This paper establishes a bridge between linear logic and mainstream graph theory, building on previous work by Retor\'e (2003). We show that the problem of correctness for MLL+Mix proof nets is equivalent to the problem of uniqueness of a perfect matching. By applying matching theory, we obtain new results for MLL+Mix proof nets: a linear-time correctness criterion, a quasi-linear sequentialization algorithm, and a characterization of the sub-polynomial complexity of the correctness problem. We also use graph algorithms to compute the dependency relation of Bagnol et al. (2015) and the kingdom ordering of Bellin (1997), and relate them to the notion of blossom which 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 forbidden transitions. In fact, it is by analyzing RB-graphs that we arrived at this construction, and thus obtained a polynomial-time algorithm for finding trails avoiding forbidden transitions; the latter is among the material covered in another paper by the author focusing on graph theory (arXiv:1901.07028).

中文翻译：

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具有混合的线性逻辑的独特完美匹配、禁止转换和证明网络

本文在 Retor\'e (2003) 先前工作的基础上，在线性逻辑和主流图论之间架起了一座桥梁。我们表明，MLL+Mix 证明网络的正确性问题等价于完美匹配的唯一性问题。通过应用匹配理论，我们获得了 MLL+Mix 证明网络的新结果：线性时间正确性标准、准线性序列化算法以及正确性问题的子多项式复杂度的表征。我们还使用图算法来计算 Bagnol 等人的依赖关系。(2015) 和 Bellin (1997) 的王国排序，并将它们与开花的概念联系起来，这是组合最大匹配算法的核心。在这个期刊版本中，我们增加了对 Retor\'e 的“RB-graphs”的解释 在具有禁止转换的图的一般构造方面。事实上，正是通过分析RB-graphs，我们得出了这种构造，从而获得了一个多项式时间算法，用于寻找避免禁止转换的轨迹；后者是作者在另一篇专注于图论的论文 (arXiv:1901.07028) 中涵盖的材料之一。