Let $G$ be a graph without any isolated vertices. Let $S$ be a minimal dominating set for $G$ (in other words, no subset of $S$ is a dominating set for $G$).
i. Prove that $V(G)$ − $S$ is also a dominating set for $G$.
ii. Prove that $V(G)$ − $S$ is not necessarily minimal.
I'm really stumped with these questions, any help would be greatly appreciated!
Hints:
i. A set $S$ is dominating if for every vertex $v \in V$, either $v \in S$ or $v$ is a neighbour of a vertex in $S$. Therefore to prove that $V \setminus S$ is dominating, you must prove that every vertex not in $V \setminus S$ is a neighbour of a vertex in $V \setminus S$. For this kind of question it can really help to draw a diagram! In particular, this diagram would have two groups of vertices: $V$ and $V \setminus S$, with some edges between them.
ii. Try constructing a counterexample on 3 vertices.