The crossing number $\text{cross} (G)$ of a graph $G$ is the fewest number of crossings needed to draw $G$ in $\mathbb{R}^{2}.$ (We impose the rule that edges are not allowed to pass through vertices.) A graph is called *planar* if it can be drawn in $\mathbb{R}^{2}$ with no edge crossings. There are natural generalisations that are really interesting, e.g. graphs that can be drawn on a torus with no edge crossings... # Properties **Planar graphs**: one can use its [[Dual Graph]] to prove [[Euler's Characteristic for Connected Planar Graphs]].