Give an example of a connected planar graph with $v$ vertices, $e$ edges and $k$ components such $e=3v-6k$.

Question

Give an example of a connected planar graph with $v$ vertices, $e$ edges and $k$ components such that $e= 3v− 6k$.

Answer 1

Recall that if $G$ is a connected planar graph with $v\geq 3$ vertices and $e$ edges then we always have

$$e\leq 3v-6$$

That means, in this case, we need an example of which equality holds in the above inequality. 

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Let us draw such an example by taking $v=3$. Note that in this case $3v-6=3$. Now we have to create a graph on three vertices such that the number of edges in it is three. So the graph is nothing but a triangle. 

e edges. Then