Braces of Perfect Matching Width 2

A graph G is called matching covered if it is connected and every edge is contained in a perfect matching. Perfect matching width is a width parameter for matching covered graphs based on a branch decomposition that can be considered a generalisation of directed treewidth. We show that the perfect matching width of every bipartite matching covered graph is within a factor of 2 of the perfect matching width of its braces. Moreover, we give characterisations for braces of perfect matching width in terms of edge maximal graphs similar to k-trees for undirected treewidth and elimination orderings. The latter allows us to identify braces of perfect matching width 2 in polynomial time and provides an algorithm to construct an optimal decomposition