# Maximum size of vertex set with no induced connected component on more than k vertices

An independent set of a graph is a collection of vertices such that the induced subgraph consists of disconnected vertices. The maximum possible cardinality of an independent set is then called the independence number of the graph.

For $$k$$ a positive integer, we define a $$k$$-independent set of a graph to be a collection of vertices such that the induced subgraph consists of components with $$k$$ vertices or fewer. (In particular, a $$1$$-independent set is the same as an independent set.) We then define the $$k$$-independence number of a graph to be the maximum possible cardinality of a $$k$$-independent set.

Have these notions of $$k$$-independent sets and $$k$$-independence numbers been studied before? If so, what names do they go by in the literature?

(Note: the term "$$k$$-independence number" does appear in the literature, but it asks for the induced subgraphs to have maximum degree at most $$k$$, rather than at most $$k$$ vertices.)

There is a version of graph colouring where instead of insisting that each colour class is an independent set, you require each colour class to be a $$c$$-independent set. This is known as a colouring with clustering $$c$$. The clustered chromatic number of a class of graphs is the minimum $$k$$ such that for some integer $$c$$, every graph in the class is $$k$$-colourable with clustering $$c$$. Clustered chromatic number has received quite a lot of attention recently. For example, van den Heuvel and Wood proved that every $$K_t$$-minor free graph is $$(2t-2)$$-colourable with clustering at most $$\lceil\frac{1}{2}(t-2) \rceil$$. Note that the analogous statement for the chromatic number is the linear Hadwidger's conjecture, and is still open.