Steinitz Representations
László Lovász
Microsoft Research
One Microsoft Way, Redmond, WA 98052
[email protected]
3-connected planar graph
Every 3-connected planar graph
is the skeleton of a convex 3-polytope.
Steinitz 1922
Coin representation Koebe (1936)
Every planar graph can be represented by touching circles
Polyhedral version
Every 3-connected planar graph
is the skeleton of a convex polytope
such that every edge
touches the unit sphere
Andre’ev
From polyhedra to circles
horizon
From polyhedra to representation of the dual
Rubber bands and planarity
Tutte (1963)
G: 3-connected planar graph
outer face fixed to
convex polygon
edges replaced by
rubber bands
Energy: E 
2
(
u

u
)
 i j
ij E
1
Equilibrium: ui 
di

j N ( i )
uj
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
<0, if ijE
Mij
0, if ij  E , i  j
Mii arbitrary
weighted adjacency matrix of G
G-matrix
1  2  ...  k 1  ...  n: eigenvalues of M
 0 WLOG
G planar, M G-matrix

corank of M is at most 3.
Colin de Verdière
Van 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!
Strong Arnold property
VxV symmetric matrices
Aij  0 for ij  E
M
rk( A)  rk( M )
transversal intersection
X  ( X ij ) symmetric, MX  0,
X ij  0 for ij  E and i  j

X=0
Nullspace representation
x1 x2 x3 :
  
x11 x21 x31
basis of nullspace of M
 u1
x12 x22 x32  u2
Representation of G in R3
x12 x22 x3n  un
M u
ij
j
0
j
 (c M
i
j
1
ij
c j )(c j u j )  0
scaling M  scaling the ui
Van der Holst’s Lemma
or…
connected
like convex polytopes?
Van der Holst’s Lemma, restated
Let Mx=0. Then
sup ( x), sup ( x)
are connected, unless…
G 3-connected planar

nullspace representation
can 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
Equilibrium:
x
M
j
ij
M ij ( x  y )
( x j  xi )  0
y
Braced polyhedra
Bars
0
Cables
M ij  0
(i, j V , ij  E )
M ii  M 0i   M ij
j V
stress-matrix
M u
ij
j V
j
0
There is no non-zero stress
on the edges of a convex polytope
Cauchy
Every braced polytope
has a nowhere zero stress (canonically)
u
q
v
Fu
p
p  q  M uv (u  v)
u

M uv v 
v N ( u )

M uv (u  v) 
v N ( u )

v N ( u )
M uv v   M uu u

pq edge
of Fu
pq  0
The stress matrix of a
nowhere 0 stress on a braced polytope
has exactly one negative eigenvalue.
The stress matrix of a
any 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?
3. Other eigenvalues?
Let Mx  k x.
Let sup ( x) span a components;

let
sup ( x) span b components.
Then
a  b  k , unless…
From another eigenvalue of the dodecahedron,
we get the great star dodecahedron.
4. 4-dimensional analogue?
 (G ) (Colin de Verdière number): maximum
corank of a G-matrix with the Strong Arnold
property
 (G )  3  G planar
 (G )  4  G is linklessly embedable
in 3-space
LL-Schrijver
Linklessly embeddable graphs
embeddable in R3 without linked cycles
homological, homotopical,…
equivalent
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)
Given a linklessly embedable graph…
Can we construct in P a
linkless embedding?
Can it be decided in P whether
a given embedding is linkless?
Is there an embedding that can
be certified to be linkless?