1. Vertex orderings

2. 2 3 4 5 6 7 8 9 10 11 12 13 14 15 1 16 Vertex ordering

3. 2 3 4 5 6 7 8 9 10 11 12 13 14 15 s = 1 t = 16 st-numbering has two neighbors j, k

4. 2 3 4 5 6 7 8 9 10 11 12 13 14 15 s = 1 t = 16 st-numbering has two neighbors j, k

5. 2 3 4 5 6 7 8 9 10 11 12 13 14 15 s = 1 t = 16 st-numbering and induce connected subgraphs. For any i, both vertices

6. Application of st-numbersing Planarity testing Visibility drawing Internet routing

10. Canonical Ordering 12 3 4 5 6 7 8 9 10 11 12 1314 15 16 Triangulated plane graph

11. Canonical Ordering 12 3 4 5 6 7 8 9 10 11 12 1314 15 16 G k : subgraph of G induced by vertices

12. Canonical Ordering 12 3 4 5 6 7 8 9 10 11 12 1314 15 16 G k : subgraph of G induced by vertices G9G9

13. Canonical Ordering 12 3 4 5 6 7 8 9 10 11 12 1314 15 16 For any (co1) G k is biconnected and internally triangulated (co2) vertices 1 and 2 are on the outer face of G k (co3) vertex k +1 is on the outer face of G k and the neighbor of k+1 is consecutive on the outer cycle C o ( G k ).

14. Canonical Ordering 12 3 4 5 6 7 8 9 10 11 12 1314 15 16 For any (co1) G k is biconnected and internally triangulated (co2) vertices 1 and 2 are on the outer face of G k (co3) vertex k +1 is on the outer face of G k and the neighbor of k+1 is consecutive on the outer cycle C o ( G k ). G3G3

15. Canonical Ordering 12 3 4 5 6 7 8 9 10 11 12 1314 15 16 For any (co1) G k is biconnected and internally triangulated (co2) vertices 1 and 2 are on the outer face of G k (co3) vertex k +1 is on the outer face of G k and the neighbor of k+1 is consecutive on the outer cycle C o ( G k ). G4G4

16. Canonical Ordering 12 3 4 5 6 7 8 9 10 11 12 1314 15 16 For any (co1) G k is biconnected and internally triangulated (co2) vertices 1 and 2 are on the outer face of G k (co3) vertex k +1 is on the outer face of G k and the neighbor of k+1 is consecutive on the outer cycle C o ( G k ). G 10

17. Canonical Ordering 12 3 4 5 6 7 8 9 10 11 12 1314 15 16 For any (co1) G k is biconnected and internally triangulated (co2) vertices 1 and 2 are on the outer face of G k (co3) vertex k +1 is on the outer face of G k and the neighbor of k+1 is consecutive on the outer cycle C o ( G k ). G 10

18. Canonical Ordering 12 3 4 5 6 7 8 9 10 11 12 1314 15 16 For any (co1) G k is biconnected and internally triangulated (co2) vertices 1 and 2 are on the outer face of G k (co3) vertex k +1 is on the outer face of G k and the neighbor of k+1 is consecutive on the outer cycle C o ( G k ). G 10

19. Plane graph Straight line grid drawing. de Fraysseix et al. ’90 Straight Line Grid Drawing

20. Initial drawing of G 3

21. Install k + 1 GkGk

22. Shift and install k + 1 Shift method