Canonical Decomposition, Realizer, Schnyder Labeling and Orderly Spanning Trees of Plane Graphs...

Canonical Decomposition, Realizer,Schnyder Labeling

and Orderly Spanning Trees of Plane Graphs

Kazuyuki Miura, Machiko Azuma and Takao Nishizeki

title

Convex grid drawingconvex drawing

1:all vertices are put on grid points 2:all edges are drawn as straight line segments 3:all faces are drawn as convex polygons

1. canonical decomposition 2. realizer3. Schnyder labeling

drawing methods

Convex grid drawing

method (cd)

V1

V2 V3

V4

V5V6

V7

V 8

Canonical Decomposition [CK97]

[CK97]

convex grid drawing

How to construct a convex grid drawing

outer triangular

ordered partition convex grid drawing but not outer triangularconvex grid drawing

method (re)

Realizer [DTV99]

How to construct a convex grid drawing

[Fe01]

outer triangular convex grid drawing

method (sl)

Schnyder labeling [Sc90,Fe01]

21

3

1 11

2233

1

2 3

3

3

3

1

233

3

2

2 2111

12

12

31

2

3

1 2 31

23

21

2

3

How to construct a convex grid drawing

outer triangular convex grid drawing

[Fe01]

relations

V1

V2 V3V4

V5V6

V7

V8

Canonical Decomposition

Realizer Schnyder labeling

213

1 11

2233

1

2 3

333

1

233

3

2

2 2111

12

12

312

3

1 2 31

23

212

3

Convex Grid Drawing

Are there any relations between these concepts?

Are there any relations between these concepts?

Question 1 Question 1

known result

[BTV99]

Realizer

[Fe01]

Schnyder labeling

213

1 11

2233

1

2 3

333

1

233

3

2

2 2111

12

12

312

3

1 2 31

23

212

3

[Fe01]

Known results

V1

V2 V3V4

V5V6

V7

V8

Canonical Decomposition Convex Grid Drawing

3-connectivity is a sufficient condition 3-connectivity is a sufficient condition

[BTV99]

Realizer

[Fe01]

Schnyder labeling

213

1 11

2233

1

2 3

333

1

233

3

2

2 2111

12

12

312

3

1 2 31

23

212

3

[Fe01]

Known results

V1

V2 V3V4

V5V6

V7

V8

Canonical Decomposition Convex Grid Drawing

What is the necessary and sufficient condition?What is the necessary and sufficient condition?

Question 2 Question 2

[BTV99]

Realizer

[Fe01]

Schnyder labeling

213

1 11

2233

1

2 3

333

1

233

3

2

2 2111

12

12

312

3

1 2 31

23

212

3

Known results

V1

V2 V3V4

V5V6

V7

V8

Canonical Decomposition[Fe01]

Convex Grid Drawing

1

2

3

4

5 6 78

9 10

11

12

13

Orderly Spanning Tree [CLL01]

applicationApplications of a canonical decomposition a realizer a Schnyder labeling an orderly spanning tree

convex grid drawing

floor-planning

graph encoding

2-visibility drawing

etc.

Realizer Schnyder labeling

213

1 11

2233

1

2 3

333

1

233

3

2

2 2111

12

12

312

3

1 2 31

23

212

3

Our results

V1

V2 V3V4

V5V6

V7

V8

Canonical Decomposition Convex Grid Drawing

1

2

3

4

5 6 78

9 10

11

12

13

Orderly Spanning Tree

our result

Are there any relations between these concepts?

Are there any relations between these concepts?

Question 1 Question 1

What is the necessary and sufficient condition?What is the necessary and sufficient condition?

Question 2 Question 2

Realizer Schnyder labeling

213

1 11

2233

1

2 3

333

1

233

3

2

2 2111

12

12

312

3

1 2 31

23

212

3

Our results

V1

V2 V3V4

V5V6

V7

V8

Canonical Decomposition Convex Grid Drawing

1

2

3

4

5 6 78

9 10

11

12

13

Orderly Spanning Tree

these five notions are equivalent with each other

What is the necessary and sufficient condition?What is the necessary and sufficient condition?

Question 2 Question 2

Realizer Schnyder labeling

213

1 11

2233

1

2 3

333

1

233

3

2

2 2111

12

12

312

3

1 2 31

23

212

3

Our results

V1

V2 V3V4

V5V6

V7

V8

Canonical Decomposition Convex Grid Drawing

1

2

3

4

5 6 78

9 10

11

12

13

Orderly Spanning Tree

these five notions are equivalent with each other

a necessary and sufficient conditionfor plane graphs for their existence

theorem

Theorem G : plane graph with each degree ≥ 3

(a) - (f) are equivalent with each other.

(a) G has a canonical decomposition.

(b) G has a realizer.

(c ) G has a Schnyder labeling.

(d) G has an outer triangular convex grid drawing.

(e) G has an orderly spanning tree.

( f ) necessary and sufficient condition•G is internally 3-connected.•G has no separation pair {u,v } such that both u and v are on the same Pi (1 ≤ i ≤ 3).

each degree ≥ 3

necessary and sufficient condition(f) Our necessary and sufficient condition

P1

P2P3

•G is internally 3-connected.•G has no separation pair {u,v} s.t. both u and v are on the same path Pi (1 ≤ i ≤ 3).

each degree ≥ 3

P1

P2P3

-1. internally 3-connected(f) Our necessary and sufficient condition

•G is internally 3-connected.•G has no separation pair {u,v} s.t. both u and v are on the same path Pi (1 ≤ i ≤ 3). For any separation pair {u,v} of G ,

1) both u and v are outer vertices, and 2) each component of G - {u,v} contains an outer vertex.

For any separation pair {u,v} of G , 1) both u and v are outer vertices, and 2) each component of G - {u,v} contains an outer vertex.

G

separation pairu

v

G - {u,v}

each degree ≥ 3

P1

P2P3

(f) Our necessary and sufficient condition

•G is internally 3-connected.•G has no separation pair {u,v} s.t. both u and v are on the same path Pi (1 ≤ i ≤ 3). For any separation pair {u,v} of G ,

1) both u and v are outer vertices, and 2) each component of G - {u,v} contains an outer vertex.

For any separation pair {u,v} of G , 1) both u and v are outer vertices, and 2) each component of G - {u,v} contains an outer vertex.

Gseparation pairu

v

G - {u,v}

each degree ≥ 3

P1

P2P3

(f) Our necessary and sufficient condition

•G is internally 3-connected.•G has no separation pair {u,v} s.t. both u and v are on the same path Pi (1 ≤ i ≤ 3). For any separation pair {u,v} of G ,

1) both u and v are outer vertices, and 2) each component of G - {u,v} contains an outer vertex.

For any separation pair {u,v} of G , 1) both u and v are outer vertices, and 2) each component of G - {u,v} contains an outer vertex.

each degree ≥ 3

P1

P2P3

(f) Our necessary and sufficient condition

•G is internally 3-connected.•G has no separation pair {u,v} s.t. both u and v are on the same path Pi (1 ≤ i ≤ 3).

-2. separation pair on Pi

v

u

each degree ≥ 3

P1

P2P3

(f) Our necessary and sufficient condition

•G is internally 3-connected.•G has no separation pair {u,v} s.t. both u and v are on the same path Pi (1 ≤ i ≤ 3). v

u

each degree ≥ 3

P1

P2P3

(f) Our necessary and sufficient condition

•G is internally 3-connected.•G has no separation pair {u,v} s.t. both u and v are on the same path Pi (1 ≤ i ≤ 3).

P3

P1

P2u

v

theorem

Theorem G : plane graph with each degree ≥ 3

(a) - (f) are equivalent with each other.

(a) G has a canonical decomposition.

(b) G has a realizer.

(c ) G has a Schnyder labeling.

(d) G has an outer triangular convex grid drawing.

(e) G has an orderly spanning tree.

( f ) necessary and sufficient condition•G is internally 3-connected.•G has no separation pair {u,v } such that both u and v are on the same Pi (1 ≤ i ≤ 3).

(outline : convexdraw=>ns)

Realizer Schnyder labeling

213

1 11

2233

1

2 3

333

1

233

3

2

2 2111

12

12

312

3

1 2 31

23

212

3

Our results

V1

V2 V3V4

V5V6

V7

V8

Canonical Decomposition[Fe01]

Convex Grid Drawing

1

2

3

4

5 6 78

9 10

11

12

13

Orderly Spanning Tree

our result

necessary and sufficient condition•G is internally 3-connected.•G has no separation pair {u,v} s.t. both u and v are on the same Pi (1 ≤ i ≤ 3).

[BTV99]

[Fe01]

G has an outer triangular convex grid drawing

•G is internally 3-connected•G has no separation pair {u,v} s.t. both u and v are on the same Pi (1 ≤ i ≤ 3)

G has no outer triangular convex grid drawing

•G is not internally 3-connected , or•G has a separation pair {u,v} such that both u and v are on the same Pi (1 ≤ i ≤ 3)

if

These faces cannot be simultaneously drawnas convex polygons.

G has no convex drawing.

G has an outer triangular convex grid drawing

•G is internally 3-connected•G has no separation pair {u,v} s.t. both u and v are on the same Pi (1 ≤ i ≤ 3)

G has no outer triangular convex grid drawing

•G is not internally 3-connected , or•G has a separation pair {u,v} such that both u and v are on the same Pi (1 ≤ i ≤ 3)

if

u

v

This face cannot be drawn as a convex polygon.

G has no outer triangularconvex grid drawing.

(outline : ns =>cd )

Realizer Schnyder labeling

213

1 11

2233

1

2 3

333

1

233

3

2

2 2111

12

12

312

3

1 2 31

23

212

3

Our results

V1

V2 V3V4

V5V6

V7

V8

Canonical Decomposition Convex Grid Drawing

1

2

3

4

5 6 78

9 10

11

12

13

Orderly Spanning Tree

our result

necessary and sufficient condition•G is internally 3-connected.•G has no separation pair {u,v} s.t. both u and v are on the same Pi (1 ≤ i ≤ 3).

[BTV99]

[Fe01]

[Fe01]

P3

P2

G has a canonical decomposition

G

•G is internally 3-connected•G has no separation pair {u,v} s.t. both u and v are on the same Pi (1 ≤ i ≤ 3)

V8

V7

V6V5

V4

V3V2

V1P1

V1

P3

P2

G has a canonical decomposition

G

•G is internally 3-connected•G has no separation pair {u,v} s.t. both u and v are on the same Pi (1 ≤ i ≤ 3)

V8

V7

V6V5

V4

V3V2

P1

(cd1) V1 consists of all vertices on the inner face containing ,　 and Vh={ }.(cd2) Each Gk (1 ≤ k ≤ h) is internally 3-connected.(cd3) All the vertices in each Vk (2 ≤ k ≤ h -1) are outer vertices of Gk , and either (a) or (b).

V1

VhG

G has a canonical decomposition

•G is internally 3-connected•G has no separation pair {u,v} s.t. both u and v are on the same Pi (1 ≤ i ≤ 3)

(cd1) V1 consists of all vertices on the inner face containing ,　 and Vh={ }.(cd2) Each Gk (1 ≤ k ≤ h) is internally 3-connected.(cd3) All the vertices in each Vk (2 ≤ k ≤ h -1) are outer vertices of Gk , and either (a) or (b).

V1

VhG

G has a canonical decomposition

•G is internally 3-connected•G has no separation pair {u,v} s.t. both u and v are on the same Pi (1 ≤ i ≤ 3)

(cd1) V1 consists of all vertices on the inner face containing ,　 and Vh={ }.(cd2) Each Gk (1 ≤ k ≤ h) is internally 3-connected.(cd3) All the vertices in each Vk (2 ≤ k ≤ h -1) are outer vertices of Gk , and either (a) or (b).

V7

V6V5

V4

V3V2

V1

G has a canonical decomposition

•G is internally 3-connected•G has no separation pair {u,v} s.t. both u and v are on the same Pi (1 ≤ i ≤ 3)

(cd1) V1 consists of all vertices on the inner face containing ,　 and Vh={ }.(cd2) Each Gk (1 ≤ k ≤ h) is internally 3-connected.(cd3) All the vertices in each Vk (2 ≤ k ≤ h -1) are outer vertices of Gk , and either (a) or (b).

G

V5

V4

V3V2

5

V1

G has a canonical decomposition

•G is internally 3-connected•G has no separation pair {u,v} s.t. both u and v are on the same Pi (1 ≤ i ≤ 3)

(cd1) V1 consists of all vertices on the inner face containing ,　 and Vh={ }.(cd2) Each Gk (1 ≤ k ≤ h) is internally 3-connected.(cd3) All the vertices in each Vk (2 ≤ k ≤ h -1) are outer vertices of Gk , and either (a) or (b).

G1

V1

G has a canonical decomposition

•G is internally 3-connected•G has no separation pair {u,v} s.t. both u and v are on the same Pi (1 ≤ i ≤ 3)

(cd1) V1 consists of all vertices on the inner face containing ,　 and Vh={ }.(cd2) Each Gk (1 ≤ k ≤ h) is internally 3-connected.(cd3) All the vertices in each Vk (2 ≤ k ≤ h -1) are outer vertices of Gk , and either (a) or (b).

G

V2

2

V1

G has a canonical decomposition

•G is internally 3-connected•G has no separation pair {u,v} s.t. both u and v are on the same Pi (1 ≤ i ≤ 3)

(cd1) V1 consists of all vertices on the inner face containing ,　 and Vh={ }.(cd2) Each Gk (1 ≤ k ≤ h) is internally 3-connected.(cd3) All the vertices in each Vk (2 ≤ k ≤ h -1) are outer vertices of Gk , and either (a) or (b).

G

V3V2

3

h

(cd2) Each Gk (1≤k≤h) is internally 3-connected.(cd3) All the vertices in each Vk (2 ≤ k ≤ h -1) are outer vertices of Gk , and either (a) or (b).

G=GVh

G k-1

(a)

Vk

G k-1

(b)

Vk

G has a canonical decomposition

•G is internally 3-connected•G has no separation pair {u,v} s.t. both u and v are on the same Pi (1 ≤ i ≤ 3)

V h-1

Gh-1

(cd2) Each Gk (1≤k≤h) is internally 3-connected.(cd3) All the vertices in each Vk (2 ≤ k ≤ h -1) are outer vertices of Gk , and either (a) or (b).

G has a canonical decomposition

•G is internally 3-connected•G has no separation pair {u,v} s.t. both u and v are on the same Pi (1 ≤ i ≤ 3)

G k-1

(a)

Vk

G k-1

(b)

Vk

Gh-2

(cd2) Each Gk (1≤k≤h) is internally 3-connected.(cd3) All the vertices in each Vk (2 ≤ k ≤ h -1) are outer vertices of Gk , and either (a) or (b).

V h-2

G has a canonical decomposition

•G is internally 3-connected•G has no separation pair {u,v} s.t. both u and v are on the same Pi (1 ≤ i ≤ 3)

G k-1

(a)

Vk

G k-1

(b)

Vk

Gh-3

(cd2) Each Gk (1≤k≤h) is internally 3-connected.(cd3) All the vertices in each Vk (2 ≤ k ≤ h -1) are outer vertices of Gk , and either (a) or (b).

Vh-3

G has a canonical decomposition

•G is internally 3-connected•G has no separation pair {u,v} s.t. both u and v are on the same Pi (1 ≤ i ≤ 3)

G k-1

(a)

Vk

G k-1

(b)

Vk

(cd2) Each Gk (1≤k≤h) is internally 3-connected.(cd3) All the vertices in each Vk (2 ≤ k ≤ h -1) are outer vertices of Gk , and either (a) or (b).

G has a canonical decomposition

•G is internally 3-connected•G has no separation pair {u,v} s.t. both u and v are on the same Pi (1 ≤ i ≤ 3)

G k-1

(a)

Vk

G k-1

(b)

Vk

GV

8

V7

V6V5

V4

V3V2

V1

(outline : ostrealizer)

Realizer Schnyder labeling

213

1 11

2233

1

2 3

333

1

233

3

2

2 2111

12

12

312

3

1 2 31

23

212

3

Our results

V1

V2 V3V4

V5V6

V7

V8

Canonical Decomposition Convex Grid Drawing

1

2

3

4

5 6 78

9 10

11

12

13

Orderly Spanning Tree

necessary and sufficient condition•G is internally 3-connected.•G has no separation pair {u,v} s.t. both u and v are on the same Pi (1 ≤ i ≤ 3).

[BTV99]

[Fe01]

[Fe01]

Schnyder labeling

213

1 11

2233

1

2 3

333

1

233

3

2

2 2111

12

12

312

3

1 2 31

23

212

3

Convex Grid Drawing

[Fe01]

Our results

V1

V2 V3V4

V5V6

V7

V8

Canonical Decomposition

necessary and sufficient condition•G is internally 3-connected.•G has no separation pair {u,v} s.t. both u and v are on the same Pi (1 ≤ i ≤ 3).

[BTV99]

[Fe01]

1

2

3

4

5 6 78

9 10

11

12

13

Orderly Spanning Tree

Realizer

ost

Orderly Spanning Tree

2

3

4

5 6

1preorder

78

9 10

11

12

n =13

parent

smaller

children

largerv

realizer

Realizer

for each vertex

(outline : re =>ost )

Realizer Schnyder labeling

213

1 11

2233

1

2 3

333

1

233

3

2

2 2111

12

12

312

3

1 2 31

23

212

3

Our results

V1

V2 V3V4

V5V6

V7

V8

Canonical Decomposition Convex Grid Drawing

1

2

3

4

5 6 78

9 10

11

12

13

Orderly Spanning Tree

necessary and sufficient condition•G is internally 3-connected.•G has no separation pair {u,v} such that both u and v are on the same path Pi (1 ≤ i ≤ 3).

[BTV99]

[Fe01]

[Fe01]

Realizer

Orderly Spanning Tree

1

2

3

4

5 6 78

9 10

11

12

13

Orderly Spanning Tree

parent

smaller

children

largerv

(outline : ost =>re )

Realizer Schnyder labeling

213

1 11

2233

1

2 3

333

1

233

3

2

2 2111

12

12

312

3

1 2 31

23

212

3

Our results

V1

V2 V3V4

V5V6

V7

V8

Canonical Decomposition Convex Grid Drawing

1

2

3

4

5 6 78

9 10

11

12

13

Orderly Spanning Tree

necessary and sufficient condition•G is internally 3-connected.•G has no separation pair {u,v} such that both u and v are on the same path Pi (1 ≤ i ≤ 3).

[BTV99]

[Fe01]

[Fe01]

Realizer

Orderly Spanning Tree

1

2

3

4

5 6 78

9 10

11

12

13

v

Realizer

Orderly Spanning Tree

1

2

3

4

5 6 78

9 10

11

12

13

vv

Orderly Spanning Tree

1

2

3

4

5 6 78

9 10

11

12

13

Realizer

vv

Orderly Spanning Tree

1

2

3

4

5 6 78

9 10

11

12

13

Realizer

vv

Orderly Spanning Tree

1

2

3

4

5 6 78

9 10

11

12

13

Realizer

vv

Orderly Spanning Tree

1

2

3

4

5 6 78

9 10

11

12

13

RealizerRealizer

vv

Realizer Schnyder labeling

213

1 11

2233

1

2 3

333

1

233

3

2

2 2111

12

12

312

3

1 2 31

23

212

3

Conclusion

V1

V2 V3V4

V5V6

V7

V8

Canonical Decomposition Convex Grid Drawing

1

2

3

4

5 6 78

9 10

11

12

13

Orderly Spanning Tree

necessary and sufficient condition•G is internally 3-connected.•G has no separation pair {u,v} such that both u and v are on the same path Pi (1 ≤ i ≤ 3).

[BTV99]

[Fe01]

[Fe01]

theorem

Theorem G : plane graph with each degree ≥ 3

(a) - (f) are equivalent with each other.

(a) G has a canonical decomposition.

(b) G has a realizer.

(c ) G has a Schnyder labeling.

(d) G has an outer triangular convex grid drawing.

(e) G has an orderly spanning tree.

( f ) necessary and sufficient condition•G is internally 3-connected.•G has no separation pair {u,v } such that both u and v are on the same Pi (1 ≤ i ≤ 3).

If a plane graph G satisfies the necessary and sufficient condition,then one can find the followings in linear time

(a) canonical decomposition, (b) realizer, (c) Schnyder labeling, (d) orderly spanning tree, and (e) outer triangular convex grid drawing of G having the size (n -1)×(n -1).

Corollary G : plane graph with each degree ≥ 3

3-connectednot 3-connected

remaining problem

The remaining problem is to characterize the class of plane graphs having convex grid drawings such that the size is (n -1)×(n -1) and the outer face is not always a triangle.

triangle pentagon

E N D

---------------------

Orderly Spanning Tree ost-def

1

2

3

4

5 6 78

9 10

11

1213

r

u u

parent

smaller

children

larger

(each subset may be empty)

N1

N2

N3

N4

(ost2) For each leaf other than u and u , neither N2 nor N4 is empty.

(ost1) For each edge not in the tree T, none of the endpoints is an ancestor of the other in T.

1u 2u

Gi

F

FP

choose subset

G has a canonical decomposition

•G is internally 3-connected•G has no separation pair {u,v} s.t. both u and v are on the same Pi (1 ≤ i ≤ 3)

(cd2) Each Gk (1≤k≤h) is internally 3-connected.(cd3) All the vertices in each Vk (2 ≤ k ≤ h -1) are outer vertices of Gk , and either (a) or (b).

G k-1

(a)

Vk

G k-1

(b)

Vk

1u 2u

Gi

FP

Case 1

G has a canonical decomposition

•G is internally 3-connected•G has no separation pair {u,v} s.t. both u and v are on the same Pi (1 ≤ i ≤ 3)

(cd2) Each Gk (1≤k≤h) is internally 3-connected.(cd3) All the vertices in each Vk (2 ≤ k ≤ h -1) are outer vertices of Gk , and either (a) or (b).

G k-1

(a)

Vk

G k-1

(b)

Vk

1u 2u

Gi

P

Case 1 Vi

G has a canonical decomposition

•G is internally 3-connected•G has no separation pair {u,v} s.t. both u and v are on the same Pi (1 ≤ i ≤ 3)

(cd2) Each Gk (1≤k≤h) is internally 3-connected.(cd3) All the vertices in each Vk (2 ≤ k ≤ h -1) are outer vertices of Gk , and either (a) or (b).

G k-1

(a)

Vk

G k-1

(b)

Vk

1u 2u

Gi

P

Case 1

G has a canonical decomposition

•G is internally 3-connected•G has no separation pair {u,v} s.t. both u and v are on the same Pi (1 ≤ i ≤ 3)

(cd2) Each Gk (1≤k≤h) is internally 3-connected.(cd3) All the vertices in each Vk (2 ≤ k ≤ h -1) are outer vertices of Gk , and either (a) or (b).

G k-1

(a)

Vk

G k-1

(b)

Vk

1u 2u

Gi

Case 2

FP

G has a canonical decomposition

•G is internally 3-connected•G has no separation pair {u,v} s.t. both u and v are on the same Pi (1 ≤ i ≤ 3)

(cd2) Each Gk (1≤k≤h) is internally 3-connected.(cd3) All the vertices in each Vk (2 ≤ k ≤ h -1) are outer vertices of Gk , and either (a) or (b).

G k-1

(a)

Vk

G k-1

(b)

Vk

1u 2u

Gi

Case 2

FP

u v

P2

G has a canonical decomposition

•G is internally 3-connected•G has no separation pair {u,v} s.t. both u and v are on the same Pi (1 ≤ i ≤ 3)

(cd2) Each Gk (1≤k≤h) is internally 3-connected.(cd3) All the vertices in each Vk (2 ≤ k ≤ h -1) are outer vertices of Gk , and either (a) or (b).

G k-1

(a)

Vk

G k-1

(b)

Vk

•G is internally 3-connected•G has no separation pair {u,v} s.t. both u and v are on the same Pi (1 ≤ i ≤ 3)

2u

Gi

Case 2

FP

1u

u vP3

G has a canonical decomposition

(cd2) Each Gk (1≤k≤h) is internally 3-connected.(cd3) All the vertices in each Vk (2 ≤ k ≤ h -1) are outer vertices of Gk , and either (a) or (b).

G k-1

(a)

Vk

G k-1

(b)

Vk

1u 2u

Gi

Case 2

FP

G has a canonical decomposition

•G is internally 3-connected•G has no separation pair {u,v} s.t. both u and v are on the same Pi (1 ≤ i ≤ 3)

(cd2) Each Gk (1≤k≤h) is internally 3-connected.(cd3) All the vertices in each Vk (2 ≤ k ≤ h -1) are outer vertices of Gk , and either (a) or (b).

G k-1

(a)

Vk

G k-1

(b)

Vk

2u

Gi

Case 2

FP

1u

Vi

G has a canonical decomposition

•G is internally 3-connected•G has no separation pair {u,v} s.t. both u and v are on the same Pi (1 ≤ i ≤ 3)

(cd2) Each Gk (1≤k≤h) is internally 3-connected.(cd3) All the vertices in each Vk (2 ≤ k ≤ h -1) are outer vertices of Gk , and either (a) or (b).

G k-1

(a)

Vk

G k-1

(b)

Vk

2u

Gi

Case 2

FP

1u

G has a canonical decomposition

•G is internally 3-connected•G has no separation pair {u,v} s.t. both u and v are on the same Pi (1 ≤ i ≤ 3)

(cd2) Each Gk (1≤k≤h) is internally 3-connected.(cd3) All the vertices in each Vk (2 ≤ k ≤ h -1) are outer vertices of Gk , and either (a) or (b).

G k-1

(a)

Vk

G k-1

(b)

Vk

chord-path

chord-path P

1. P connects two outer vertices u and v.2. u,v is a separation pair of G.3. P lies on an inner face. 4. P does not pass through any outer edge and any outer vertex other than u and v.

u v

G has a canonical decomposition

•G is internally 3-connected•G has no separation pair {u,v} s.t. both u and v are on the same Pi (1 ≤ i ≤ 3)

chord-path P

u

v

u

vu

v

u

v

G has a canonical decomposition

•G is internally 3-connected•G has no separation pair {u,v} s.t. both u and v are on the same Pi (1 ≤ i ≤ 3)

1. P connects two outer vertices u and v.2. u,v is a separation pair of G.3. P lies on an inner face. 4. P does not pass through any outer edge and any outer vertex other than u and v.

chord-path P

w1

w2

wt

minimal

G has a canonical decomposition

•G is internally 3-connected•G has no separation pair {u,v} s.t. both u and v are on the same Pi (1 ≤ i ≤ 3)

1. P connects two outer vertices u and v.2. u,v is a separation pair of G.3. P lies on an inner face. 4. P does not pass through any outer edge and any outer vertex other than u and v.

chord-path P

w1

w2

wt

minimal

find a minimal chord-path P,then choose such a vertex set.

plan of proof :

G has a canonical decomposition

•G is internally 3-connected•G has no separation pair {u,v} s.t. both u and v are on the same Pi (1 ≤ i ≤ 3)

1. P connects two outer vertices u and v.2. u,v is a separation pair of G.3. P lies on an inner face. 4. P does not pass through any outer edge and any outer vertex other than u and v.

3-connected plane graph3-connected plane graph

3-connected plane graph

3-connected plane graph