  Matching minors are a specialisation of minors fit for the study of graph
with perfect matchings. The notion of matching minors has been used to give a
structural description of bipartite graphs on which the number of perfect
matchings can becomputed 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 introducea version of Erdos-Posa property for matching minors and
find a direct link between this property and planarity. From this, it follows
that a class of bipartite graphs withperfect matchings has bounded perfect
matching width if and only if it excludes aplanar 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 graphGcontains afixed planar
graphHas 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.
