1. Vector representations of graphs
László Lovász
Microsoft Research
One Microsoft Way, Redmond, WA
Cim

2. Every planar graph can be drawn in the plane with straight edges
Fáry-Wagner

3. 3-connected planar graph
Steinitz
Every 3-connected planar graph
is the skeleton of a polytope.

4. Rubber bands and planarity
Tutte (1963)
G: 3-connected planar graph
outer face fixed to
convex polygon
edges replaced by
rubber bands
Energy:
Equilibrium:

5. rubber band embedding is planar
G 3-connected planar
rubber band embedding is planar
(Easily) polynomial time
computable
Lifts to polyhedral
representation
Maxwell
Tutte

6. Largest has multiplicity 1.
The Colin de Verdière number
G: connected graph
Roughly: multiplicity of second largest eigenvalue
of adjacency matrix
Largest has multiplicity 1.
But: maximize over weighting the edges and diagonal entries
But: non-degeneracy condition on weightings

7. Strong Arnold Property
M=(Mij): symmetric VxV matrix
Mij
<0, if ijE
0, if
Mii arbitrary
normalization
M has =1 negative eigenvalue
symmetric, 
X=0
Strong Arnold Property

8. Basic Properties μk is polynomial time decidable for fixed k
μ(G) is minor monotone
deleting and
contracting edges
for μ>2, μ(G) is invariant under subdivision
for μ>3, μ(G) is invariant under Δ-Y transformation

9. Special values
rank=1

10. Special values
Violates the Strong Arnold property!
rank=2

11. Special values
μ(G)1  G is a path
non-singular

12. Special values μ(G)1  G is a path μ(G)2  G is outerplanar
μ(G)3  G is a planar
Colin de Verdière, using pde’s
Van der Holst, elementary proof
μ(G)4  G is linklessly embeddable in 3-space …
μ(G)n-4  complement G is planar _ ~
Kotlov-L-Vempala

13. homological, homotopical,…
Linklessly embeddable graphs
embeddable in R3 without linked cycles
homological, homotopical,…
equivalent
Apex graph

14.  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

15. The Petersen family
(graphs arising from K6 by Δ-Y and Y- Δ)

16. G is linklessly embeddable
: follows from Robertson-Seymour-Thomas
L-Schrijver

17. Nullspace representation
basis of nullspace of M
Representation of G in Rd

18. Van der Holst’s Lemma
connected
or…
like convex polytopes?

19. Linked Borsuk Theorem P R5 convex polytope A,B: faces of P
A, B opposite:  parallel supporting hyperplanes
H, H’ such that A  H, B  H’.

20. Linked Borsuk Theorem  embedding φ: P1R3
 opposite 2-dimensional faces A,B,
such that φ(A) and φ(B) are linked
L-Schrijver
Special case: K6 is not linklessly embeddable

21. ?     G path nullspace representation gives embedding in R1
properly normalized
G 2-connected
outerplanar 
nullspace representation gives
outerplanar embedding in R2
G 3-connected
planar 
nullspace representation gives
planar embedding in S2
L-Schrijver
G 4-connected
linkless embed. 
nullspace representation gives
linkless embedding in R3 ?

22. planar embedding
nullspace representation

23.  G 3-connected nullspace representation gives planar
planar embedding in S2
The vectors can be rescaled so that
we get a convex polytope.

24. Colin de Verdière matrix M Steinitz representation Pq p u v

25. Every planar graph can be represented by touching circles
Coin representation
Koebe (1936)
Every planar graph can be represented by touching circles

26. Polyhedral version Every 3-connected planar graph
is the skeleton of a convex polytope
such that every edge
touches the unit sphere
Andre’ev

27. From polyhedra to circles
horizon

28. From polyhedra to representation of the dual

29. The Gram representation
Kotlov – L - Vempala
pos semidefinite
Gram representation

30. Properties of the Gram representation
Assume: G has no twin nodes, and
exceptional
 ui is a vertex of P
is an edge of P 
 0  int P
If G has no twin nodes, and μ(G)n-4, then
is planar.

31. Vectors to spheres ui Ci ui Ci Cj uj representation of by orth circles
planar

32. of hyperbolic geometry
Projective distance
Hilbert d
distance: a b c
Cayley-Klein model
of hyperbolic geometry

33. Projective “distance”:b a b a
“distance”: 1
“distance”: p q r s d b c
“distance”: a C D

34. G has euclidean distance 1 representation in R1  G is a path
G: connected graph
G has euclidean distance 1 representation in R1 
G is a path
G has projective “distance” >1 representation in R2 
G is outerplanar
G has projective “distance” 1 representation in R3 
G is planar
Koebe

37. Rubber bands and connectivity
G: arbitrary graph
A,B  V, |A|=|B|=k
A: affine indep in Rd
edges: rubber bands

38.  For almost all choices of edge strengths: B affine indep Linial-L-
k disjoint (A,B)-paths
Linial-L-
Wigderson
()
cutset

39.  For almost all choices of edge strengths: B affine indep Linial-L-
k disjoint (A,B)-paths
Linial-L-
Wigderson
()
strengthen

40.   edges strength s.t. B is independent no algebraic relation
for a.a. choices of edge strengths, B is independent
no algebraic relation
between edge strength
G is k-connected 
nodes in the generic rubber band
embedding, with A fixed, are in general position

41. Rubber bands and maximum cuts
maximize

42. Polynomial with 12% error
Max Cut: NP-hard
Approximations?
Easy with 50% error Erdős
NP-hard with 6% error Hastad
Polynomial with 12% error
Goemans-Williamson

43. spring (repulsive)
Energy:
How to find minimum energy position?
dim=1: Max Cut
Min energy  4 Max Cut
dim=2: probably hard 
dim=n: Poly time solvable!
semidefinite optimization

44. Solvable in polynomial time
Introduce new variables:
These satisfy:
linear!
convex!
The objective function is:
Solvable in polynomial time

45. minimum energy
in n dimension
random hyperplane
Probability of edge ij cut:
Expected number of edges cut: