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

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 $\gamma \le n-3$.