Why it matters

Planar graphs have special algorithms (5-colorable). Understanding shapes graph classification.

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The architecture

Kuratowski: no K5 or K3,3 minor.

Linear-time: HT or BL algorithm.

Planarity testingTest graphinputLinear-time checkHT / BL algorithmResult + drawingplanar or notPlanar graphs: at most 3V-6 edges; Euler's formula V - E + F = 2
Planarity.
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How it works end to end

Kuratowski's theorem: planar iff no K5 or K3,3 minor.

Modern algorithms: linear-time.

Applications: circuit design, graph drawing.