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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...

draks ...
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  • For a connected graph, the number of connected components is $1$, whereas the Euler characteristic is number of connected components + 1. –  Dec 07 '13 at 23:37
  • @user17762 isn't $\chi$ related rather to the genus of the surface where the gaph lives on than just its number of connected components + 1? – draks ... Dec 08 '13 at 22:45

1 Answers1

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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.

J W
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