Matching complexes of polygonal line tilings

The matching complex of a simple graph $G$ is a simplicial complex consisting of the matchings on $G$. Jeli'c Milutinovi'c et al. studied the matching complexes of the polygonal line tilings, and they gave a lower bound for the connectivity of the matching complexes of polygonal line tilings. In this paper, we determine the homotopy types of the matching complexes of polygonal line tilings recursively, and determine their connectivities.