On the homotopy type of complexes of graphs with bounded domination number

TL;DR

This paper investigates the topological structure of complexes of graphs with bounded domination number, proving that for certain parameters, these complexes are homotopy equivalent to wedges of spheres, using discrete Morse theory techniques.

Contribution

It establishes the homotopy type of complexes of graphs with high domination number and extends the analysis to infinite graphs, introducing new acyclic matching methods.

Findings

$D_{n,n-2}$ is homotopy equivalent to a wedge of 2-spheres

The approach extends to infinite graphs

Homotopy equivalences for certain domination numbers do not generalize for lower values.

Abstract

Let $D_{n,\gamma}$ be the complex of graphs on $n$ vertices and domination number at least $\gamma$. We prove that $D_{n,n-2}$ has the homotopy type of a finite wedge of 2-spheres. This is done by using discrete Morse theory techniques. Acyclicity of the needed matching is proved by introducing a relativized form of a well known method for constructing acyclic matchings on suitable chunks of simplices. Our approach allows us to extend our results to the realm of infinite graphs. In addition, we give evidence supporting the assertion that the homotopy equivalences $D_{n,n-1}\simeq\bigvee S^0$ and $D_{n,n-2}\simeq\bigvee S^2$ do not seem to generalize for $D_{n,\gamma}$ with $\gamma\leq n-3$.