PROFESSOR: So connectivity is more than just
an all or nothing affair.
We can talk about how connected a graph is.
So let's begin with two vertices.
Two vertices are said to be k-edge connected
if they remain connected if you remove fewer
than k edges from the graph.
Let's look at an example.
So here's a graph, and let's focus on those two vertices
that I've highlighted in magenta.
They are 1-edge connected because they're connected,
and if you remove one edge they become disconnected.
So they're 1-edge connected, but they're not 2-edge connected.
In particular, if I delete that edge,
then they no longer is a path between the two magenta
vertices.
Here's an example of two vertices, these two
green vertices, that are 2-edge connected.
That means that I can remove any number of edges less than two,
which is to say one edge, and they'll stay connected.
But if I do remove two vertices, they become disconnected.
So with those two edges judiciously selected,
there will no longer be a path between the two green vertices.
For a 3-edge connected example, we
can look at these two vertices highlighted in purple
and discover that we can remove any two edges,
and they're still going to be stuck together.
You can kind of see that because they're in a cycle here
and in a cycle there.
And in order to break them, you're
going to have to break both cycles, which means
two edges to break two cycles.
And if you remove three, on the other hand,
you can separate them.
So if I cut these three edges-- might
have been more obvious to cut that one,
but I'm doing it this way to make it interesting--
if I cut these three edges, then in fact I
wind up with no path between those two vertices.
So they are 3-edge connected but not 4-edge connected.
Meaning you can remove two, but you can't remove three
and keep them connected.
For a whole graph, a whole graph is k-edge
connected if every two vertices are k-edge connected.
And the point of this degree of connectivity
is to measure-- one motivation of it
is if you think of this graph as indicating a communication
network, where vertices are centers that are sending
and receiving data, and an edge corresponds
to having some kind of a channel or cable
between adjacent centers.
Then connectivity is a measure of how many of your channels
or cables can go down and still have the property
that every center of information can
communicate with every other center of information.
So how many connections can fail without cutting off
communication?
Here's an example of a graph that globally-- this
was the graph that we saw before that has some vertices that
are 1-edge connected, others that are 2, others that are 3.
But the graph as a whole is only 1-edge
connected because there's this one edge that can be removed
that splits it in two.
This graph, on the other hand, a modification of it,
is 2-edge connected.
Now I can remove any one edge, and no matter
what single edge I remove, this will
remain a connected graph, which makes it 2-edge connected.
Now, there's a corresponding definition of vertex connected,
meaning it's k-vertex connected if you
can remove any number of vertices up to k
and it will stay connected.
If it's k-vertex connected, it's certainly k-edge connected,
but not conversely.
Here's a graph that is 2-edge vertex connected but 1-vertex
connected.
If you remove one vertex, it breaks.
If you remove that vertex, it breaks.
But on the other hand, it needs two edges
to be cut in order to split the graph in two.
The complete graph on n vertices is-- to cut it up,
it requires cutting n minus 1 edges
and, in fact, n minus 1 vertices to break it up
because everything is connected to everything else.
So Kn is as connected a graph on n
vertices as you can possibly have.
But of course, it's got a lot of edges.
It's got n choose 2 edges, or about n
times n minus 1 over 2 edges.
Another interesting structure is the n-dimensional hypercube.
So the square is a 2-dimensional,
and the cube, the ordinary cube is the H3, has eight vertices
and looks like a cube.
And in general, the way we define Hn
is that its vertices are the binary strings of length n.
And two vertices are adjacent if they differ in one bit.
They're the same except that in one coordinate
one of them has a 1 and the other has a 0.
In that case, you can make the edges adjacent.
It turns out that the n-dimensional hypergraph
is n-vertex connected.
And this is worked out in a homework or a class problem.
So to summarize-- well, actually before we summarize,
there's a basic theorem that we're not going to prove.
It's just a little bit too much of a challenge
for us to get into.
It would take maybe a couple of lectures--
is to prove Menger's theorem, which says that a graph is
k-vertex connected if and only if you can connect any two
vertices with completely non-overlapping different paths
k of them.
So if two vertices are k-connected,
then there are completely separate, k separate paths
connecting them, which means, of course,
that you have to cut each of the paths
in order to separate them.
And a similar theorem goes for edge connected.
So to summarize about some graphs
that we know about, given that connectivity measures fault
tolerance in a network, we're interested in, well,
how many edges did it take to achieve
this level of connectivity?
So the number of edges is kind of a measure of the cost.
And the complete graph on n vertices is n minus 1
connected, but it has about n squared over 2 edges.
The hypercube.
Now, if we're going to be talking about graphs of size n
to be uniform, the hypercube on Hn has 2 to the n vertices.
So if we want an n-vertex version,
we're talking about h sub log n, which has n vertices.
And then it has about n log n over 2 edges.
So it's got fewer edges than the complete graph
but significantly less, exponentially
less connectivity.
The grid is simply like the x, y axis or the x,
y plane with points at integers so that every node is connected
to four adjacent nodes-- one up, one down, one right,
and one left.
Now of course, when you get to the edge of it,
if you make it finite, the ones on the edge
are only going to be 2-connected.
So you have to wrap it around in the shape of a donut
to really make it work with a finite graph.
But if you do that, then the thing
is 4-connected-- 4-edge connected and only has
2n edges.
And then a cycle is 2-connected.
A cycle of size n is 2-connected with n vertices.
And finally, trees, which we'll talk about next time,
are the smallest possible connected graphs.
They are 1-connected.
Indeed everything in a tree is connected,
and they only have n minus 1 edges.
But if you remove one edge, it splits.
And that's sort of the summary of the k-connectivity story.