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In a finite simple planar graph, is the number of faces the same regardless of how you draw the graph?

I know that for such a graph we have Euler's formula: $v-e+f=2$. However does this formula only apply to those drawings of the planar graph that don't have any edge crossings?

I think so. Because with any graph I can do something stupid and draw all vertices on one line and have all edges overlap. So is it correct to say that in any drawing of the graph without edge crossings, the number of faces is the same?

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  • $\begingroup$ Yes, that's correct. $\endgroup$ – B. Mehta May 10 '18 at 3:47
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Yes, what you wrote is correct, and the example you give in the question is an excellent argument for why this should be the case. I'd also like to note that, since a planar graph is defined as one which can be drawn with no edge intersections, when we consider a drawing of such a graph, we usually only consider drawings with this property (I'm sure there are exceptions though).

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