Constrained path-finding and structure from acyclicity
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This note presents several results in graph theory inspired by the author's work in the proof theory of linear logic; these results are purely combinatorial and do not involve logic. We show that trails avoiding forbidden transitions and rainbow paths for complete multipartite color classes can be found in linear time, whereas finding rainbow paths is NP-complete for any other restriction on color classes. For the tractable cases, we also state new structural properties equivalent to Kotzig's theorem on bridges in unique perfect matchings. We also exhibit a connection between blossoms and bridge deletion orders in unique perfect matchings.
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