Recognise planar graphs and apply Euler's formula relating vertices, edges and faces.

What this dot point is asking

A graph drawing often has edges that cross, but the crossings may be an accident of how it was drawn. The real question is whether the same connections can be redrawn flat with no crossings at all.

Counting faces

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.

Why planarity matters

Planar graphs model situations where crossings are physically awkward or forbidden, such as circuit boards where wires must not touch, road networks without overpasses, or seating plans. Knowing a graph is planar tells you a clean, crossing-free layout exists.

Using Euler's formula to find any missing 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.

Testing for non-planarity

Some graphs simply cannot be drawn without crossings no matter how they are rearranged. The two classic non-planar graphs are $K_5$ (five vertices each joined to every other) and $K_{3,3}$ (the "three houses, three utilities" graph). When a TASC question asks you to show a graph is non-planar, demonstrating that it contains one of these structures, or that no crossing-free redrawing is possible, is the expected argument. For such graphs Euler's formula in the simple form does not apply, because the formula assumes a crossing-free planar drawing exists.

Connected requirement

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.

A complete answer states whether the graph is planar (and redraws it without crossings if needed), counts vertices and edges, includes the outer face when counting faces, and uses $v - e + f = 2$ to confirm or to find a missing count.