  Perfect matching width is a treewidth-like parameter designed for graphs with
perfect matchings. The concept was originally introduced by Norine for the
study of non-bipartite Pfaffian graphs. Additionally, perfect matching width
appears to be a useful structural tool for investigating matching minors, a
specialised version of minors related to perfect matchings. In this paper we
lay the groundwork for understanding the interaction of perfect matching width
and matching minors by establishing tight connections between the perfect
matching width of any matching covered graph $G$ and the perfect matching width
of its bricks and braces (a matching theoretic version of blocks) and proving
that perfect matching width is almost monotone under the matching minor
relation. As an application, we give several characterisations for braces of
perfect matching width two, including one that allows for a polynomial time
recognition algorithm.
