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We describe the application of edge-local complementation (ELC) to a Tanner graph associated with a binary linear code, C. Various properties of ELC are described, mainly the special case of isomorphic ELC operations and the relationship to the automorphism group of the code, Aut(C), and the generalization of ELC to weight-bounding ELC (WB-ELC) operations under which the number of edges remains upper-bounded. ELC generates all systematic parity-check matrices (the orbit) of the code, so WB-ELC facilitates a restriction to low-weight matrices of this orbit. We propose using ELC and WB-ELC as a source of diversity, to improve iterative soft-input soft-output decoding of high-density parity-check (HDPC) codes, with the sum-product algorithm (SPA). A motivation of ELC-enhanced SPA decoding is locality; that diversity is achieved by local graph action, and is well-suited to the local actions that constitute the SPA and allows for parallel software implementation. Simulation data on the error-rate performance of the proposed SPA-ELC and SPA-WBELC iterative decoding algorithms is shown for several HDPC codes. A gain is reported over SPA decoding, and over a recently proposed algorithm to decode HDPC codes using permutations from Aut(C). ELC-enhanced decoding extends the scope of iterative decoding to codes with trivial Aut(C).