# Graphs “weak” in context of cutting subgraphs

Lately I've been looking into graphs (simple, undirected, finite) that are in some way weak when it comes to connectivity, that is:

Let $$G$$ be a graph of order $$n$$. We'll say that $$G$$ is $$k$$-weak if for every induced connected subgraph $$H$$ of order $$k$$, $$G \setminus H$$ is disconnected.

I wish to find a good (or any) characterization of such graphs. It is fairly easy to show that trees are $$2$$-weak iff. every leaf is a neighbour of a vertex of a degree greater than $$2$$, and the condition is necessary for all graphs, but it isn't sufficient in general. However, I haven't been able to find any publications on the topic, even for $$k=2$$.