Title: Steinitz Representations
1 Steinitz Representations László Lovász
Microsoft Research One Microsoft Way, Redmond,
WA 98052 lovasz_at_microsoft.com
23-connected planar graph
3Coin representation
Koebe (1936)
Every planar graph can be represented by touching
circles
4Polyhedral version
Every 3-connected planar graph is the skeleton
of a convex polytope such that every edge
touches the unit sphere
Andreev
5From polyhedra to circles
horizon
6From polyhedra to representation of the dual
7Rubber bands and planarity
Tutte (1963)
8Tutte
9G(V,E) connected graph
M(Mij) symmetric VxV matrix
Mii arbitrary
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11Proof.
(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!
12Strong Arnold property
VxV symmetric matrices
13Nullspace representation
14Van der Holsts Lemma
or
like convex polytopes?
15Van der Holsts Lemma, restated
16- G 3-connected planar
- ?
- nullspace representation
- can be scaled to convex polytope
17nullspace representation
planar embedding
18Stresses of tensegrity frameworks
bars
struts
cables
19Braced polyhedra
stress-matrix
20Every braced polytope has a nowhere zero stress
(canonically)
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22The 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)
23Proof Given a 3-connected planar G, true for
- for some Steinitz representation
- and the canonical stress
(b) every Steinitz representation and the
canonical stress
(c) every Steinitz representation and every
stress
24Problems
- Find direct proof that the canonical
- stress matrix has only 1 negative eigenvalue
- Directed analog of Steinitz Theorem
- recently proved by Klee and Mihalisin.
- Connection with eigensubspaces of
- non-symmetric matrices?
253. Other eigenvalues?
Let . Let span
a components let span b
components. Then , unless
From another eigenvalue of the dodecahedron, we
get the great star dodecahedron.
264. 4-dimensional analogue?
(Colin de Verdière number) maximum corank of a
G-matrix with the Strong Arnold property
? G planar
? G is linklessly embedable in 3-space
LL-Schrijver
27Linklessly embeddable graphs
embeddable in R3 without linked cycles
28Basic facts about linklessly embeddable graphs
Closed under - subdivision
- minor
- ?-Y and Y- ? transformations
29The Petersen family
(graphs arising from K6 by ?-Y and Y- ?)
30Given 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?
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