The crossing number is the minimum number of crossing points among plane graph drawings of a finite graph. Use drawings with vertices at distinct points, edge interiors avoiding vertices, finitely many crossings, and no three edge interiors meeting at one crossing. A crossing is a point where two edge interiors pass through one another.
Count crossing points, rather than just pairs of edges that cross: if the same pair crosses twice, it contributes twice. A graph is a planar graph exactly when its crossing number is zero. Deleting one selected edge at each crossing leaves a planar graph, so a drawing with crossings can be made planar by deleting at most edges. The crossing lemma gives a quantitative lower bound for dense simple graphs.
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