Characterizations of the Rigidity of Graphsahigashi/tanigawa_rigidity.pdf · Characterizations of Generic Rigidity algebraic combinatorial rigidity Asimov-Roth 78 d 2 : Laman 70 d

Characterizations of the Rigidity of Graphs

Shin-ichi Tanigawa1

1Kyoto University, Japan

February 22, 2015

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Rigidity of Graphs

flexible rigid

rigidglobally rigid

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Rigidity of Bar-joint FrameworksI (Bar-joint) framework: (G , p)

I G = (V ,E)I p : V → Rd , a joint configuration

I (G , p) ∼ (G , q)def⇔ ∀ij ∈ E , ‖p(i)− p(j)‖ = ‖q(i)− q(j)‖

I p ≡ qdef⇔ ∀i , j ∈ V , ‖p(i)− p(j)‖ = ‖q(i)− q(j)‖

I (G , p) is globally rigid in Rd def⇔∀q ∈ (Rd)V , (G , p) ∼ (G , q) ⇒ p ≡ q

I (G , p) is (locally) rigid in Rd def⇔∃neighborU of p in (Rd)V s.t. ∀q ∈ U, (G , q) ∼ (G , p) ⇒ q ≡ p

I (G , p) is universally rigid in Rd def⇔ (G , p) is globally rigid in Rd′for

all d ′ ≥ d .

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all d ′ ≥ d .

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all d ′ ≥ d .

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all d ′ ≥ d .

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all d ′ ≥ d .

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Characterization of Rigidity

I Our goal is to give a good characterization of the local/globalrigidity of frameworks.

I Example. 1-dimensional rigidityI (G , p) is rigid in R1 ⇔ G is connected.

I Checking global/local rigidity is in general a hard problemI (Saxe 79) 1-dimensional global rigidityI (Abbot 08) 2-dimensional rigidity

I Theorem. Let (G , p) be a 1-dimensional framework with generic p(i.e., the set of coordinates is algebraically independent over Q).Then (G , p) is globally rigid in R1 iff G is 2-connected.

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Characterizations of Generic Rigidity

algebraic combinatorial

rigidity Asimov-Roth 78d ≤ 2 : Laman 70

d ≥ 3 : open

global rigidity Gortler-Healy-Thurston 10d ≤ 2 : Jackson-Jordan 05

d ≥ 3 : openuniversal rigidity Gortler-Thurston 14 open

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Algebraic Characterization: RigidityI rigidity map of G :

`G : RdV 3 p 7→ (· · · , ‖p(i)− p(j)‖2, · · · )> ∈ RE

I rigidity matrix R(G , p) of (G , p)def⇔ The Jacobian of `G at p

I Proposition(Asimov-Roth 78) If p is generic, (G , p) is rigid in Rd iff

rank R(G , p) = d |V | −(d + 1

2

).

I Intuition:I The rank represents the number of ”independent” constraintsI Each vertex has d degree of freedomsI Each rigid framework has

(d+12

)degree of freedoms for congruent

motions (which cannot be eliminated by adding bars)

I ∵ I `−1G (`G (p)) ⊇ q ∈ RdV : q ≡ p ' Eucl(d)

I dimEucl(d) =(d+12

)I If p is generic, (G , p) is rigid iff dim `−1

G (`G (p)) = dimEucl(d), i.e.,dim kerR(G , p) =

(d+12

)

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`G : RdV 3 p 7→ (· · · , ‖p(i)− p(j)‖2, · · · )> ∈ RE



rank R(G , p) = d |V | −(d + 1

2

).


(d+12




I dimEucl(d) =(d+12



(d+12

)

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`G : RdV 3 p 7→ (· · · , ‖p(i)− p(j)‖2, · · · )> ∈ RE



rank R(G , p) = d |V | −(d + 1

2

).


(d+12




I dimEucl(d) =(d+12



(d+12

)

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`G : RdV 3 p 7→ (· · · , ‖p(i)− p(j)‖2, · · · )> ∈ RE



rank R(G , p) = d |V | −(d + 1

2

).


(d+12




I dimEucl(d) =(d+12



(d+12

)6 / 38

Rigidity of Graphs


rank R(G , p) = d |V | −(d + 1

2

).

I Remark:I rank R(G , p) is maximized for any generic pI Rigidity is a generic property of graphs (i.e., rigid for some generic p

⇔ rigid for every generic p).

I G is rigid in Rd def⇔ (G , p) is rigid in Rd for some/all generic pI Can be checked by a randomized algorithmI Open if it is in P.

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Rigidity of Graphs


rank R(G , p) = d |V | −(d + 1

2

).

I Remark:I rank R(G , p) is maximized for any generic pI Rigidity is a generic property of graphs (i.e., rigid for some generic p

⇔ rigid for every generic p).

I G is rigid in Rd def⇔ (G , p) is rigid in Rd for some/all generic pI Can be checked by a randomized algorithmI Open if it is in P.

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Infinitesimal Rigidity

I (G , p) is infinitesimally rigiddef⇔ rank R(G , p) = d |V | −

(d+12

).

I Proposition. If (G , p) is infinitesimally rigid, then (G , p) is rigid.I If p is generic, infinitesimal rigidity ⇔ rigidity.

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Stresses

I ω : E → R is self-stressdef⇔ ω ∈ kerR(G , p)>

⇔ ∀i ∈ V ,∑

j∈NG (i)

ω(ij)(p(i)− p(j)) = 0.

1

1

1

1−1

−1

I (G , p) is stress-freedef⇔ (G , p) has no non-zero self-stress.

I ⇔ R(G , p) is row independentI (G , p) is rigid iff G contains a spanning stress-free subgraph H with

|E(H)| = d |V (H)| −(d+12

).

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Stress-freeness of Convex Polytopes

I Theorem (Dehn 1916). The 1-skelton of a strictly convex polytopein R3 is stress-free.

I Corollary. The 1-skelton of a strictly convex polytope in R3 isinfinitesimally rigid iff it is simplicial.

I Corollary. A planar graph is rigid in R3 if and only if it is atriangulation.

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Combinatorial Characterization: Rigidity

I Maxwell’s condition. If (G , p) is stress-free, then

∀F ⊆ E with |V (F )| ≥ d , |F | ≤ d |V (F )| −(d + 1

2

)

I d = 1: the converse is true; G is forest iff ∀F ⊆ E , |F | ≤ |V (F )| − 1

I d = 2 : the converse is true if p is generic;

Theorem(Laman 1970) Suppose p is generic. Then (G , p) isstress-free iff ∅ 6= ∀F ⊆ E , |F | ≤ 2|V (F )| − 3

I f2,3 : 2E 3 F 7→ 2|V (F )| − 3 is submodular, i.e.,

f2,3(X ) + f2,3(Y ) ≥ f2,3(X ∩ Y ) + f2,3(X ∪ Y ) for any X ,Y ∈ 2E .

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∀F ⊆ E with |V (F )| ≥ d , |F | ≤ d |V (F )| −(d + 1

2

)






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∀F ⊆ E with |V (F )| ≥ d , |F | ≤ d |V (F )| −(d + 1

2

)






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∀F ⊆ E with |V (F )| ≥ d , |F | ≤ d |V (F )| −(d + 1

2

)






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Proof of Laman’s Theorem: Henneberg Construction

I (Combinatorial part) A graph satisfies Laman’s condition iff it canbe built up from K2 by a sequence of 0-extension and 1-extension.

0-extension 1-extension

I (Algebraic part) Both 0-extension and 1-extension preserve thestress-freeness in R2.

I The d-dimensional version also holds

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A Proof for the Combinatorial Part

I Theorem(Edmonds 70) For a monotone submodular functionf : 2E → Z,

I := I ⊆ E : ∅ 6= ∀F ⊆ I , |F | ≤ f (F )

is the family of independent sets of a matroid on E .

I f2,3 induces a matroid M on the complete graph on V .

I Claim If G satisfies Laman’s condition, then G can be reduced to asmaller graph by the inverse of 0-extension or 1-extension.

I the average degree of G is less than fourI ∃v with dG (v) ≤ 2 ⇒ the inverse of 0-extension at v is admissible.I ∃v with dG (v) = 3 ⇒ the inverse of 1-extension at v is admissible.

I Let NG (v) = a, b, c.I If not admissible, ab, bc, ca ∈ clM(E − va, vb, vc)I va, vb, vc, ab, bc, ca is dependent in M, so

va ∈ clM(vb, vc, ab, bc, ca) ⊆ clM(E − va+ ab, bc, ca) =clM(E − va)

I E is dependent, a contradiction.

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I := I ⊆ E : ∅ 6= ∀F ⊆ I , |F | ≤ f (F )








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I := I ⊆ E : ∅ 6= ∀F ⊆ I , |F | ≤ f (F )








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Generic Rigidity in R3

I d ≥ 3: Maxwell’s condition is not sufficient in general

I 0-extension and 1-extension are not sufficient

I Partial resultsI sparse graphs (Jackson-Jordan 05)I degree-bounded graphs (Jackson-Jordan 05)I K5-minor-free graphs (Nevo 07)I molecular graphs (the squares of graphs) (Katoh-T 11)

I Conjecture(Lovasz-Yemini 81) If G is 11-connected, then G is rigidin R3.

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Sparse Graphs

I Theorem(Jackson-Jordan 05) Let G = (V ,E ) be a graph and d bean even integer. If

|F | ≤ d

2|V (F )| − (d + 1)

∅ 6= ∀F ⊆ E , then G is stress-free in Rd .I f d

2,d+1 is integer-valued monotone submodular, hence inducing a

matroid.I The remaining part of the proof is the same as the case in Laman’s

theorem

I Conjecture(Jackson-Jordan 05) This holds even for odd d .

I Theorem(Jackson-Jordan 05) If |F | ≤ 52 |V (F )| − 7

3 for anynonempty F ⊆ E , then G is stress-free in R3.

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Sparse Graphs

I Theorem(Jackson-Jordan 05) Let G = (V ,E ) be a graph and d bean even integer. If

|F | ≤ d

2|V (F )| − (d + 1)

∅ 6= ∀F ⊆ E , then G is stress-free in Rd .I f d

2,d+1 is integer-valued monotone submodular, hence inducing a

matroid.I The remaining part of the proof is the same as the case in Laman’s

theorem

I Conjecture(Jackson-Jordan 05) This holds even for odd d .

I Theorem(Jackson-Jordan 05) If |F | ≤ 52 |V (F )| − 7

3 for anynonempty F ⊆ E , then G is stress-free in R3.

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Molecular Frameworks

I A molecular framework is a bar-joint framework whose underlyinggraph is G 2 of some G .

I Theorem (Katoh-T11) Suppose p is generic. Then (G 2, p) is rigid inR3 if and only if 5G contains six edge-disjoint spanning trees.

G G 2

5G

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Global RigidityI ω ∈ RE is a self-stress of (G , p)

def⇔

∀i ∈ V ,∑

j∈NG (i)

ω(ij)(p(i)− p(j)) = 0.

I stress matrix Ω (of ω)def⇔ the Laplacian of G weighted by ω, i.e.,

∑i 6=1 ω(1i) −ω(12) . . . −ω(1n)−ω(12)

∑i 6=2 ω(2i) . . . −ω(2n)

......

. . ....

−ω(1n) −ω(2n) . . .∑

i 6=n ω(1n)

I Proposition. Suppose p(V ) affinely spans Rd . Then

dim ker Ω ≥ d + 1.

I ∵I

(1 . . . 1

)> ∈ ker ΩI The projection of p to each coordinate is in ker Ω.I They are linearly independent if p(V ) affinely spans Rd

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Global RigidityI ω ∈ RE is a self-stress of (G , p)

def⇔

∀i ∈ V ,∑

j∈NG (i)

ω(ij)(p(i)− p(j)) = 0.

I stress matrix Ω (of ω)def⇔ the Laplacian of G weighted by ω, i.e.,

∑i 6=1 ω(1i) −ω(12) . . . −ω(1n)−ω(12)

∑i 6=2 ω(2i) . . . −ω(2n)

......

. . ....

−ω(1n) −ω(2n) . . .∑

i 6=n ω(1n)

I Proposition. Suppose p(V ) affinely spans Rd . Then

dim ker Ω ≥ d + 1.

I ∵I

(1 . . . 1

)> ∈ ker ΩI The projection of p to each coordinate is in ker Ω.I They are linearly independent if p(V ) affinely spans Rd

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Characterizing Global Rigidity

I Theorem (Connelly 06, Gortler-Healy-Thurston 10) If p is generic,then (G , p) is globally rigid in Rd iff

∃ω ∈ ker R(G , p)> s.t. dim ker Ω = d + 1.

I Corollary. Global rigidity is a generic property!!

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Proof Sketch for the Sufficiency

I Lemma Suppose p is generic and (G , p) ∼ (G , q). Then a stress of(G , p) is a stress of (G , q).

I By the inverse function theorem, there is a diffeomorphismψ : Uq → Up from a neighbor Uq of q to that Up of p such thatψ(q) = p and fG (ψ(x)) = fG (x) for x ∈ Uq.

I ωR(G , q) = 2ωdfG |x=q = 2ωdfG |x=p · dψ|x=q = ωR(G , p)dψ|x=q = 0.

I Lemma. Suppose ω is a self-stress of (G , p) with rankΩ = n− d − 1.Then any (G , q) having ω as a self-stress is an affine image of(G , p), i.e., ∃A ∈ M(d), t ∈ Rd s.t. q(i) = Ap(i) + t (∀i ∈ V ).

I Let P be the (d +1)× n-matrix whose i-th row is

(p(i)1

), and let Q

be the corresponding matrix for q.I ω is a self-stress of (G , q) ⇒ QΩ = 0

I The row vectors of P span ker Ω ⇒ ∃(A, t) s.t.(A t0 1

)P = Q

I Now, if (G , p) has a stress ω with rank Ω = n − d − 1, then (G , q)is an affine image of (G , p) for any equivalent (G , q)

I The genericity implies that A is actually orthogonal, implying p ≡ q.

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(p(i)1

), and let Q



)P = Q



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(p(i)1

), and let Q



)P = Q



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Combinatorial Characterization of Global RigidityTheorem (Hendrickson 92)If G is globally rigid in Rd , then G is (d + 1)-connected and redundantlyrigid in Rd .

Proof Sketch (for d = 2)I Take a generic p, and suppose (G , p) is rigid but (G − ij , p) is notI Take uv ∈ E − ij and let

W = q ∈ RdV : q(u) = p(u), q(v) = p(v)I By the inverse function theorem, f −1

G (fG (p)) ∩W is a compact1-manifold. So a component C containing p is a cycle.

I gij : C 3 x 7→ ‖x(i)− x(j)‖2 is continous, and hence ∃q ∈ C \ ps.t. gij(q) = gij(p), i.e., (G , q) ∼ (G , p).

I How can we ensure that q is not congruent to p??

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Combinatorial Characterization of Global RigidityTheorem (Hendrickson 92)If G is globally rigid in Rd , then G is (d + 1)-connected and redundantlyrigid in Rd .

Proof Sketch (for d = 2)I Take a generic p, and suppose (G , p) is rigid but (G − ij , p) is notI Take uv ∈ E − ij and let

W = q ∈ RdV : q(u) = p(u), q(v) = p(v)I By the inverse function theorem, f −1

G (fG (p)) ∩W is a compact1-manifold. So a component C containing p is a cycle.

I gij : C 3 x 7→ ‖x(i)− x(j)‖2 is continous, and hence ∃q ∈ C \ ps.t. gij(q) = gij(p), i.e., (G , q) ∼ (G , p).

I How can we ensure that q is not congruent to p??20 / 38

Combinatorial Characterization of Global Rigidity

Theorem (Hendrickson 92)If G is globally rigid in Rd , then G is (d + 1)-connected and redundantlyrigid in Rd .

I d = 1: the converse is true; G is globally rigid in R1 iff G is2-connected.

I d = 2: the converse is true;Theorem (Jackson-Jordan 05) G is globally rigid in R2 iff G is3-connected and redundantly rigid in R2.

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Combinatorial Characterization of Global Rigidity

Theorem (Hendrickson 92)If G is globally rigid in Rd , then G is (d + 1)-connected and redundantlyrigid in Rd .

I d = 1: the converse is true; G is globally rigid in R1 iff G is2-connected.

I d = 2: the converse is true;Theorem (Jackson-Jordan 05) G is globally rigid in R2 iff G is3-connected and redundantly rigid in R2.

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Proof Sketch

Theorem (Jackson-Jordan 05)G is globally rigid in R2 iff G is 3-connected and redundantly rigid in R2.

I Algebraic Part (Connelly 05) 1-extension preserves the global rigidity

1-extension

I Combinatorial Part (Berg-Jordan 03, Jackson-Jordan 05) A graph is3-connected and redundantly rigid in R2 if and only if it can be builtfrom K4 by a sequence of 1-extension and edge addition.

I The proof also implies the theorem by GHT for d = 2.

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A Simpler Proof (T14)

I Algebraic Part. Suppose that G − v is rigid and G − v + K (NG (v))is globally rigid. Then G is globally rigid.

I Combinatorial Part. Suppose G is 3-connected and redundantly rigidin R2. Then ∃v ∈ V with dG (v) = 3 or ∃e ∈ E s.t. G − e is3-connected and redundantly rigid.

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A Key Tool: Ear-decompostion (Jackson-Jordan05)

I A graph is 2-connected iff it can be built up from a cycle byattaching ”ears”

I ⇒ If G is minimally 2-connected, then ∃v with dG (v) = 2I ⇒ A 2-connected graph can be built up from a triangle by a

sequence of 1-extension and edge-addition

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I A graph is 2-connected iff it can be built up from a cycle byattaching ”ears”

I ⇒ If G is minimally 2-connected, then ∃v with dG (v) = 2I ⇒ A 2-connected graph can be built up from a triangle by a

sequence of 1-extension and edge-addition

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I A matroid M = (E , I) is (2-)connected def⇔ for any pair of elementsthere is a circuit containing them

I 2-connectivity of G ⇔ connectivity of R1(G)

I Theorem(JJ05) If G is 3-connected, redundant rigidity of G in R2 ⇔connectivity of R2(G)

I Theorem(Coullard and Hellerstein 96) M is connected iff ∃an eardecomposition, that is, a sequence C1,C2, . . . ,Ct of circuits in Msatisfying, for any 2 ≤ i ≤ t,

I Ci ∩ Di−1 6= ∅ and Ci \ Di−1 6= ∅, where Di =⋃i

j=1 Cj ;I for any circuit C with this property, C \ Di−1 6⊂ Ci \ Di−1;I E = Dt .

I Claim(JJ05) If R2(G ) is minimally connected, then ∃v withdG (v) = 3.

I Let V ′ = V \ V (Dt−1). By the minimality of R2(G), V ′ 6= ∅.I The rank increases by |Ct \ Dt−1| − 1 when attaching Ct , i.e.,

2|V | − 3− (2|V \ V ′| − 3) = |Ct \ Dt−1| − 1. So|Ct \ Dt−1| = 2|V ′|+ 1

I the average degree of vertices in V ′ is2|Ct\Dt−1|−dG (V ′)

|V ′| < 4

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I 2-connectivity of G ⇔ connectivity of R1(G)I Theorem(JJ05) If G is 3-connected, redundant rigidity of G in R2 ⇔

connectivity of R2(G)






2|V | − 3− (2|V \ V ′| − 3) = |Ct \ Dt−1| − 1. So|Ct \ Dt−1| = 2|V ′|+ 1


|V ′| < 4

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2|V | − 3− (2|V \ V ′| − 3) = |Ct \ Dt−1| − 1. So|Ct \ Dt−1| = 2|V ′|+ 1


|V ′| < 4

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2|V | − 3− (2|V \ V ′| − 3) = |Ct \ Dt−1| − 1. So|Ct \ Dt−1| = 2|V ′|+ 1


|V ′| < 4

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Generic Global Rigidity in R3

I d ≥ 3: Hendrickson’s necessary condition is not sufficientI Kd+2,d+2 (Connelly)I Infinite examples for d ≥ 5 (Frank-Jiang 11)I Infinite examples for d ≥ 3 (Jordan-Kiraly-T14)

Not redundantly rigid Redunduntly rigid

I Theorem(T14) If G is vertex-redundantly rigid in Rd , then G isglobally rigid in Rd .

I G − v is rigidI G − v + K(NG (v)) is vertex-redundantly rigid, thus globally rigid by

induction

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Recent Topics

I Generic local/global rigidity in 3-space (or d-space)

I Rigidity of symmetric frameworks (non-generic frameworks)

I Extension to local/global unique completability

I Universal rigidity

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Rigidity of Symmetric FrameworksI A framework (G , p) is Γ-symmetric if

I Γ acts on G through θ : Γ → Aut(G) andI p(θ(γ)v) = γp(v) for any γ ∈ Γ and v ∈ V

Cπ rr′

id

id

r

I Extension of Laman’s theorem (Malestein-Theran2010,2012, Ross2011, T12, Jordan-Kaszanitzky-T12, Schulze-T13),

I combinatorial condition is described in terms of sparsity counts onthe underlying quotient Γ-labeled graphs graded by using subgroupsinduced by fundamental circuits.

I Much less is known about global/universal rigidity

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A Common Approach

I Compute the block-diagonalization of R(G , p)

T>R(G , p)S :=

R0(G , p) 0. . .

0 Rr (G , p)

I Characterizes the rank of each block in terms of combinatorial

conditions of the underlying quotient graph G/Γ.I In each block, a row and a column are associated with an edge orbit

and a vertex orbit, respectively.I The zero-nonzero pattern of each block is the same as I (G/Γ)⊗ Rd ,

where I (G/Γ) denotes the incidence matrix of G/Γ.

I (Schulze 10) The rank of R0(G , p) characterizes the existence ofsymmetry-forced motions of (G , p) if p is generic under thesymmetry

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1-dimensional case

I (G , p): Γ-symmetric framework with Γ = −,+I p : V → R1 with p((−)v) = −p(v) for each v ∈ V

I

T>R(G , p)S :=

(R0(G , p) 0

0 R1(G , p)

)I R1(G , p) = I (G/Γ)

I The row-independence of R0(G , p) cannot be described by G/Γ...

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Group-labled Graphs

I Quotient group labeled graph (G/Γ, ψ)I G/Γ: a directed quotient graphI ψ : E(G/Γ) → Γ encoding the covering map

1

C1

C21

C31

2

C2

C32

C22

3

C3C23

C33

G

1

2

3

id

C2

C3

id

C

(G/Γ, ψ)

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1-dimensional case

I (G , p): Z2-symmetric framework

I (G , p): Γ-symmetric frameworkI p : V → R1 with p((−)v) = −p(v) for each v ∈ V

I

T>R(G , p)S :=

(R0(G , p) 0

0 R1(G , p)

)I R1(G , p) = I (G/Γ)

I R0(G , p) = I (G/Γ, ψ)I where I (G/Γ, ψ) is row independent iff each connected component

contains no cycle or just one cycle, which is unbalanced if existsI ⇔ independent in the frame matroid of (G , ψ).I ⇔ ∀F ⊆ E/Γ,

|F | ≤ |V (F )| − 1 +

0 if F is balanced

1 if F is unbalanced

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2-dimensional reflection symmetry (Schulze-T 13)I (G , p): 2-dimensional framework with reflection symmetry

I p : V → R2 is ”generic” under the reflection symmetry

I

T>R(G , p)S :=

(R0(G , p) 0

0 R1(G , p)

)I Ri (G , p) is row-independent iff ∀F ⊆ E/Γ,

|F | ≤ 2|V (F )| − 3 +

0 if F is balanced

2− i if F is unbalanced

−

−

+ +

++

+

+

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Count Conditions on Group Labeled Graphs

I (G/Γ, ψ): Γ-labeled graph

I ψ(W ): the total gain though a closed walk W

I 〈F 〉v := 〈ψ(W ) : a closed walk W ⊆ F ⊆ E/Γ starting at v〉I F is balanced iff 〈F 〉v is trivial

I Examples of counts

|F | ≤ 2|V (F )|−3+

0 〈F 〉v is trivial

2 〈F 〉v is nontrivial and cyclic

3 otherwise

(F ⊆ E/Γ)

Given a representation ρ : Γ → GL(Rd),

|F | ≤ d |V (F )|−(d+1)+dim spanρ(γ) : γ ∈ 〈F 〉v (F ⊆ E/Γ)

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Count Conditions on Group Labeled Graphs

I µ : 2Γ → R is symmetric polymatroidal ifI µ(∅) = 0;I µ(X ) ≤ µ(Y ) for any X ⊆ Y ⊆ Γ;I µ(X ) + µ(Y ) ≥ µ(X ∪ Y ) + µ(X ∩ Y ) for any X ,Y ⊆ Γ;I µ(X ) = µ(〈X 〉) for any nonempty X ⊆ Γ;I µ(X ) = µ(γXγ−1) for any nonempty X ⊆ Γ and γ ∈ Γ.

I Theorem(T12) Let µ : 2Γ → 0, 1, . . . , k be a symmetricpolymatroidal function, and (G , ψ) a Γ-labaled graph. Let

Iµ := I ⊆ E : ∀connected F ⊆ I , |F | ≤ k|V (F )| − `+ µ(〈F 〉v ).

Then, (E , Iµ) is a matroid.

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Universal Rigidity

I (G , p) is universally rigid in Rd def⇔ (G , p) is globally rigid in Rd′

with d ′ ≥ d .

I ⇔ the uniqueness of the optimal solution of the following sdp:

max 0s.t. 〈Eij ,X 〉 = `G (p)(ij) (ij ∈ E )

X 0

where Eij = (ei − ej)(ei − ej)>.

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Characterizing Universal Rigidity

Theorem (Connelly 82, Gortler-Thurston 14)Suppose p is generic. Then (G , p) is universally rigid in Rd iff there is astress ω of (G , p) such that Ω is PSD with rank n − (d + 1).


X 0

min 〈`G (p), ω〉 = p>(Ω⊗ Id)ps.t. Ω =

∑ij∈E ω(ij)Eij 0

”⇐”

I ω is a minimizer of the dual

I q | (G , q) ∼ (G , p) = feasible XI By CS condition along with the rank condition,

feasible X ⊆ Face(`Kn(p)) in `Kn(x) | x ∈ (Rn)V I Face(`Kn(p)) = affine image of pI By genericity, affine image of p = q | q ≡ p

”⇒”

I Genericity implies the strong complementarity.

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Characterizing Universal Rigidity

Theorem (Connelly 82, Gortler-Thurston 14)Suppose p is generic. Then (G , p) is universally rigid in Rd iff there is astress ω of (G , p) such that Ω is PSD with rank n − (d + 1).


X 0

min 〈`G (p), ω〉 = p>(Ω⊗ Id)ps.t. Ω =

∑ij∈E ω(ij)Eij 0

”⇐”

I ω is a minimizer of the dual

I q | (G , q) ∼ (G , p) = feasible XI By CS condition along with the rank condition,

feasible X ⊆ Face(`Kn(p)) in `Kn(x) | x ∈ (Rn)V I Face(`Kn(p)) = affine image of pI By genericity, affine image of p = q | q ≡ p

”⇒”

I Genericity implies the strong complementarity.37 / 38

Lots of problems remain...

I Characterizing graphs whose generic realizations in Rd are alwaysuniversally rigid.

I Exact algorithm for checking universal rigidityI Facial reduction (Connelly-Gortler 14)I Polynomial-time solvability

I Exact algorithm for checking global rigidity

I Extension to symmetric frameworks

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Documents

Characterizations of the Rigidity of Graphsahigashi/tanigawa_rigidity.pdf · Characterizations of Generic Rigidity algebraic combinatorial rigidity Asimov-Roth 78 d 2 : Laman 70 d