# Attempting to understand what a "Face" in planar graph to count faces correctly

How does a graph with one vertex have one face? I understand Euler's theorem polyhedron $$V-E+F=2$$. However, I don't understand what a polytope and hyperplane is so I can't understand the topologically complicated definition of face. If I lookup a dumbed-down version of what a "face" is, it's too hand-wavy.

Let us say I have the following graph with $$6$$ vertices, $$6$$ edges, and therefore $$2$$ faces. I see how the triangular-like region formed by $$4$$ vertices makes up a face. I however don't understand how the extra 1 face was counted. See image drawn below:

Next, let's remove an edge below. Now the transformed graph below has $$1$$ face total. How the heck did this happen? How does it have $$1$$ face and not $$2$$?

See below:

Recall wolfram's:

Generally, a face is a component polygon, polyhedron, or polytope. A two-dimensional face thus has vertices and edges, and can be used to make cells. More formally, a face is the intersection of an $$n$$-dimensional polytope with a tangent hyperplane. Zero-dimensional faces are known as polyhedron vertices (nodes), one-dimensional faces as polyhedron edges, $$(n-2)-D$$ faces as ridges, and $$(n-1)$$-dimensional faces as facets.

and the "Face" in dumbed-down speak

 is regions bounded by edges, including the outer, infinitely large region


What infinitely large region does wikipedia talk about on https://en.wikipedia.org/wiki/Planar_graph? Is every graph bounded by an infinitely larger graph? Does that mean that an infinitely large graph is a non-existent because it is an element and subset of itself and therefore not an element of itself? Russell's paradox

• Isn't a face something of the form $\overline C$ where $C$ is a connected component of $\mathbb R^2$ minus the embedding? Commented Jun 11, 2017 at 3:58
• Your second image has two faces. Note that the graph has 5 vertices and 5 edges, giving equality for $(5) - (5) + F = 2$. Also, the "infinitely large region" Wikipedia mentions in the region "surrounding" the graph. Commented Jun 11, 2017 at 4:06
• The second image has 6 vertices. I removed one edge without removing the endpoints of the edge.
– user420360
Commented Jun 11, 2017 at 4:09
• I can remove an edge without removing endpoints. that smudge is a vertex. 6-(6-1)+x=2. x=1=faces.
– user420360
Commented Jun 11, 2017 at 4:09
• Ah, I see what you mean. Euler's Formula only applies to connected graphs. If we try to apply it to non-connected graphs, we have the situation where a group of $n$ vertices with no edges would imply there exist $2 - n$ faces, and we would have to make sense of "negative" faces. Commented Jun 11, 2017 at 4:12

In your second image, the floating disconnected vertex is throwing off your total. Euler's Theorem only applies to connected graphs -- otherwise you could arbitrarily add as many isolated vertices as you want and make $F-E+V$ come out to any arbitrary whole number greater than or equal to $2$.

• A disconnected vertex has zero lines coming out of it. In the picture, every node has degree at least equal to one. That is, the degree of a disconnected vertex is zero and the degree of every node shown in the diagram is at least one. The graph is connected. Let $G.VS$ be the vertex set of graph $G$. We have that $\forall V, W \in G.VS$, there exists a path from vertex $V$ to vertex $W$. Thus, graph $G$ is connected by the definition of connectivity. There are no isolated vertices in the diagram provided. An isolated vertex has degree zero. Commented Oct 5, 2023 at 10:34
• @ToothpickAnemone the original image in the problem had an actually disconnected vertex. The problem was edited at some point in the past and the original hand-drawn image was replaced with the current one. You can see evidence of this in the comments under OP: "that smudge is a vertex". Commented Oct 6, 2023 at 20:16

The last face is the entire plane that isn’t in the other faces you mentioned, which they called the outer, infinitely large region. In the case of a single point, it is the entire plane. A face can be thought of as a region of space such that you ca go anywhere in the region without having to cross over a line. Also, Euler's Theorem only applies to connected graphs.

As an example, here is a planar graph representing a cube. You'll notice that the left image tells you to remove the bottom, but it should say that the area that isn’t labeled "side" or "top" is the bottom of the cube.

## First Observation

I think you might have miscounted the number of vertices and edges for one of your examples.

Let us count the number of vertices and edges.

## Figure One

$$\begin{matrix} \mathtt{VERTEX\_COUNT} & - & \mathtt{EDGE\_COUNT} & + & \mathtt{FACE\_COUNT} & = & 2 \\ |G.VS| & - & |G.ES| & + & |G.FS| & = & 2 \\ V & - & E & + & F & = & 2 \\ 6 & - & 6 & + & 2 & = & 2 \\ \end{matrix}$$

## Figure Two

Figure two has one less edge and one less vertex than figure one.

$$\begin{matrix} \mathtt{VERTEX\_COUNT} & - & \mathtt{EDGE\_COUNT} & + & \mathtt{FACE\_COUNT} & = & 2 \\ |G.VS| & - & |G.ES| & + & |G.FS| & = & 2 \\ V & - & E & + & F & = & 2 \\ 5 & - & 5 & + & 2 & = & 2 \\ \end{matrix}$$

## Some Remarks

Let $$G$$ be a large complicated graph and let $$C$$ be a cycle in graph $$G$$.

Usually, cycle $$C$$ is a face if and only if for every pair of vertices in cycle $$C$$, if there is an edge between those two vertices in graph $$G$$, then there is an edge between those same two vertices in cycle $$C$$.

In other words, we cannot have an edge from from one side of a face, to the other side of the face, crossing through the middle of the face.

However, a lot of graph theorists talk about the "outer face".

The outer* face is usually what you get if you go around the outside of the whole thing and ignore the inner parts.

Alternatively, the outer face is the bottom of a three dimensional shape that has flat slides.

If you were like a bird, or in a helicopter, looking down on the Egyptian pyramid, you would see four faces. However, a pyramid has five faces, because the fifth face is the bottom of the pyramid.

## Figure Three

Here is one last example of a graph with some faces. Here, the faces are labeled.

Although seven faces are obvious, most mathematicians would say that there are 8 faces, because they include the "outer face" or the bottom of the shape.