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

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 $2$.


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.

Planar cube

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.

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Graph Theory

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