Complex of disconnected graphs

Naturally, the examples are endless, and easy to make up. One instance, which has appeared in knot theory, comes from the graph property of being disconnected. It can be shown that the complex of disconnected graphs on n vertices is homotopy equivalent to the order complex of the partition lattice iT see Proposition 13.15. 

Also, for the complexes of combinatorial properties, the description can be more succinct when it uses the forbidden patterns instead of the allowed ones. For example, for the complex of disconnected graphs, the minimal non-simplices correspond to spanning trees. For the complex of directed forests, the minimal nonsimplices are all pairs of directed edges that have the same end vertex, together with all directed cycles. 

For the connection between complexes of disconnected graphs and knot theory, see Vassiliev, [Va93]. Complexes of directed trees first appeared in [Ko99]. 

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