Let $G=(V,E)$ be a planar graph. Suppose a planar representation of $G$ has been chosen and that $$v-e+f=2,$$ where $v,e$ and $f$ are the number of vertices, edges and faces respectively. See Wikipedia.
Does this imply that $G$ must be connected?
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Suppose that the graph is not connected but still is planar. Then there are two graphs that are disconnected such that $v_1-e_1+f_1=2$ and $v_2-e_2+f_2=2$. On the other hand we have $v_1+v_2=v$ and $e_1+e_2=e$ but $f_1+f_2=f+1$. So we get: $$ v_1-e_1+f_1+ v_2-e_2+f_2=4\implies v-e+f=3 $$ which means that the original graph will not be planar and therefore we arrived at a contradiction. So the graph should be connected.