  The perfect matching complex of a simple graph G is a simplicial complex
having facets (maximal faces) as the perfect matchings of G. This article
discusses the perfect matching complex of polygonal line tiling and the
$\left(2 \times n\right)$-grid graph in particular. We use tools from discrete
Morse theory to show that the perfect matching complex of any polygonal line
tiling is either contractible or homotopically equivalent to a wedge of
spheres. While proving our results, we also characterise all the matchings that
can not be extended to form a perfect matching.
