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I wanted to clarify if this is possible, so I thought about a possible vertex cover that can also serve as an independent set:

Independent Set/Vertex Cover
So, to clarify, am I right to say that the nodes in red are both (i) a vertex cover, and (ii) an independent set?

Edit: I was confused by the theorem that states "If G = (V,E) is a graph, then S is an independent set <-> V-S is a vertex cover." So I thought that a vertex cover and an independent set are mutually exclusive.

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If something satisfies the definition of $X$ and the definition of $Y$, then it is both an $X$ and a $Y$.

Although it's true that, whenever $S$ is independent, $V\setminus S$ is a vertex cover, it's possible for both $S$ and $V\setminus S$ to be independent: this happens when $S$ is one side of a bipartite graph, as is the case in the example in the question.

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