Canonical Decomposition, Realizer, Schnyder Labeling and Orderly Spanning Trees of Plane Graphs Kazuyuki Miura, Machiko Azuma and Takao Nishizeki title

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)

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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]


[Fe01]

outer triangular convex grid drawing

method (sl)

Schnyder labeling [Sc90,Fe01]

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outer triangular convex grid drawing

[Fe01]

relations

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Canonical Decomposition

Realizer Schnyder labeling

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Convex Grid Drawing

Are there any relations between these concepts?


Question 1 Question 1

known result

[BTV99]

Realizer

[Fe01]

Schnyder labeling

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Known results

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Canonical Decomposition Convex Grid Drawing

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

[BTV99]

Realizer

[Fe01]

Schnyder labeling

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[Fe01]

Known results

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


[BTV99]

Realizer

[Fe01]

Schnyder labeling

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Known results

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Canonical Decomposition[Fe01]

Convex Grid Drawing

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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.


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Our results

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Orderly Spanning Tree

our result







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Our results

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these five notions are equivalent with each other




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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




Gseparation pairu

v

G - {u,v}

each degree ≥ 3

P1

P2P3





each degree ≥ 3

P1

P2P3



-2. separation pair on Pi

v

u

each degree ≥ 3

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). v

u

each degree ≥ 3

P1

P2P3



P3

P1

P2u

v

theorem









(outline : convexdraw=>ns)


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Our results

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Canonical Decomposition[Fe01]

Convex Grid Drawing

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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 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 )


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Our results

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our result


[BTV99]

[Fe01]

[Fe01]

P3

P2

G has a canonical decomposition

G


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V3V2

V1P1

V1

P3

P2


G


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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




V1

VhG




V7

V6V5

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V3V2

V1




G

V5

V4

V3V2

5

V1




G1

V1




G

V2

2

V1




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



V h-1

Gh-1




G k-1

(a)

Vk

G k-1

(b)

Vk

Gh-2


V h-2



G k-1

(a)

Vk

G k-1

(b)

Vk

Gh-3


Vh-3



G k-1

(a)

Vk

G k-1

(b)

Vk




G k-1

(a)

Vk

G k-1

(b)

Vk

GV

8

V7

V6V5

V4

V3V2

V1

(outline : ostrealizer)


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Our results

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5 6 78

9 10

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[BTV99]

[Fe01]

[Fe01]

Schnyder labeling

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Convex Grid Drawing

[Fe01]

Our results

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Canonical Decomposition


[BTV99]

[Fe01]

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Realizer

ost


2

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5 6

1preorder

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n =13

parent

smaller

children

largerv

realizer

Realizer

for each vertex

(outline : re =>ost )


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Our results

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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


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5 6 78

9 10

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12

13


parent

smaller

children

largerv

(outline : ost =>re )


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1 11

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Our results

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V2 V3V4

V5V6

V7

V8


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5 6 78

9 10

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[BTV99]

[Fe01]

[Fe01]

Realizer


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v

Realizer


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vv


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Realizer

vv


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Realizer

vv


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Realizer

vv


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RealizerRealizer

vv


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Conclusion

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[BTV99]

[Fe01]

[Fe01]

theorem









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

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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 k-1

(a)

Vk

G k-1

(b)

Vk

1u 2u

Gi

FP

Case 1




G k-1

(a)

Vk

G k-1

(b)

Vk

1u 2u

Gi

P

Case 1 Vi




G k-1

(a)

Vk

G k-1

(b)

Vk

1u 2u

Gi

P

Case 1




G k-1

(a)

Vk

G k-1

(b)

Vk

1u 2u

Gi

Case 2

FP




G k-1

(a)

Vk

G k-1

(b)

Vk

1u 2u

Gi

Case 2

FP

u v

P2




G k-1

(a)

Vk

G k-1

(b)

Vk


2u

Gi

Case 2

FP

1u

u vP3



G k-1

(a)

Vk

G k-1

(b)

Vk

1u 2u

Gi

Case 2

FP




G k-1

(a)

Vk

G k-1

(b)

Vk

2u

Gi

Case 2

FP

1u

Vi




G k-1

(a)

Vk

G k-1

(b)

Vk

2u

Gi

Case 2

FP

1u




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



chord-path P

u

v

u

vu

v

u

v




chord-path P

w1

w2

wt

minimal




chord-path P

w1

w2

wt

minimal

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

plan of proof :




3-connected plane graph3-connected plane graph

3-connected plane graph

3-connected plane graph

Documents

Canonical Decomposition, Realizer, Schnyder Labeling and Orderly Spanning Trees of Plane Graphs Kazuyuki Miura, Machiko Azuma and Takao Nishizeki title