constant resistance networks with self-dual structures are the classical lattice and bridged-T networks. In this paper, we investigate the topological aspect of the problem, with the aim of obtaining new constant resistance network configurations. Let be a self-dual one-terminal-pair graph with respect to vertices ( ), and with the degrees of ( ) both equal to . It is proved that for can be realized with edges, but not with fewer edges, if the union of and an edge joining ( ) is to be 3-connected. Using these graphs as the basis, a class of constant resistance networks are generated, which include the classical lattice and bridged-T networks as special cases for . The generation of a constant resistance network for is shown in detail, with a numerical example illustrating its application in transfer function synthesis.