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This paper establishes a bridge between linear logic and mainstream graphtheory, building on previous work by Retor\'e (2003). We show that the problemof correctness for MLL+Mix proof nets is equivalent to the problem ofuniqueness of a perfect matching. By applying matching theory, we obtain newresults for MLL+Mix proof nets: a linear-time correctness criterion, aquasi-linear sequentialization algorithm, and a characterization of thesub-polynomial complexity of the correctness problem. We also use graphalgorithms to compute the dependency relation of Bagnol et al. (2015) and thekingdom ordering of Bellin (1997), and relate them to the notion of blossomwhich is central to combinatorial maximum matching algorithms.  In this journal version, we have added an explanation of Retor\'e's"RB-graphs" in terms of a general construction on graphs with forbiddentransitions. In fact, it is by analyzing RB-graphs that we arrived at thisconstruction, and thus obtained a polynomial-time algorithm for finding trailsavoiding forbidden transitions; the latter is among the material covered inanother paper by the author focusing on graph theory (arXiv:1901.07028).

Unique perfect matchings, forbidden transitions and proof nets for linear logic with Mix