Quick questions on Planar Graphs and Euler's Formula - TCE Mathematics Applications (Tasmania)

Faces are the enclosed regions of a planar drawing, plus the single infinite region outside the graph. Beginners forget the outer face, so always add one for the outside when you count.

Because the formula links three quantities, knowing any two gives the third. Rearrange to suit the unknown: $f = 2 - v + e$ for faces, $e = v + f - 2$ for edges, or $v = e - f + 2$ for vertices. TASC questions exploit this by giving, for instance, the number of faces and edges and asking for the vertices, as in the worked exam answer above. The key is to substitute carefully and watch the signs, since the $-e$ and $-v$ terms are where arithmetic slips occur.

Euler's formula in the form $v - e + f = 2$ applies to connected planar graphs. If the graph splits into separate pieces, the formula must be adjusted (each extra component subtracts one from the total), so first confirm the graph is in one connected piece before applying it.