Characterizing graphs of maximum matching width at most 2
  The maximum matching width is a width-parameter that is defined on a branch-decomposition over the vertex set of a graph. The size of a maximum matching in the bipartite graph is used as a cut-function. In this paper, we characterize the graphs of maximum matching width at most 2 using the minor obstruction set. Also, we compute the exact value of the maximum matching width of a grid. 
Jisu Jeong, Seongmin Ok, Geewon Suh
math.CO cs.DM