AI 中文总结
该研究在网格模型中定义二分图的特征m，将n-可扩展性概念扩展到不平衡二分图，基于m将二分图划分为11类，提供了统一的几何分析框架。
AI 中文摘要
我们在网格模型中研究有限、连通、简单的二分图，其中图被绘制为矩形数组，其结构可从称为“洞”的空子矩形中读取。在该模型中，我们为每个块（brick）附加一个数值不变量，即其特征m，它是行数与最大真独立集的大小之差。若m>0，则该块为过度块（excessive）。我们的主要结果涉及该不变量：我们从其各个分量的特征中确定了不连通过度块的特征，表明m = min_i min{m_i, imb(W_i)}，而不平衡性是可加的；并且我们证明了一个m-过度块是m-可扩展的，即每个大小为m的匹配都可扩展为最大匹配。由于Plummer提出的n-可扩展性概念仅针对具有完美匹配的图定义，而我们的证明从未使用平衡性，因此该特征将该概念规范地扩展到了不平衡二分图。利用该特征，我们将二分图划分为11个结构类。底层的原子块分解是经典的基本分量分解，而通过块偏序集的理想对最大真独立集的描述同样是经典的；本文明确指出了哪些结果是经典的，哪些不是本文所主张的。网格模型所提供的是一个单一的几何框架，其中洞、特征和块三角形式都可从一张图中读取。
英文摘要
We study finite, connected, simple bipartite graphs in a grid model, in which a graph is drawn as a rectangular array and its structure is read off from empty subrectangles, called holes. In this model we attach to every brick a numerical invariant, its characteristic m, the difference between the number of rows and the largest proper independent set. A brick is excessive if m > 0. Our main results concern this invariant. We determine the characteristic of a disconnected excessive brick from those of its components, showing that m = min_i min{m_i, imb(W_i)} while the imbalance is additive; and we prove that an m-excessive brick is m-extendable, that is, every matching of size m extends to a maximum matching. Since Plummer's notion of n-extendability is defined only for graphs carrying a perfect matching, and our proof nowhere uses balance, the characteristic extends that notion canonically to unbalanced bipartite graphs. Using the characteristic we partition bipartite graphs into eleven structural classes. The underlying decomposition into atomic blocks is the classical decomposition into elementary components, and the description of the maximum proper independent sets by ideals of the block poset is likewise classical; the paper states precisely which results are classical and are not claimed here. What the grid model adds is a single geometric framework in which holes, characteristics and the block triangular form are read off from one picture.
Comments60 pages. Structural decomposition of bipartite graphs via a grid model. Introduces the characteristic m of a brick, determines it for disconnected excessive bricks, shows that m-excessiveness implies m-extendability with the same m, extending Plummer's notion to unbalanced graphs, and classifies bipartite graphs into eleven structural classes
