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

Jesus Gonzalez & Teresa I. Hoekstra-Mendoza
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...
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