Of Face and Circuit Rank

The circuit rank of a graph $G$ is given by

$$r = m - n + c,$$

where $m$ is the number of edges in $G$, $n$ is the number of vertices, and $c$ is the number of connected components.

Doesn't Euler's formula say the same?

$$ \#\text{faces} = \#\text{edges}-\#\text{vertices} +\#\text{components} \;\;\;(+\chi), $$ where $\chi$ is Euler's characteristic of the surface where the graph lives on.

So the circuit rank is just the number of faces, right?

I just wonder since I can't find any face on the Wiki page...

Solution 1:

We have

$$ \#\text{circuit rank} = \#\text{edges} - \#\text{vertices} + \#\text{components} $$

and, for a planar graph (or a graph on the surface of a sphere),

$$ \#\text{vertices} - \#\text{edges} + \#\text{faces} = \#\text{components} + 1 \text{.} $$

This leads to

$$ \#\text{circuit rank} = \#\text{faces} - 1 \text{.} $$

The number of faces includes the outer region. If you do not include it then the circuit rank does equal the number of faces.