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Communication plays a key role in the overall performance of a parallel system. In the general case, the communication network is required to transfer information from any set of sources to to any set of destinations. In practice, parallel applications have various communication needs. Many applications restrict their communication requests to a more limited number of possible connections between sources and destinations. The communication requests of parallel applications are represented by graphs. Parallel graph contraction has been used to embed these graphs into a reconfigurable parallel architecture. We review some of the previous work done in this area showing how various useful communication graphs (e.g., ring, kary trees, mesh, CCC, hypercube, etc.), can be contracted for an efficient embedding. We also present some results for the parallel contraction of other useful graphs such as de Bruijn, shuffle-exchange, mesh of trees, pyramid, and multi-stage communication structures. Lower bounds are proved for all graphs. A heuristic approach for parallel graph contraction is reviewed for irregular graphs based on simulated annealing.