Planar

This is a planar graph.

Puzzle game involving untangling planar graphs

Here's an example of a planar graph.

Here's a planar graph on five nodes.

We had a planar graph. amp gt amp gt About the planar graphs. amp gt amp gt Yup.

Here's a planar graph a bunch of nodes and edges.

They can be planar formations, they can be three dimensional formations.

We can build any planar graph by iteratively adding nodes and edges.

To do that we're going to need another concept about planar graphs.

For any planar graph, n m or is equal to 2. Really?

What about coloring a graph with four colors if its a planar graph.

Let's think about the colorability of planar graphs. In particular, here's a decision problem.

There's going to be some graphs that can't be drawn as a planar graph. If we add this edge between this node and this node, the resulting graph is no longer planar.

As you can see here, they collapse from a three dimensional formation into planar formation.

We're also going to now talk about a notion of regions in a planar graph.

Any lines drawn across co planar surfaces can be erased to form one larger surface.

We're given a graph G and a number K. Is planar graph G K colorable?

Another category of graphs that are very important and come up a lot are planar graphs.

A three coloring of planar graph isn't any easier than three coloring in general graph, it's NP complete.

And then so you said draw yourself a planar graph but we know the relation n m r 2.

It becomes a planar graph. I didn't explain how to do this but it turns out to be true.

Let's consider this first case where we had a new node and an edge between them, and it's still a planar graph.

How do you do that? You know, we're supposed to have a planar foot, so we can't get off on the springboard.

What we're going to do is we're going to add to this graph so that it's still planar and see what happens to this formula.

Given that we're doing a proof by induction, we're going to assume that we have some planar graph and that Euler's formula holds for that graph.

Now we're going to use Euler's formula to give us a handle on how fast edges grow relative to the number of nodes in a planar graph.

To try to understand the relationship between edges and nodes in a planar graph we're going to use a result due to Euler same Euler as before.

One is that every region in a planar graph has to be encapsulated has to be bounded by at least three edges for it to be a region.

It still would be a planar graph even if I drew it with a crossing in it, because the issue is that there is some way to do it so that the edges don't cross.

Well, remarkably, the problems gotten harder and harder from constant to linear to basically it seems like it requires exponential time, it goes back to constant time, and that's because of the celebrated 4 color map theorem that says that any map can be colored with four colors and in this particular case, any map can be translated into a planar graph and so we know that every planar graph can be colored with four colors.

You give me a graph and tell me this is planar and I say, Yup, it's four colorable, and therefore, it's also going to be five, six, seven colorable because you can just use the first four.

Now, in this particular example, we've got five nodes and five edges, but what we're interested in is what is the relationship in general between the number of nodes in a planar graph and the number of edges.

In case three, well 3 colorability we showed was NP complete, but we didn't do that for planar graphs, we did that for general graphs. but it turns out that you can actually introduce another gadget into the mix here that actually let you deal with edge crossings.

It could be as bad as N M. If (M) is dense matrix, then this is (N)², so this algorithm ended up being in cubed again, but if it's a sparse matrix, so the number of edges is linear like in a planar graph, for example, then this is (N)² which is actually a lot better.