Rigidity of graphs
Viet Hang NGUYEN, G-SCOP
February 25, 2011
1 / 18
Preliminaries
Known results
Open problems
Preliminaries
2 / 18
Bar-and-joint frameworks
Preliminaries
Structure S : rigid rods (bars) connected at their ends (joints).
Known results
Open problems
3 / 18
Bar-and-joint frameworks
Preliminaries
Structure S : rigid rods (bars) connected at their ends (joints).
Known results
Open problems
Question : Can S deform continuously to another shape?
3 / 18
Bar-and-joint frameworks
Preliminaries
Structure S : rigid rods (bars) connected at their ends (joints).
Known results
Open problems
Question : Can S deform continuously to another shape?
Non-rigid
Rigid
3 / 18
Bar-and-joint frameworks
Preliminaries
Structure S : rigid rods (bars) connected at their ends (joints).
Known results
Open problems
Question : Can S deform continuously to another shape?
Non-rigid
Rigid
• S ∼ (G, p).
⋄ V : set of joints of S; E(G) : set of bars of S.
⋄ p : V → Rd , embedding.
3 / 18
Bar-and-joint frameworks
Preliminaries
Structure S : rigid rods (bars) connected at their ends (joints).
Known results
Open problems
Question : Can S deform continuously to another shape?
Non-rigid
Rigid
• S ∼ (G, p).
⋄ V : set of joints of S; E(G) : set of bars of S.
⋄ p : V → Rd , embedding.
• (G, p) : a d-dim. bar-and-joint framework.
3 / 18
Preliminaries
Known results
Open problems
• (G, p) is congruent to (G, q)
⇔ ||p(u) − p(v)|| = ||q(u) − q(v)||,
∀u, v ∈ V.
4 / 18
Preliminaries
Known results
Open problems
• (G, p) is congruent to (G, q)
⇔ ||p(u) − p(v)|| = ||q(u) − q(v)||,
∀u, v ∈ V.
• (G, p) is rigid
⇔ every continuous motion of (G, p) preserving the
length of edges results in a framework congruent to
(G, p).
4 / 18
Preliminaries
Known results
Open problems
• An infinitesimal motion of (G, p) is a µ : V → Rd s.t.
(p(u) − p(v))(µ(u) − µ(v)) = 0,
uv ∈ E(G).
5 / 18
Preliminaries
Known results
Open problems
• An infinitesimal motion of (G, p) is a µ : V → Rd s.t.
(p(u) − p(v))(µ(u) − µ(v)) = 0,
uv ∈ E(G).
The instantaneous velocity of (G, p) is an infinitesimal motion.
5 / 18
Rigidity matrix
Preliminaries
Known results
• Rigidity matrix R(G, p) : |E| × d|V | matrix.
Open problems
.. . .
.
.
uv 
0 · · ·
.. . .
.
.

u
..
.
···
p(u) − p(v) · · ·
..
.
···
v
..
..
.
.
p(v) − p(u) · · ·
..
..
.
.
..
.
0
.
..
.

6 / 18
Rigidity matrix
Preliminaries
Known results
• Rigidity matrix R(G, p) : |E| × d|V | matrix.
Open problems
.. . .
.
.
uv 
0 · · ·
.. . .
.
.

u
..
.
···
p(u) − p(v) · · ·
..
.
···
v
..
..
.
.
p(v) − p(u) · · ·
..
..
.
.
..
.
0
.
..
.

• µ is an infinitesimal motion of (G, p) if and only if
R(G, p)µ = 0 (i.e µ ∈ ker R(G, p)).
• The space of infinitesimal motions induced by
translations and rotations is of dimension d(d + 1)/2.
⇒ rank R(G, p) ≤ d|V | − d(d + 1)/2.
6 / 18
Determining rigidity
Preliminaries
Known results
Determine rigidity by calculating rank R(G, p)?
Open problems
7 / 18
Determining rigidity
Preliminaries
Known results
Determine rigidity by calculating rank R(G, p)?
Open problems
Degenerate
Generic
a
a
d
b
e
d
c
b
c
e
Rigid but
rankR(G, p) < d|V | − d(d + 1)/2
OK!
(d=2).
7 / 18
Determining rigidity
Preliminaries
Known results
Determine rigidity by calculating rank R(G, p)?
Open problems
Degenerate
Generic
a
a
d
b
e
d
c
b
c
e
Rigid but
rankR(G, p) < d|V | − d(d + 1)/2
OK!
(d=2).
Theorem (Asimow, Roth 1979). For generic p,
(G, p) is rigid ⇔ rankR(G, p) = d|V | − d(d + 1)/2.
7 / 18
Determining rigidity
Preliminaries
Known results
Determine rigidity by calculating rank R(G, p)?
Open problems
Degenerate
Generic
a
a
d
b
e
d
c
b
c
e
Rigid but
rankR(G, p) < d|V | − d(d + 1)/2
OK!
(d=2).
Theorem (Asimow, Roth 1979). For generic p,
(G, p) is rigid ⇔ rankR(G, p) = d|V | − d(d + 1)/2.
Almost all embeddings are generic and for all generic p,
rank R(G, p) depends uniquely on G.
7 / 18
Determining the rigidity of generic
frameworks
Preliminaries
Known results
Calculating rank R(G, p) for some generic p?
Open problems
8 / 18
Determining the rigidity of generic
frameworks
Preliminaries
Known results
Calculating rank R(G, p) for some generic p?
Open problems
Not efficient!
8 / 18
Determining the rigidity of generic
frameworks
Preliminaries
Known results
Calculating rank R(G, p) for some generic p?
Open problems
Not efficient!
• How to generate generic embedding?
8 / 18
Determining the rigidity of generic
frameworks
Preliminaries
Known results
Calculating rank R(G, p) for some generic p?
Open problems
Not efficient!
• How to generate generic embedding?
Solution:
Using combinatorial structure of the rigidity matrix
⇔ a matroid on E(G) (linear matroid defined on the rows of
R(G, p)).
8 / 18
Preliminaries
Known results
Open problems
• G is rigid (in dimension d) if (G, p) is rigid for some
generic p (and hence for all generic p).
• G (or E) is independent (in dimension d) if the rows of
R(G, p) is independent for some generic p (and hence
for all generic p).
• Minimally rigid graphs = Maximally independent graphs.
9 / 18
Preliminaries
Known results
Open problems
Known results
10 / 18
Characterizations in dimension 2
Preliminaries
Known results
Open problems
Theorem (Laman 1970). Graph (V, E) is rigid in dimension 2
if and only if there exists E ′ ⊆ E s.t.
• |E ′ | = 2|V | − 3,
• |F | ≤ 2|V (F )| − 3,
∅=
6 F ⊆ E′.
11 / 18
Characterizations in dimension 2
Preliminaries
Known results
Open problems
Theorem (Laman 1970). Graph (V, E) is rigid in dimension 2
if and only if there exists E ′ ⊆ E s.t.
• |E ′ | = 2|V | − 3,
• |F | ≤ 2|V (F )| − 3,
∅=
6 F ⊆ E′.
F={
}
V(F) ={ }
11 / 18
Characterizations in dimension 2
Preliminaries
Known results
Open problems
Theorem (Laman 1970). Graph (V, E) is rigid in dimension 2
if and only if there exists E ′ ⊆ E s.t.
• |E ′ | = 2|V | − 3,
• |F | ≤ 2|V (F )| − 3,
∅=
6 F ⊆ E′.
11 / 18
Characterizations in dimension 2
Preliminaries
Known results
Open problems
Theorem (Laman 1970). Graph (V, E) is rigid in dimension 2
if and only if there exists E ′ ⊆ E s.t.
• |E ′ | = 2|V | − 3,
• |F | ≤ 2|V (F )| − 3,
∅=
6 F ⊆ E′.
Equivalently,
A graph G is independent in dim. 2 if and only if it satisfies
|F | ≤ 2|V (F )| − 3 for all ∅ =
6 F ⊆E.
11 / 18
Characterizations in dimension 2
Preliminaries
Known results
Open problems
Theorem (Lovász&Yemini 1982). A graph G is independent
in dim. 2 if and only if G + e is the union of two forests for
every e ∈ E.
12 / 18
Characterizations in dimension 2
Preliminaries
Known results
Open problems
Theorem (Lovász&Yemini 1982). A graph G is independent
in dim. 2 if and only if G + e is the union of two forests for
every e ∈ E.
Theorem (Lovász&Yemini 1982). G is rigid in dim. 2 if and
only if
t
X
(2|V (Gi )| − 3) ≥ 2|V | − 3,
i=1
for every G1 , . . . , Gt
(E(Gi ) 6= ∅) s.t. G1 ∪ G2 ∪ · · · ∪ Gt = G.
12 / 18
Connectivity and rigidity
Preliminaries
Known results
Open problems
Theorem (Lovász&Yemini 1982). Every 6-connected graph is
rigid in dim. 2.
13 / 18
Connectivity and rigidity
Preliminaries
Known results
Open problems
Theorem (Lovász&Yemini 1982). Every 6-connected graph is
rigid in dim. 2.
6 is the best possible!
13 / 18
Connectivity and rigidity
Preliminaries
Known results
Open problems
Theorem (Lovász&Yemini 1982). Every 6-connected graph is
rigid in dim. 2.
6 is the best possible!
X
5n
19n
(2|V (Gi )| − 3) = 7n +
=
< 10n − 3 = 2|V | − 3.
2
2
13 / 18
Rigidity in dimension 3
Preliminaries
No simple counting condition!
Known results
Open problems
14 / 18
Rigidity in dimension 3
Preliminaries
No simple counting condition!
Known results
Open problems
The “double banana” satisfies
|F | ≤ 3|V (F )|−6 for all
F ⊆ E, |F | ≥ 2,
and
E = 3|V | − 6
but is not rigid.
14 / 18
Rigidity in dimension 3
Preliminaries
Known results
Decidable for special classes?
Open problems
Yes, for square graphs!
15 / 18
Rigidity in dimension 3
Preliminaries
Known results
Decidable for special classes?
Open problems
Yes, for square graphs!
15 / 18
Rigidity in dimension 3
Preliminaries
Known results
Decidable for special classes?
Open problems
Yes, for square graphs!
15 / 18
Rigidity in dimension 3
Preliminaries
Known results
Decidable for special classes?
Open problems
Yes, for square graphs!
Theorem (Kato&Tanigawa 2009). Let G be a graph of
minimum degree at least 2. Then the graph G2 is rigid in dim.
3 if and only if 5G contains 6 disjoint spanning trees.
15 / 18
Preliminaries
Known results
Open problems
Open problems
16 / 18
Characterization
Preliminaries
Known results
Open problems
Open problem
Characterize rigid/independent graphs in dimension d ≥ 3.
17 / 18
Connectivity and rigidity
Preliminaries
Are highly connected graphs rigid?
Known results
Open problems
Conjecture (Lovász &Yemini 1982) :
There is a constant kd such that every kd -connected graph is
rigid in dimension d. (kd = d(d + 1)?)
18 / 18
Connectivity and rigidity
Preliminaries
Are highly connected graphs rigid?
Known results
Open problems
Conjecture (Lovász &Yemini 1982) :
There is a constant kd such that every kd -connected graph is
rigid in dimension d. (kd = d(d + 1)?)
Theorem (Jordán 2010). Every 7-connected square graph is
rigid in dim. 3.
18 / 18