# Number of faces in a planar graph

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?

• Yes, that's correct. – B. Mehta May 10 '18 at 3:47