1. Vertex orderings

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

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

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

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

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

10. Canonical Ordering 16 13 14 15 10 12 11 6 5 9 4 8 7 3 1 2 Triangulated plane graph

11. Canonical Ordering 16 13 14 15 10 12 11 6 5 9 4 8 7 3 1 2 Gk: subgraph of G induced by vertices

12. Canonical Ordering 16 13 14 15 10 12 11 6 G9 5 9 4 8 7 3 1 2 Gk: subgraph of G induced by vertices

13. For any Canonical Ordering 16 13 14 15 10 12 11 6 5 (co1) Gk is biconnected and internally triangulated 9 4 8 7 3 1 2 (co2) vertices 1 and 2 are on the outer face of Gk (co3) vertex k+1 is on the outer face of Gk and the neighbor of k+1 is consecutive on the outer cycle Co(Gk).

14. For any Canonical Ordering 16 13 14 15 10 12 11 6 5 (co1) Gk is biconnected and internally triangulated 9 4 8 7 3 1 2 G3 (co2) vertices 1 and 2 are on the outer face of Gk (co3) vertex k+1 is on the outer face of Gk and the neighbor of k+1 is consecutive on the outer cycle Co(Gk).

15. For any Canonical Ordering 16 13 14 15 10 12 11 6 5 (co1) Gk is biconnected and internally triangulated 9 4 8 7 3 1 2 G4 (co2) vertices 1 and 2 are on the outer face of Gk (co3) vertex k+1 is on the outer face of Gk and the neighbor of k+1 is consecutive on the outer cycle Co(Gk).

16. For any Canonical Ordering 16 13 14 15 10 12 11 6 5 (co1) Gk is biconnected and internally triangulated 9 4 8 7 3 1 2 G10 (co2) vertices 1 and 2 are on the outer face of Gk (co3) vertex k+1 is on the outer face of Gk and the neighbor of k+1 is consecutive on the outer cycle Co(Gk).

17. For any Canonical Ordering 16 13 14 15 10 12 11 6 5 (co1) Gk is biconnected and internally triangulated 9 4 8 7 3 1 2 G10 (co2) vertices 1 and 2 are on the outer face of Gk (co3) vertex k+1 is on the outer face of Gk and the neighbor of k+1 is consecutive on the outer cycle Co(Gk).

18. For any Canonical Ordering 16 13 14 15 10 12 11 6 5 (co1) Gk is biconnected and internally triangulated 9 4 8 7 3 1 2 G10 (co2) vertices 1 and 2 are on the outer face of Gk (co3) vertex k+1 is on the outer face of Gk and the neighbor of k+1 is consecutive on the outer cycle Co(Gk).

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

20. Initial drawing of G3

21. Install k + 1 Gk

22. Shift and install k + 1 Shift method

25. Thank You