Steinitz Representations

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Steinitz Representations László Lovász
Microsoft Research One Microsoft Way, Redmond,
WA 98052 lovasz_at_microsoft.com
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3-connected planar graph
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Coin representation
Koebe (1936)
Every planar graph can be represented by touching
circles
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Polyhedral version
Every 3-connected planar graph is the skeleton
of a convex polytope such that every edge
touches the unit sphere
Andreev
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From polyhedra to circles
horizon
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From polyhedra to representation of the dual
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Rubber bands and planarity
Tutte (1963)
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Tutte
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G(V,E) connected graph
M(Mij) symmetric VxV matrix
Mii arbitrary
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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!
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Strong Arnold property
VxV symmetric matrices
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Nullspace representation
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Van der Holsts Lemma
or
like convex polytopes?
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Van der Holsts Lemma, restated
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• G 3-connected planar
• ?
• nullspace representation
• can be scaled to convex polytope

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nullspace representation
planar embedding
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Stresses of tensegrity frameworks
bars
struts
cables
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Braced polyhedra
stress-matrix
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Every braced polytope has a nowhere zero stress
(canonically)
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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)
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Proof 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
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Problems

• 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?

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3. 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.
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4. 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
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Linklessly embeddable graphs
embeddable in R3 without linked cycles
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Basic facts about linklessly embeddable graphs
Closed under - subdivision
- minor
- ?-Y and Y- ? transformations
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The Petersen family
(graphs arising from K6 by ?-Y and Y- ?)
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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?
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