 Solution

The source's explicit parenthetical condition is domination: every unselected vertex has a neighbour in the selected set. Together with independence this defines an independent dominating set. A first-order sentence expressing that definition with size at most three isThe finite conjunctions and disjunctions abbreviate three explicitly fixed terms, so this is a first-order formula. Repeated witnesses allow one- or two-element choices; the first conjunction enforces independence only between distinct selected vertices. The last clause says that every vertex is selected or adjacent to a selected vertex. With the usual nonempty-domain convention, any such choice has size at least one, so repeated witnesses cover all possibilities of size at most three.

For the standard meaning of independent vertex cover, replace the domination clause byThis instead demands that every edge has a selected endpoint. These interpretations differ: in a triangle in a graph, any singleton is an independent dominating set, but no independent vertex cover exists. Giving both clauses avoids silently treating the source's two definitions as equivalent.