Can an independent set (of vertices) be a vertex cover as well? I wanted to clarify if this is possible, so I thought about a possible vertex cover that can also serve as an independent set:



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 &lt;-> V-S is a vertex cover." So I thought that a vertex cover and an independent set are mutually exclusive.
If something satisfies the definition of&nbsp;$X$ and the definition of&nbsp;$Y$, then it is both an&nbsp;$X$ and a&nbsp;$Y$.

Although it's true that, whenever $S$&nbsp;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 $S$&nbsp;is one side of a bipartite graph, as is the case in the example in the question.