Maximum independent set of a residual graph

Question

Given a graph $G$ and a maximum independent set (MIS) $I$, can we comment on the size of a maximum independent set of the residual graph $G\setminus I$? Note that $G\setminus I$ denotes the induced graph obtained from $G$ after deleting the set of vertices in $I$.

A trivial upper bound is: $|\text{MIS}(G\setminus I)| \le \text{MIS}(G)=|I|$

Do we have a reasonable lower bound?

Also, instead of deleting a MIS, what happens if we delete some other subset of vertices, say a maximal independent set of $G$?

Any pointers to existing works or how to approach this problem are appreciated.

Answer 1

Not much can be said in general: your graph could consist of some number of independent sets of a fixed size, and then a lot of edges between these. For instance, the graph $K_{t,t}$ consisting of two independent sets $I_1, I_2$ of size $t$ and all other edges present, satisfies $|MIS(K_{t,t})| = t = |MIS(K_{t,t}\backslash I_1)|$.