planar graphs in 2 dimensions - don sheehydonsheehy.net/talks/lifting-planar-graphs-long.pdf ·...

Planar Graphs in 21/2 Dimensions

Don Sheehy

1

21/2 Dimensions

2

Cast of Characters

3

Cast of Characters

James Clerk Maxwell

Luigi Cremona

3

Cast of Characters

James Clerk Maxwell

Luigi Cremona

Ernst Steinitz

3

Cast of Characters

James Clerk Maxwell

Luigi Cremona

Ernst Steinitz

W. T. Tutte

3

Planar Graphs

4

Planar Graphs

5

Duality

6

Polar Polytopes

7

Polar Polytopes

A!= {x ! R

d | a · x " 1,#a ! A}

7

The Maxwell-Cremona Correspondence

8

Equilibrium Stresses

9


There is a 1-1 correspondence between “proper” liftings and equilibrium stresses

of a planar straight line graph.

10


11

Reciprocal Diagrams from Equilibrium Stresses

12

Reciprocal Diagrams from Liftings

13

The Maxwell-Cremona Corresondence

Equilibrium Stresses

Reciprocal Diagrams

Liftings

14

Other Famous Reciprocal Diagrams

Delaunay Triangulation

Voronoi Diagram

15



Voronoi Diagram

Weighted

Weighted

15



Voronoi Diagram

2½ dimensional polarity

Weighted

Weighted

15

How to Draw a Graph

16

Tutte’s Algorithm

1. Fix one face of a simple, planar, 3-connected graph in convex position.

2. Place each other vertex at the barycenter (centroid) of its neighbors.

The result is a non-crossing, convex drawing.

17

Spring Interpretation

18

Computing Forces

v ! R2

v

19

Computing Forces

v ! R2

Fv =!

u!v

(v ! u)

v

19

Computing Forces

v ! R2

Fv =!

u!v

(v ! u)

= dvv !

!

u!v

uv

19

Computing Forces

v ! R2

Fv =!

u!v

(v ! u)

= dvv !

!

u!v

u

F = LV

v

19

Computing Forces

v ! R2

Fv =!

u!v

(v ! u)

L = D ! A

= dvv !

!

u!v

u

F = LV

v

19

Computing Forces

v ! R2

Fv =!

u!v

(v ! u)

L = D ! A

= dvv !

!

u!v

u

F = LV

degrees

v

19

Computing Forces

v ! R2

Fv =!

u!v

(v ! u)

L = D ! A

= dvv !

!

u!v

u

F = LV

degrees adjacency

v

19

Computing Forces

v ! R2

Fv =!

u!v

(v ! u)

L = D ! A

= dvv !

!

u!v

u

F = LV

degrees adjacency

The Laplacian!

v

19

Computing Forces

LV = F

20

Computing Forces

LV = F = 0?

20

Computing Forces

LV = F = 0? V1: boundaryV2: interior

20

Computing Forces

LV = F = 0?

!

L1 BT

B L2

" !

V1

V2

"

=

!

F !

0

"

V1: boundaryV2: interior

20

Computing Forces

LV = F = 0?

!

L1 BT

B L2

" !

V1

V2

"

=

!

F !

0

"

BV1 + L2V2 = 0


20

Computing Forces

LV = F = 0?

!

L1 BT

B L2

" !

V1

V2

"

=

!

F !

0

"

BV1 + L2V2 = 0

V2 = !L!1

2B V1


20

Computing Forces

LV = F = 0?

!

L1 BT

B L2

" !

V1

V2

"

=

!

F !

0

"

BV1 + L2V2 = 0

V2 = !L!1

2B V1( )


20

Monotone PathsPick a direction and a vertex. There is a monotone path in that direction from the vertex to the boundary.

21

Planar, 3-Connected Graphs

22


➡ No K5 or K3,3 minors

22



➡ Removing a face does not disconnect the graph.

22



➡ Removing a face does not disconnect the graph.

➡ No face has a diagonal.

22

Double Crossing a FaceLemma: No two disjoint paths have interleaved endpoints on a face.

23

Tutte’s AlgorithmNo ZigZags

24

Tutte’s AlgorithmNo Crossings

25

Tutte’s AlgorithmNo Overlaps

26

Tutte and Maxwell-Cremona

27


➡ Weirdness on the outer face.

27



➡ Lifting still works, except outer face.

27



➡ Lifting still works, except outer face.

➡ Lifting is convex.

27

Steinitz’s Theorem

28


A graph G is the 1-skeleton of a3-polytope if and only if it is

simple, planar, and 3-connected.

29


Claim: If the graph has a triangle, then the Tutte embedding followed by the Maxwell-Cremona lifting gives the desired polytope.

30



Fix the triangle as the outer face.

30



Fix the triangle as the outer face.

After the lifting, the triangle must lie on a plane.

30


Question: What if there is no triangle?

31


Question: What if there is no triangle?Answer: Dualize (the dual has a triangle)

31

Steinitz’s TheoremLemma: Every 3-connected, planar graph has a triangle or a vertex of degree 3.

32

Steinitz’s TheoremLemma: Every 3-connected, planar graph has a triangle or a vertex of degree 3.|V |! |E| + |F | = 2

32


|E| =1

2

!

v!V

!(v)

32


|E| =1

2

!

v!V

!(v)

|E| =1

2

!

f!F

|f |

32


|E| =1

2

!

v!V

!(v)

|E| =1

2

!

f!F

|f |

!v !(v) " 4 # |E| " 2|V | (No degree 3)

32


|E| =1

2

!

v!V

!(v)

|E| =1

2

!

f!F

|f |

!v !(v) " 4 # |E| " 2|V |

!f |f | " 4 # |E| " 2|F |

(No degree 3)

(No triangles)

32


|E| =1

2

!

v!V

!(v)

|E| =1

2

!

f!F

|f |

!v !(v) " 4 # |E| " 2|V |

!f |f | " 4 # |E| " 2|F |

(No degree 3)

(No triangles)

|E|

2! |E| +

|E|

2" 2

32


|E| =1

2

!

v!V

!(v)

|E| =1

2

!

f!F

|f |

!v !(v) " 4 # |E| " 2|V |

!f |f | " 4 # |E| " 2|F |

(No degree 3)

(No triangles)

|E|

2! |E| +

|E|

2" 2

0 ! 2

32


So, with the Tutte embedding and the Maxwell-Cremona Correspondence, we can construct a polytope with 1-skeleton isomorphic to either the graph or its dual.

33



If we have the dual, polarize.

33



If we have the dual, polarize.

[Eades, Garvan 1995]

33

A Tour of Other Stuff

34

Rigidity and Unfolding

[Connelly, Demaine, Rote, 2000]

35

Greedy Routing

[Papadimitriou, Ratajczak, 2004]

[Morin, 2001]

36

Robust Geometric Computing

[Hopcroft and Kahn 1992]

37

Spectral Embedding

[Lovasz, 2000]

Correspondence between Colin de Verdiere matrices and Steinitz representations

38

Spectral Embedding

[Lovasz, 2000]

Correspondence between Colin de Verdiere matrices and Steinitz representations

It’s Maxwell-Cremona

38

...

39

Thank you.

40

Thank you.Questions?

40

planar graphs in 2 dimensions - don sheehydonsheehy.net/talks/lifting-planar-graphs-long.pdf ·...

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