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This article presents a graph-theoretic method for constructing low-density parity-check (LDPC) codes from connected graphs without the requirement of large girth. This method is based on finding a set of paths in a connected graph, which satisfies the constraint that any two paths in the set are either disjoint or cross each other at one and only one vertex. Two trellis-based algorithms for finding these paths are devised. Good LDPC codes of practical lengths are constructed and they perform well with iterative decoding.