The notion of matching minors is a specialisation of minors fit for the study of graphs with perfect matchings. Matching minors have been used to give a structural description of bipartite graphs on which the number of perfect matchings can be computed efficiently, based on a result of Little, by McCuaig et al. in 1999.

In this paper we generalise basic ideas from the graph minor series by Robertson and Seymour to the setting of bipartite graphs with perfect matchings. We introduce a version of Erdős-Pósa property for matching minors and find a direct link between this property and planarity. From this, it follows that a class of bipartite graphs with perfect matchings has bounded perfect matching width if and only if it excludes a planar matching minor. We also present algorithms for bipartite graphs of bounded perfect matching width for a matching version of the disjoint paths problem, matching minor containment, and for counting the number of perfect matchings. From our structural results, we obtain that recognising whether a bipartite graph G contains a fixed planar graph H as a matching minor, and that counting the number of perfect matchings of a bipartite graph that excludes a fixed planar graph as a matching minor are both polynomial time solvable.

匹配次要概念是次要的专门化，适用于完美匹配图的研究。匹配次要已被用来给出二分图的结构描述，基于 Little 的结果，可以有效地计算完美匹配的数量，由 McCuaig 等人提出。1999年。

在本文中，我们将 Robertson 和 Seymour 的图小级数的基本思想推广到具有完美匹配的二部图的设置。我们引入了用于匹配未成年人的 Erdős-Pósa 属性的版本，并找到了该属性与平面性之间的直接联系。由此可见，一类具有完美匹配的二分图当且仅当它排除平面匹配次要时才具有有界完美匹配宽度。我们还提出了有界完美匹配宽度的二分图算法，用于不相交路径问题的匹配版本、匹配次要包含以及计算完美匹配的数量。从我们的结构结果中，我们可以识别二部图G是否包含固定平面图H作为匹配次要项，以及计算排除固定平面图作为匹配次要项的二部图的完美匹配数量都是多项式时间可解的。