Steinitz Representations

László Lovász

Microsoft Research

One Microsoft Way, Redmond, WA 98052

lovasz@microsoft.com

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…