Lauren Williams pointed me toward https://arxiv.org/abs/math/0609764 (the original reference for Postnikov's work) as well as https://arxiv.org/abs/0706.2501 (an article by Postnikov, Speyer, and Williams with more of an emphasis on matchings and flows). In this setting one has Plucker relations that are essentially the same thing as Kuo's graphical condensation.
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replaced the link to the arXiv front end; see https://meta.mathoverflow.net/questions/5124/is-it-time-to-replace-links-to-the-ucdavis-arxiv-frontend
Martin Sleziak
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James Propp
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