Divide the vertices (arbitrarily) into V<sub>1</sub> and V<sub>2</sub> each of n
vertices.  Respond to each question of the algorithm as follows:

<ul>

<li>a) If the algorithm asks about u,v in V<sub>1</sub> say "yes, (u,v) in E."
<li>b) If the algorithm asks about u,v in V<sub>2</sub> say "yes, (u,v) in E."
<li>c) If the algorithm asks about u in V<sub>1</sub> and v in V<sub>2</sub> say "no, (u,v) not in E."
</ul>

Notice that that algorithm cannot know G is not connected until it
has asked about every edge fitting under rule (c) above (since the
existence of any one of those edges in G would connect the graph).
The number of edges fitting rule (c) above is n<sup>2</sup> and hence at least
n<sup>2</sup> vertex pairs must be tested.