This is an updated cross-post from MO: https://mathoverflow.net/questions/340215/graphs-weak-in-context-of-cutting-subgraphs where it didn't get much attention.
Lately we'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.
We wish to find a good characterization of such graphs or any reference to that problem at all. What we know so far:
Trees are $2$-weak iff. there does not exist a pair of neighbours $(v_1, v_2)$ such that $deg_G(v_1) = 1$ and $deg_G(v_2) = 2$.
If there does not exist a pair of neighbours $(v_1, v_2)$ in a tree $T$ such that $deg_T(v_1) = 1$ and $deg_T(v_2) \leq k$, then $T$ is $k$-weak.
If a graph $G$ is $k$-weak, then the number of vertices of degree $1$ is greater or equal to the length of longest cycle.
The smallest $k$-weak tree is the star of order $k+1$, the smallest $2$-weak graph containing a cycle is a graph with cycle of length $3$ with the degree sequence $(4, 4, 2, 1, 1, 1, 1)$.
28.09.19 - if a graph $G$ is $2$-weak, then every leaf lies on an induced $K_{1, 3}$ (claw) subgraph.
We're also interested in graphs such that deleting any subgraph of a given diameter (instead of order) is the cutset, however that is not the main part of that question.