A recursive construction of an acyclic matching on the independence complex of a graph with a simplicial vertex

TL;DR

This paper introduces a recursive method to construct acyclic matchings on independence complexes of graphs with simplicial vertices, aiding in homotopy type determination and homology computation.

Contribution

It provides a new recursive construction technique for acyclic matchings on independence complexes, simplifying homotopy analysis of certain graph classes.

Findings

Determined homotopy types for chordal graphs and certain comparability graphs.

Enabled efficient homology computation through the recursive construction.

Extended previous results obtained by complex homotopy theoretic methods.

Abstract

We provide a recursive construction of an acyclic matching (also known as a gradient vector field, an equivalent notion to a discrete Morse function) on the independence complex of a graph with a simplicial vertex using given acyclic matchings on the independence complexes of specific subgraphs. As an application, we determine the homotopy type of the independence complexes of the family of chordal graphs and of a class of graphs generalising the comparability graphs of grid posets in an algorithmic and combinatorial manner via discrete Morse theory, some of which were previously obtained by sophisticated homotopy theoretic techniques. Even when the homotopy type is not easily determinable, our construction may be applied to obtain a pre-processing framework for efficient homology computation.