When can a graph be drawn without edges crossing, and what relation links its parts?
Identify planar graphs, count vertices, edges and faces, and verify and apply Euler's formula v minus e plus f equals 2.
How to recognise a planar graph, redraw it without crossings, count faces including the outer region, and apply Euler's formula linking vertices, edges and faces.
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What this dot point is asking
You must recognise when a graph is planar, count faces correctly, and use Euler's formula to verify a drawing or find a missing count.
Planar graphs
A graph is planar if it can be drawn in the plane so that no two edges cross. A graph may look tangled yet still be planar, because what matters is whether some redrawing removes all crossings.
Euler's formula
For any connected planar graph, the three counts are tied together.
The formula gives a quick check: if your counts do not satisfy it, you have miscounted (often the outer face) or the graph is not connected.
Using the formula in reverse
Euler's formula works for any one unknown. Given two of , and you find the third. It also confirms whether a proposed graph is consistent before you spend time analysing it.
Connected requirement
Euler's formula in this form assumes the graph is connected. If a graph has separate pieces, it does not apply directly, which is itself a useful diagnostic when the numbers refuse to balance.