Let $G$ be an undirected graph with no multiple edges or loops. Recall that the independece system $\mathcal{I}(G)$ consists of all those subsets $A$ of the vertex set such that the induced subgraph $G[A]$ is totall disconnected. The independence system is an abstract simplicial complex and a lot of its topological invariants are related to combinatorial properties (like, chromatic number, domination number etc.) of the underlying graph.

I have following two generalization of the independence system in mind. A simple Google search did not yield any work that explores these complexes.

- For a natural number $j\geq 2$ denote by $\mathcal{C}_j(G)$ the collection of subsets of vertices of $G$ such that for $A$ in that collection the induced subgraph $G[A]$ does not contain a $j$-clique.
- For a natural number $r\geq 1$ denote by $\mathcal{E}_r(G)$ the collection such that for every $A\in \mathcal{E}_r(G)$ each connectedd component of $G[A]$ has at most $r$ vertices.

It is clear from above definitions that $\mathcal{I}(G) = \mathcal{C}_2(G) = \mathcal{E}_1(G)$. Can anybody point me to a reference where either of these generalizations are studied? I would like to know if any of topological invariants of these complexes relate to graph theoretic information.