And the solution that we hear is that you can do a vertex cover
with just three vertices, so here's one example.
You could select this one which would cover all of those edges here.
Now we could select this one, covering all those edges.
And, finally, we could select the one down here.
So all edges are covered selecting just three vertices,
and there are alternative solutions but none that contain less than three vertices.
Now what about over here--here we can find
a very small vertex cover using just two vertices.
This vertex here, we just don't even need to care about
because it has no edges that it's connected to.
We can select this one here--it covers all of those edges.
And we can select, for example, this one here.
Or not only for example because this is actually the only possible smallest solution.
So now we did this exercise to see if vertex cover could have any relation
to clique or independent set.
So let's have a look back at which vertices were contained
in the largest possible clique in this network,
and those were four vertices--it was this one here, this one here,
this one, and this one.
Not any really apparent relation between the two problems.
This one over here, however, is more interesting
because in the largest possible independent set, we had those four here.
So it could seem like vertex cover--those two green vertices here--
is exactly the opposite of independent set,
so if you have found the largest possible independent set,
then you have found the smallest possible vertex cover.
And, indeed, it's actually not that difficult to see that this is always the case.
