I am studying discrete Morse theory and as an example, discrete Morse theory is used to obtain the homotopy type of the complex of non-connected graphs of $n$ vertices. I also read that this kind of complex can be obtained from any graph property that remains valid when you remove edges. So I thought what about other graph properties? I did some research and found results for some graph properties like being bipartite. One property that I thought about is having domintion number grater than $k$. I did it for $n=4$ and $k=2$, I obtained that this complex has the homotopy type of the wedge of three 3-spheres. I tried to do the same for $n=5$ and $k=3$ or to get a general argument for any $n$ and $k=n-2$ following the idea of the case of the complex of non connected graphs but got nowhere with it. What I tried was constructing a discrete vector field on the complex but I couldn't get one that was good enough. I tried constructing it inductively like the case of the non-connected graphs. Does anyone have an idea or has this been done in any way?
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