Perfect Matching Complexes of Polygonal Line Tilings

TL;DR

This paper investigates the topological structure of perfect matching complexes in polygonal line tilings and grid graphs, revealing they are either contractible or homotopy equivalent to wedges of spheres, using discrete Morse theory.

Contribution

It characterizes the homotopy types of perfect matching complexes for polygonal line tilings and grid graphs, and identifies non-extendable matchings in grid graphs.

Findings

Perfect matching complexes are either contractible or wedge of spheres.

Characterization of non-extendable matchings in (2 x n)-grid graphs.

Application of discrete Morse theory to topological analysis.

Abstract

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 tilings 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 homotopy equivalent to a wedge of spheres. While proving our results, we also characterize all the matchings of $\left(2 \times n\right)$-grid graph that cannot be extended to form a perfect matching.