2-Matching Complexes

TL;DR

This paper investigates the topological properties of 2-matching complexes in graphs, using discrete Morse theory and algorithms to analyze their homotopy types, especially in specific graph classes like wheel and caterpillar graphs.

Contribution

It introduces homotopical results for 2-matching complexes, including transformations of their homotopy types and new concepts like k-matching sequences.

Findings

Homotopy type of 2-matching complexes can change from a sphere to a point with added leaves.

Applied discrete Morse theory and Matching Tree Algorithm for topological analysis.

Analyzed 1- and 2-matching complexes of wheel and perfect caterpillar graphs.

Abstract

A $2$-matching complex is a simplicial complex which captures the relationship between $2$-matchings of a graph. In this paper, we will use discrete Morse Theory and the Matching Tree Algorithm to prove homotopical results. We will consider a class of graphs for which the homotopy type of the $2$-matching complex transforms from a sphere to a point with the addition of leaves. We end the paper by defining $k$-matching sequences and looking at the $1$- and $2$-matching complexes of wheel graphs and perfect caterpillar graphs.