Braces of Perfect Matching Width 2

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.