A question about the maximum independent set in a graph $G$

Question

Let $A$ and $B$ be two disjoint independent subsets in a graph $G$ such that $|A|=|B| \ge 1$, is it true that there must exist a maximum independent set $S_G$ of $G$ which satisfies $S_G\cap(A\cup B)=A \text{ or }B$? Here the maximum independent set means an independent set with the possible largest cardinality.

I'm trying to use contradiction,i.e., suppose that for every maximum independent set $S$ of $G$ we have $S\cap(A\cup B)\ne A$ and $S\cap(A\cup B)\ne B$, then I couldn't figure out what to do next, could any one help me out here?

Answer 1

To make a counter example we need a graph $G$ and two sets $A$ and $B$ such that every maximum independent set contains elements from $A$ and elements from $B$. This will be a counter example because then the intersection will contain elements from $A$ (and therefore cannot equal $B$) and also contain elements from $B$ (and therefore cannot equal $A$).

If $G$ contains an isolated vertex, that must be in any maximum independent set. If there are two isolated vertices we can put one in $A$ and one in $B$ and therefore force elements of both sets to be in every maximum independent set. Therefore there are graphs for which there are sets $A$ and $B$ such that there does not exist a $S_G$ with the required properties.