Steinitz Representations László Lovász Microsoft Research One Microsoft Way, Redmond, WA 98052...
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Transcript of Steinitz Representations László Lovász Microsoft Research One Microsoft Way, Redmond, WA 98052...
Steinitz Representations
László Lovász
Microsoft Research
One Microsoft Way, Redmond, WA 98052
Steinitz 1922
Every 3-connected planar graphis the skeleton of a convex 3-polytope.
3-connected planar graph
Coin representation
Every planar graph can be represented by touching circles
Koebe (1936)
Polyhedral version
Andre’ev
Every 3-connected planar graph
is the skeleton of a convex polytope
such that every edge
touches the unit sphere
From polyhedra to circles
horizon
From polyhedra to representation of the dual
Rubber bands and planarity
G: 3-connected planar graph
outer face fixed toconvex polygon
edges replaced byrubber bands
2( )i jij E
u u
EEnergy:
Equilibrium:( )
1i j
j N ii
u ud
Tutte (1963)
G 3-connected planar
rubber band embedding is planar
Tutte
(Easily) polynomial time computable
Lifts to Steinitz representation
Maxwell-Cremona
G=(V,E): connected graph
M=(Mij): symmetric VxV matrix
Mii arbitraryMij
<0, if ijE
0, if ,ij E i j
weighted adjacency matrix of GG-matrix
: eigenvalues of M1 2 1... ...k n 0
WLOG
G planar, M G-matrix
corank of M is at most 3.
Colin de VerdièreVan der Holst
G has a K4 or K2,3 minor
G-matrix M such that
corank of M is 3.
Colin de Verdière
Proof.
(a) True for K4 and K2,3.
(b) True for subdivisions of K4 and K2,3.
(c) True for graphs containing subdivisions of K4 and K2,3.
Induction needs stronger assumption!
rk( ) rk( )A M
0 forijA ij E
transversal intersection
M
VxV symmetric matrices
Strong Arnold property
( )ijX X symmetric,
X=00ijX ij E i j for and
0,MX
Representation of G in 3
Nullspace representation
0ij jj
M u
basis of nullspace of M1 2 3 :x x x
11 21 31
12 22
1
232
12 22 3n n
x x x
x x x
ux
u
u
x x
1( )( ) 0i ij j j jj
c M c c u scaling M scaling the ui
Van der Holst’s Lemma
connected
like convex polytopes?
or…
Van der Holst’s Lemma, restated
Let Mx=0. Then
sup ( ), sup ( )x x
are connected, unless…
G 3-connected planar
nullspace representationcan be scaled to convex polytope
G 3-connected planar
nullspace representation,scaled to unit vectors,gives embedding in S2
L-Schrijver
planar embedding nullspace representation
Stresses of tensegrity frameworks
bars
struts
cables x y( )ijM x y
( ) 0ij j ij
M x x Equilibrium:
Cables
Braced polyhedra
Bars
0
0
0 ( , , )ij
ii i ijj V
i j V ij E
M
M
M M
0ij jj V
M u
stress-matrix
There is no non-zero stress on the edges of a convex polytope
Cauchy
Every braced polytopehas a nowhere zero stress (canonically)
( )uvMp q u v
( ) ( )
( ) 0edge
of u
uv uvv N u v N u pq
F
u M v M u v p q
( )uv
v N uuuM v uM
q
p
uFu v
The stress matrix of anowhere 0 stress on a braced polytope
has exactly one negative eigenvalue.
The stress matrix of aany stress on a braced polytope
has at most one negative eigenvalue.
(conjectured by Connelly)
Proof: Given a 3-connected planar G, true for
(a) for some Steinitz representation and the canonical stress;
(b) every Steinitz representation and the canonical stress;
(c) every Steinitz representation and every stress;
Problems
1. Find direct proof that the canonical stress matrix has only 1 negative eigenvalue
2. Directed analog of Steinitz Theorem recently proved by Klee and Mihalisin. Connection with eigensubspaces of non-symmetric matrices?
Let .
Let span a components;
let span b components.
Then , unless…
3. Other eigenvalues?
sup ( )x
kMx x
sup ( )x
a b k
From another eigenvalue of the dodecahedron,we get the great star dodecahedron.
4. 4-dimensional analogue?
(Colin de Verdière number): maximumcorank of a G-matrix with the Strong Arnoldproperty
( )G
( ) 3G G planar
( ) 4G G is linklessly embedable in 3-space
LL-Schrijver
Linklessly embeddable graphs
homological, homotopical,…equivalent
embeddable in 3 without linked cycles
Apex graph
Basic facts about linklessly embeddable graphs
Closed under:
- subdivision
- minor
- Δ-Y and Y- Δ transformations
G linklessly embeddable
G has no minor in the “Petersen family”
Robertson – Seymour - Thomas
The Petersen family
(graphs arising from K6 by Δ-Y and Y- Δ)
Can it be decided in P whethera given embedding is linkless?
Can we construct in P a linkless embedding?
Is there an embedding that canbe certified to be linkless?
Given a linklessly embedable graph…