 Remember, the key property of a graph is the connections between the vertices. It doesn't matter how the graph is drawn unless we want it to matter. A planar graph is one that could be represented in the plane with no edges crossing. So is the graph shown planar? Since there are crossing edges, the graph isn't planar. Or is it? Remember, the appearance of a graph isn't relevant unless it is. If we move the vertices while maintaining the edge connections or stretch the edges without disconnecting them, the graph doesn't change. And in this case it's possible to represent the graph without crossing edges. And what this means is that it can be very difficult to decide if an arbitrary graph is planar. If you decide, we'll want to find either features that every planar graph must have, or features that no planar graph has. That way we can maybe possibly decide if a given graph is planar. If it has a feature that no planar graph has, it can't be planar. And if it has a feature that every planar graph has, it might be. Since edges don't cross in a planar graph, they have a new feature. A face is a region in the plane bounded by the edges of a graph. There's so a planar graph has three features, vertices and edges, which all graphs have, and this new feature faces. Now before we continue, we have an obligatory joke. How does a mathematician get a tiger in a cage? They build a cage around themselves. Then they define inside to be where they are not. This joke explains two things. First, why I don't have my own comedy special on HBO. And also why the outside of a closed path is considered a face. Now if that's too strange, here's another way to look at it. Two faces regions are separated by some boundary. So anything separated by a boundary can be considered a region. The outside of a closed path is separated from the inside by the edges, so the outside should also be considered a face. So we might want to identify the faces in the graph shown, and remember a face is any region that's bounded by the edges of a graph, as well as the outside of the graph. So we can start by finding the regions bounded by the edges of the graph, and then the region that's outside the graph is also a face. Now suppose we have a connected planar graph. Is there a relationship between the vertices, edges, and faces? Let's gather some data. While we could select a bunch of random graphs, we'll take a more organized approach. So let's keep track of the number of vertices, edges, and faces. So we can start with one vertex, zero edges, and only one face, namely everything. And let's add a vertex, and because we have to have a connected graph, we have to join it with an edge. And now there's two vertices, one edge, and one face. Let's add another edge and vertex. And now we have three vertices, two edges, and still one face. Now remember if we're using simple graphs, we don't have any vertex connected to itself and at most one edge connects to vertices. But now that we have three vertices, we could join them. And now we have a graph with three vertices, three edges, and now we have two faces, one on the inside and one on the outside. Let's add another edge and vertex. So now we're up to four vertices, four edges, still two faces. And finally, let's close off and make one more face. So we'll make a few observations. Because we have to maintain a connected graph, every time we add a vertex, we have to add an edge. And if we add an edge without adding a vertex, in other words we're joining two existing vertices, we add a face. Now because something always happens when we do something, this suggests we might be able to find an invariant for planar graphs. Let's consider this. Suppose we have some formula that's a function of the vertices, edges, and faces. And if it's constant for planar graphs. That's the idea behind it being invariant. Now since adding a vertex also adds an edge, then f must include the difference, b minus e, or e minus v. That way if you increase one, you also have to increase the other. Similarly, since adding an edge also adds a face, f must also include the difference e minus f, or f minus e, for the same reason. And since f includes both, it has to be of the form plus or minus e minus v minus f. Because this is the only way we can include both differences. Now it really doesn't matter whether we choose plus or minus, but since we expect most graphs to have more vertices than edges or faces, we'll use v plus f minus e. And this suggests that v plus f minus e is some constant k for planar graphs. Well, let's figure out what it is. And this leads to something called Euler's formula. A planar graph with v vertices, e edges, and f faces satisfies v plus f equals e plus 2.