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. 


1. 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.
2. 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. 



Generalization of independence complex of graphs
Although I am not sure if these concepts been studied much as simplicial complexes, at least the second has been studied quite extensively in graph theory and theoretical computer science (including by me).


The earliest studies I am aware of are


Sampathkumar. Discrete Math., 1993. <https://doi.org/10.1016/0012-365X(93)90493-D>


Edwards and Farr. JCTB, 2001. <https://doi.org/10.1006/jctb.2000.2018>


These come at the problem from independent perspectives, but the concept is natural enough that maybe it gets reintroduced regularly.


For instance, it has some connection to vaccination protocols:


Britton, Janson, Martin-Löf. Adv. Appl. Prob., 2007. <https://doi.org/10.1239/aap/1198177233>


There is a rather extensive study of efficient partitions of the vertex set of a graph into such (and related) types of sets, a partial (dynamic) survey of which is given by Wood in the Electr. J. Comb.


As an act of pure vanity, my own study of this type of independence concerns the binomial random graph:


Broutin and Kang. J. Comb., 2019. <http://dx.doi.org/10.4310/JOC.2018.v9.n3.a1>


As for the first concept, it has also been studied a bit I believe, but less so (probably because it is much harder to make progress on).


