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Every planar graph can be drawn in the plane with straight edges

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Presentation on theme: "Every planar graph can be drawn in the plane with straight edges"— Presentation transcript:

1 Every planar graph can be drawn in the plane with straight edges
Fáry-Wagner

2 3-connected planar graph
Steinitz Every 3-connected planar graph is the skeleton of a polytope.

3 Rubber bands and planarity
Tutte (1963) G: 3-connected planar graph outer face fixed to convex polygon edges replaced by rubber bands Energy: Equilibrium:

4 Disc representation Koebe (1936)
Every planar graph can be represented by touching circles

5 Polyhedral version Every 3-connected planar graph
is the skeleton of a convex polytope such that every edge touches the unit sphere Andre’ev

6 From polyhedra to circles
horizon

7 From polyhedra to representation of the dual

8 Graph dual horizon

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