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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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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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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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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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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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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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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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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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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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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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
)6 / 38
Rigidity of Graphs
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 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
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 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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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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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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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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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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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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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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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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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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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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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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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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A Key Tool: Ear-decompostion (Jackson-Jordan05)
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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A Key Tool: Ear-decompostion (Jackson-Jordan05)
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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A Key Tool: Ear-decompostion (Jackson-Jordan05)
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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A Key Tool: Ear-decompostion (Jackson-Jordan05)
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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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).
max 0s.t. 〈Eij ,X 〉 = `G (p)(ij) (ij ∈ E )
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).
max 0s.t. 〈Eij ,X 〉 = `G (p)(ij) (ij ∈ E )
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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