Vector representations of graphs László Lovász Microsoft Research One Microsoft Way, Redmond, WA 98052 lovasz@microsoft.com Cim

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

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

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

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

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

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

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

Special values rank=1

Special values Violates the Strong Arnold property! rank=2

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

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

homological, homotopical,… 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)

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

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

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

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’.

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

?     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 ?

planar embedding nullspace representation

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

Colin de Verdière matrix M Steinitz representation P q p u v

Every planar graph can be represented by touching circles 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

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

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.

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

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

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

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

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

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

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

  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

Rubber bands and maximum cuts maximize

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

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

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

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