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Planar Graphs in 21/2 Dimensions
Don Sheehy
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21/2 Dimensions
2
21/2 Dimensions
2
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
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Planar Graphs
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Planar Graphs
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Planar Graphs
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Planar Graphs
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Planar Graphs
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Planar Graphs
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Planar Graphs
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Duality
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Duality
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Duality
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Polar Polytopes
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Polar Polytopes
A!= {x ! R
d | a · x " 1,#a ! A}
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The Maxwell-Cremona Correspondence
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Equilibrium Stresses
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Equilibrium Stresses
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Equilibrium Stresses
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Equilibrium Stresses
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Equilibrium Stresses
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The Maxwell-Cremona Correspondence
There is a 1-1 correspondence between “proper” liftings and equilibrium stresses
of a planar straight line graph.
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The Maxwell-Cremona Correspondence
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The Maxwell-Cremona Correspondence
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The Maxwell-Cremona Correspondence
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The Maxwell-Cremona Correspondence
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The Maxwell-Cremona Correspondence
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The Maxwell-Cremona Correspondence
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Reciprocal Diagrams from Equilibrium Stresses
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Reciprocal Diagrams from Equilibrium Stresses
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Reciprocal Diagrams from Equilibrium Stresses
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Reciprocal Diagrams from Equilibrium Stresses
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Reciprocal Diagrams from Equilibrium Stresses
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Reciprocal Diagrams from Equilibrium Stresses
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Reciprocal Diagrams from Equilibrium Stresses
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Reciprocal Diagrams from Equilibrium Stresses
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Reciprocal Diagrams from Liftings
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Reciprocal Diagrams from Liftings
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Reciprocal Diagrams from Liftings
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Reciprocal Diagrams from Liftings
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Reciprocal Diagrams from Liftings
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The Maxwell-Cremona Corresondence
Equilibrium Stresses
Reciprocal Diagrams
Liftings
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Other Famous Reciprocal Diagrams
Delaunay Triangulation
Voronoi Diagram
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Other Famous Reciprocal Diagrams
Delaunay Triangulation
Voronoi Diagram
Weighted
Weighted
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Other Famous Reciprocal Diagrams
Delaunay Triangulation
Voronoi Diagram
2½ dimensional polarity
Weighted
Weighted
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How to Draw a Graph
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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.
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Spring Interpretation
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Spring Interpretation
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Computing Forces
v ! R2
v
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Computing Forces
v ! R2
Fv =!
u!v
(v ! u)
v
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Computing Forces
v ! R2
Fv =!
u!v
(v ! u)
= dvv !
!
u!v
uv
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Computing Forces
v ! R2
Fv =!
u!v
(v ! u)
= dvv !
!
u!v
u
F = LV
v
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Computing Forces
v ! R2
Fv =!
u!v
(v ! u)
L = D ! A
= dvv !
!
u!v
u
F = LV
v
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Computing Forces
v ! R2
Fv =!
u!v
(v ! u)
L = D ! A
= dvv !
!
u!v
u
F = LV
degrees
v
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Computing Forces
v ! R2
Fv =!
u!v
(v ! u)
L = D ! A
= dvv !
!
u!v
u
F = LV
degrees adjacency
v
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Computing Forces
v ! R2
Fv =!
u!v
(v ! u)
L = D ! A
= dvv !
!
u!v
u
F = LV
degrees adjacency
The Laplacian!
v
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Computing Forces
LV = F
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Computing Forces
LV = F = 0?
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Computing Forces
LV = F = 0? V1: boundaryV2: interior
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Computing Forces
LV = F = 0?
!
L1 BT
B L2
" !
V1
V2
"
=
!
F !
0
"
V1: boundaryV2: interior
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Computing Forces
LV = F = 0?
!
L1 BT
B L2
" !
V1
V2
"
=
!
F !
0
"
BV1 + L2V2 = 0
V1: boundaryV2: interior
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Computing Forces
LV = F = 0?
!
L1 BT
B L2
" !
V1
V2
"
=
!
F !
0
"
BV1 + L2V2 = 0
V2 = !L!1
2B V1
V1: boundaryV2: interior
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Computing Forces
LV = F = 0?
!
L1 BT
B L2
" !
V1
V2
"
=
!
F !
0
"
BV1 + L2V2 = 0
V2 = !L!1
2B V1( )
V1: boundaryV2: interior
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Monotone PathsPick a direction and a vertex. There is a monotone path in that direction from the vertex to the boundary.
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Monotone PathsPick a direction and a vertex. There is a monotone path in that direction from the vertex to the boundary.
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Monotone PathsPick a direction and a vertex. There is a monotone path in that direction from the vertex to the boundary.
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Planar, 3-Connected Graphs
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Planar, 3-Connected Graphs
➡ No K5 or K3,3 minors
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Planar, 3-Connected Graphs
➡ No K5 or K3,3 minors
➡ Removing a face does not disconnect the graph.
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Planar, 3-Connected Graphs
➡ No K5 or K3,3 minors
➡ Removing a face does not disconnect the graph.
➡ No face has a diagonal.
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Double Crossing a FaceLemma: No two disjoint paths have interleaved endpoints on a face.
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Double Crossing a FaceLemma: No two disjoint paths have interleaved endpoints on a face.
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Double Crossing a FaceLemma: No two disjoint paths have interleaved endpoints on a face.
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Double Crossing a FaceLemma: No two disjoint paths have interleaved endpoints on a face.
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Double Crossing a FaceLemma: No two disjoint paths have interleaved endpoints on a face.
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Tutte’s AlgorithmNo ZigZags
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Tutte’s AlgorithmNo ZigZags
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Tutte’s AlgorithmNo ZigZags
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Tutte’s AlgorithmNo ZigZags
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Tutte’s AlgorithmNo ZigZags
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Tutte’s AlgorithmNo ZigZags
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Tutte’s AlgorithmNo ZigZags
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Tutte’s AlgorithmNo Crossings
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Tutte’s AlgorithmNo Crossings
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Tutte’s AlgorithmNo Crossings
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Tutte’s AlgorithmNo Crossings
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Tutte’s AlgorithmNo Crossings
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Tutte’s AlgorithmNo Crossings
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Tutte’s AlgorithmNo Crossings
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Tutte’s AlgorithmNo Overlaps
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Tutte’s AlgorithmNo Overlaps
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Tutte’s AlgorithmNo Overlaps
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Tutte’s AlgorithmNo Overlaps
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Tutte’s AlgorithmNo Overlaps
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Tutte’s AlgorithmNo Overlaps
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Tutte’s AlgorithmNo Overlaps
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Tutte and Maxwell-Cremona
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Tutte and Maxwell-Cremona
➡ Weirdness on the outer face.
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Tutte and Maxwell-Cremona
➡ Weirdness on the outer face.
➡ Lifting still works, except outer face.
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Tutte and Maxwell-Cremona
➡ Weirdness on the outer face.
➡ Lifting still works, except outer face.
➡ Lifting is convex.
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Steinitz’s Theorem
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Steinitz’s Theorem
A graph G is the 1-skeleton of a3-polytope if and only if it is
simple, planar, and 3-connected.
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Steinitz’s Theorem
Claim: If the graph has a triangle, then the Tutte embedding followed by the Maxwell-Cremona lifting gives the desired polytope.
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Steinitz’s Theorem
Claim: If the graph has a triangle, then the Tutte embedding followed by the Maxwell-Cremona lifting gives the desired polytope.
Fix the triangle as the outer face.
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Steinitz’s Theorem
Claim: If the graph has a triangle, then the Tutte embedding followed by the Maxwell-Cremona lifting gives the desired polytope.
Fix the triangle as the outer face.
After the lifting, the triangle must lie on a plane.
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Steinitz’s Theorem
Question: What if there is no triangle?
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Steinitz’s Theorem
Question: What if there is no triangle?Answer: Dualize (the dual has a triangle)
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Steinitz’s TheoremLemma: Every 3-connected, planar graph has a triangle or a vertex of degree 3.
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Steinitz’s TheoremLemma: Every 3-connected, planar graph has a triangle or a vertex of degree 3.|V |! |E| + |F | = 2
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Steinitz’s TheoremLemma: Every 3-connected, planar graph has a triangle or a vertex of degree 3.|V |! |E| + |F | = 2
|E| =1
2
!
v!V
!(v)
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Steinitz’s TheoremLemma: Every 3-connected, planar graph has a triangle or a vertex of degree 3.|V |! |E| + |F | = 2
|E| =1
2
!
v!V
!(v)
|E| =1
2
!
f!F
|f |
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Steinitz’s TheoremLemma: Every 3-connected, planar graph has a triangle or a vertex of degree 3.|V |! |E| + |F | = 2
|E| =1
2
!
v!V
!(v)
|E| =1
2
!
f!F
|f |
!v !(v) " 4 # |E| " 2|V | (No degree 3)
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Steinitz’s TheoremLemma: Every 3-connected, planar graph has a triangle or a vertex of degree 3.|V |! |E| + |F | = 2
|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)
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Steinitz’s TheoremLemma: Every 3-connected, planar graph has a triangle or a vertex of degree 3.|V |! |E| + |F | = 2
|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
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Steinitz’s TheoremLemma: Every 3-connected, planar graph has a triangle or a vertex of degree 3.|V |! |E| + |F | = 2
|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
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Steinitz’s Theorem
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.
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Steinitz’s Theorem
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.
If we have the dual, polarize.
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Steinitz’s Theorem
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.
If we have the dual, polarize.
[Eades, Garvan 1995]
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A Tour of Other Stuff
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Rigidity and Unfolding
[Connelly, Demaine, Rote, 2000]
35
Greedy Routing
[Papadimitriou, Ratajczak, 2004]
[Morin, 2001]
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Robust Geometric Computing
[Hopcroft and Kahn 1992]
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Spectral Embedding
[Lovasz, 2000]
Correspondence between Colin de Verdiere matrices and Steinitz representations
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Spectral Embedding
[Lovasz, 2000]
Correspondence between Colin de Verdiere matrices and Steinitz representations
It’s Maxwell-Cremona
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...
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Thank you.
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Thank you.Questions?
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