Past exam of the year 1 of the computer science course of the University of Oxford / 2023 / Digital systems, linear algebra and introduction to formal proof / 8 / a

The nonempty simple graphs having an independent vertex cover of size at most two. The witnesses $x,y$ select the cover. The clause $\neg E(x,y)$ makes the selected vertices nonadjacent; the universal clause forbids every edge whose two endpoints both lie outside the selected set.

The witnesses need not be distinct, so a one-vertex cover is permitted by choosing $x=y$. A nonempty edgeless graph also satisfies the formula. The empty graph does not, because the existential quantifiers have no witnesses. Here independent vertex cover has its standard vertex cover meaning; 8(b)'s differing parenthetical definition is addressed separately below.