Independent vertex cover

An independent vertex cover is a vertex cover that is also an independent set. Its complement is an independent set as well, since the cover condition forbids edges entirely in that complement. Thus such a cover is a side of a bipartition of the graph. This standard definition is stronger than independent dominating set on graphs without isolated vertices, and remains a different notion when isolated vertices are present. In a nonempty simple graph, two possibly repeated witnesses $x,y$ with $\neg E(x,y)$ and no edge outside $\{x,y\}$ express existence of an independent vertex cover of size at most two.