vertex orderings
DESCRIPTION
Vertex orderings. Vertex ordering. 16. 13. 14. 12. 15. 11. 10. 4. 9. 8. 5. 6. 3. 7. 2. 1. 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. has two neighbors j , k. st -numbering. t = 16. 13. 14. - PowerPoint PPT PresentationTRANSCRIPT
Vertex orderings
2
3
45
67
8 9
1011
1213
1415
1
16Vertex ordering
2
3
45
67
8 9
1011
1213
1415
s = 1
t = 16st-numbering
i s t , has two neighbors j, k
.kij
2
3
45
67
8 9
1011
1213
1415
s = 1
t = 16st-numbering
10i
i s t , has two neighbors j, k
.kij
2
3
45
67
8 9
1011
1213
1415
s = 1
t = 16st-numbering
{ , , , }1 2 i and
{ , , , }i i n 1 2 induce connected subgraphs.
For any i, both vertices
9i
Application of st-numbersing
Planarity testing
Visibility drawing
Internet routing
Canonical Ordering
1 23
45
6
789
1011
12
13 1415
16
Triangulated plane graph
Canonical Ordering
1 23
45
6
789
1011
12
13 1415
16
Gk : subgraph of G induced by vertices k,,2,1
Canonical Ordering
1 23
45
6
789
1011
12
13 1415
16
Gk : subgraph of G induced by vertices k,,2,1
G9
Canonical Ordering
1 23
45
6
789
1011
12
13 1415
16
For any nkk 3,
(co1) Gk is biconnected and internally triangulated
(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).
Canonical Ordering
1 23
45
6
789
1011
12
13 1415
16
For any nkk 3,
(co1) Gk is biconnected and internally triangulated
(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).
G3
Canonical Ordering
1 23
45
6
789
1011
12
13 1415
16
For any nkk 3,
(co1) Gk is biconnected and internally triangulated
(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).
G4
Canonical Ordering
1 23
45
6
789
1011
12
13 1415
16
For any nkk 3,
(co1) Gk is biconnected and internally triangulated
(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).
G10
Canonical Ordering
1 23
45
6
789
1011
12
13 1415
16
For any nkk 3,
(co1) Gk is biconnected and internally triangulated
(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).
G10
Canonical Ordering
1 23
45
6
789
1011
12
13 1415
16
For any nkk 3,
(co1) Gk is biconnected and internally triangulated
(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).
G10
Plane graph
Straight line grid drawing.
de Fraysseix et al. ’9022nHW
nW 2
nH
Straight Line Grid Drawing
Initial drawing of G3
Install k + 1
Gk
Shift and install k + 1Shift method
