#### Finite and Infinitesimal Rigidity with Polyhedral Norms

Finite and Infinitesimal Rigidity with Polyhedral Norms
Derek Kitson 0 1
0 Department of Mathematics and Statistics, Lancaster University , Lancaster LA1 4YF , UK
1 Derek Kitson
We characterise finite and infinitesimal rigidity for bar-joint frameworks in Rd with respect to polyhedral norms (i.e. norms with closed unit ball P , a convex d-dimensional polytope). Infinitesimal and continuous rigidity are shown to be equivalent for finite frameworks in Rd which are well-positioned with respect to P . An edge-labelling determined by the facets of the unit ball and placement of the framework is used to characterise infinitesimal rigidity in Rd in terms of monochrome spanning trees. An analogue of Laman's theorem is obtained for all polyhedral norms on R2. Editor in Charge: Gnter M. Ziegler
Bar-joint framework; Infinitesimally rigid; Laman's theorem; Polyhedral norm
1 Introduction
A bar-joint framework in Rd is a pair (G, p) consisting of a simple undirected graph
G = (V (G), E (G)) (i.e. no loops or multiple edges) and a placement p : V (G) Rd
of the vertices such that pv and pw are distinct whenever vw is an edge of G. The graph
Supported by the Engineering and Physical Sciences Research Council [grant number EP/J008648/1].
G may be either finite or infinite. Given a norm on Rd we are interested in determining
when a given framework can be continuously and non-trivially deformed without
altering the lengths of the bars. A well-developed rigidity theory exists in the Euclidean
setting for finite bar-joint frameworks (and their variants), which stems from classical
results of Cauchy [6], Maxwell [17], Alexandrov [1] and Laman [14]. Of particular
relevance is Lamans landmark characterisation for generic minimally infinitesimally
rigid finite bar-joint frameworks in the Euclidean plane. Asimow and Roth proved
the equivalence of finite and infinitesimal rigidity for regular bar-joint frameworks
in two key papers [2,3]. A modern treatment can be found in works of Graver et
al. [9] and Whiteley [24,26]. More recently, significant progress has been made in
topics such as global rigidity [7,8,11] and the rigidity of periodic frameworks [5,16,
20,21] in addition to newly emerging themes such as symmetric frameworks [22] and
frameworks supported on surfaces [19]. In this article, we consider rigidity properties
of both finite and infinite bar-joint frameworks (G, p) in Rd with respect to polyhedral
norms. A norm on Rd is polyhedral (or a block norm) if the closed unit ball {x Rd :
x 1} is the convex hull of a finite set of points. Such norms form an important
class as they are computationally easy to use and are dense in the set of all norms on
Rd . While classical rigidity theory is strongly linked to statics, it has also provided
valuable new connections between different areas of pure mathematics and this latter
property is one of the emerging features of non-Euclidean rigidity theory. In particular,
the rigidity theory obtained with polyhedral norms is distinctly different from the
Euclidean setting in admitting new edge-labelling and spanning tree methods. There
are potential applications of this theory to physical networks with inherent directional
constraints, or to abstract networks with a suitable notion of distance imposed.
NonEuclidean norms, and in particular polyhedral norms, have been applied in this way to
optimisation problems in location modelling (see the industry which has resulted from
[23]) and, more recently, machine learning with submodular functions [4]. A study
of rigidity with respect to the classical non-Euclidean p norms was initiated in [12]
for finite bar-joint frameworks and further developed for infinite bar-joint frameworks
in [13]. Among these norms the 1 and norms are simple examples of polyhedral
norms and so the results obtained here extend some of the results of [12].
In Sect. 2, we provide the relevant background material on polyhedral norms and
finite and infinitesimal rigidity. In Sect. 3, we establish the role of support functionals
in determining the space of infinitesimal flexes of a bar-joint framework (Theorem 5).
We then distinguish between general bar-joint frameworks and those which are
wellpositioned with respect to the unit ball. The well-positioned placements of a finite
graph are open and dense in the set of all placements, and we show that finite and
infinitesimal rigidity are equivalent for these bar-joint frameworks (Theorem 7). We
then introduce the rigidity matrix for a general finite bar-joint framework, the
nonzero entries of which are derived from extreme points of the polar set of the unit
ball. In Sect. 4, we apply an edge-labelling to G which is induced by the placement
of each bar in Rd relative to the facets of the unit ball. With this edge-labelling we
identify necessary conditions for infinitesimal rigidity and obtain a sufficient condition
for a subframework to be relatively infinitesimally rigid (Proposition 12). We then
characterise the infinitesimally rigid bar-joint frameworks with d induced framework
colours as those which contain monochrome spanning trees of each framework colour
(Theorem 13). This result holds for both finite and infinite bar-joint frameworks and
does not require the framework to be well-positioned. In Sect. 5, we apply the spanning
tree characterisation to show that certain graph moves preserve minimal infinitesimal
rigidity for any polyhedral norm on R2. We then show that in two dimensions a
finite graph has a well-positioned minimally infinitesimally rigid placement if and
only if it satisfies the counting conditions |E (G)| = 2|V (G)| 2 and |E (H )|
2|V (H )| 2 for all subgraphs H (Theorem 23). This is an analogue of Lamans
theorem [14] which characterises the finite graphs with minimally infinitesimally
rigid generic placements in the Euclidean plane as those which satisfy the counting
conditions |E (G)| = 2|V (G)| 3 and |E (H )| 2|V (H )| 3 for subgraphs H with
at least two vertices. Many of the results obtained hold equally well for both finite and
infinite bar-joint frameworks.
2 Preliminaries
Let P be a convex symmetric d-dimensional polytope in Rd where d 2. Following
[10] we say that a proper face of P is a subset of the form P H , where H is a
supporting hyperplane for P. A facet of P is a proper face which is maximal with
respect to inclusion. The set of extreme points (vertices) of P is denoted by ext(P).
The polar set of P, denoted by P , is also a convex symmetric d-dimensional polytope
in Rd :
= {y Rd : x y 1 for all x P}.
Moreover, there exists a bijective map which assigns to each facet F of P a unique
extreme point F of P such that
The polar set of P is P.
The Minkowski functional (or gauge) for P defines a norm on Rd ,
F = {x P : x F = 1}.
This is what is known as a polyhedral norm or a block norm. The dual norm of P
is also a polyhedral norm and is determined by the polar set P ,
y P = mxaPx x y = inf{ 0 : y P } =
In general, a linear functional on a convex polytope will achieve its maximum value at
some extreme point of the polytope and so the polyhedral norm P is characterised
by
x P =
x P =
A point x Rd belongs to the conical hull cone(F ) of a facet F if x = nj=1 j x j
for some non-negative scalars j and some finite collection x1, x2 . . . , xn F . By
formulas (1), (2) and (3) the following equivalence holds:
x cone(F )
x P = x F .
with the property that x (0) = x and for every pair x , y Rd the distance x (t )
y (t ) remains constant for all values of t . In the case of a polyhedral norm P , if
is sufficiently small, then the isometries t : x x (t ) are necessarily translational
since by continuity the linear part must equal the identity transformation. Thus we
may assume that a continuous rigid motion of (Rd , P ) is a family of continuous
paths of the form
for some continuous function c : (, ) Rd (cf. [13, Lemma 6.2]).
An infinitesimal rigid motion of a normed space (Rd , ) is a vector field on
Rd which arises from the velocity vectors of a continuous rigid motion. For a
polyhedral norm P , since the continuous rigid motions are of translational type, the
infinitesimal rigid motions of (Rd , P ) are precisely the constant maps
for some a Rd (cf. [12, Lemma 2.3]).
Let (G, p) be a (finite or infinite) bar-joint framework in a normed vector space
(Rd , ). A continuous (or finite) flex of (G, p) is a family of continuous paths
such that v(0) = pv for each vertex v V (G) and v(t ) w(t ) = pv pw for
all |t | < and each edge vw E (G). A continuous flex of (G, p) is regarded as trivial
if it arises as the restriction of a continuous rigid motion of (Rd , ) to p(V (G)). If
every continuous flex of (G, p) is trivial then we say that (G, p) is continuously rigid.
An infinitesimal flex of a (finite or infinite) bar-joint framework (G, p) in a normed
space (Rd , ) is a map u : V (G) Rd , v uv which satisfies
( pv + t uv) ( pw + t uw)
pv pw
= o(t ) as t 0
for each edge vw E (G). We will denote the collection of infinitesimal flexes of
(G, p) by F (G, p). An infinitesimal flex of (G, p) is regarded as trivial if it arises as
the restriction of an infinitesimal rigid motion of (Rd , ) to p(V (G)). In other words,
in the case of a polyhedral norm, an infinitesimal flex of (G, p) is trivial if and only
if it is constant. A bar-joint framework is infinitesimally rigid if every infinitesimal
flex of (G, p) is trivial. Regarding F (G, p) as a real vector space with
componentwise addition and scalar multiplication, the trivial infinitesimal flexes of (G, p) form
a d-dimensional subspace T (G, p) of F (G, p).
The interior of a subset A Rd will be denoted by A.
3 Support Functionals and Rigidity
In this section, we begin by highlighting the connection between the infinitesimal flex
condition (5) for a general norm on Rd and support functionals on the normed space
(Rd , ). We then characterise the space of infinitesimal flexes for a general (finite or
infinite) bar-joint framework in (Rd , P ) in terms of support functionals and prove
the equivalence of finite and infinitesimal rigidity for finite bar-joint frameworks which
are well-positioned in (Rd , P ). Following this, we describe the rigidity matrix for
general finite bar-joint frameworks in (Rd , P ) and compute an example.
3.1 Support Functionals
Let be an arbitrary norm on Rd , and denote by B the closed unit ball in (Rd , ).
A linear functional f : Rd R is a support functional for a point x0 Rd if
f (x0) = x0 2 and f = x0 . Equivalently, f is a support functional for x0 if the
hyperplane
H = {x Rd : f (x ) =
is a supporting hyperplane for B which contains xx00 .
for all y Rd .
f (y)
f (y)
x0 + t y
t
x0 + t y
t
x0
x0
Proof Since f is linear and f (x0) = x0 2, we have for all y Rd ,
If t > 0, then since f (x) x0 x for all x Rd we have
f (y) x0
If t < 0, then applying the above inequality
f (y) = f (y) x0
= x0
Let (G, p) be a (finite or infinite) bar-joint framework in (Rd , ), and fix an
orientation for each edge vw E(G). We denote by supp(vw) the set of all support
functionals for pv pw. (The choice of orientation on the edges of G is for convenience
only and has no bearing on the results that follow. Alternatively, we could avoid
choosing an orientation by defining supp(vw) to be the set of all linear functionals
which are support functionals for either pv pw or pw pv.)
Proposition 2 If (G, p) is a (finite or infinite) bar-joint framework in (Rd , ) and
u : V (G) Rd is an infinitesimal flex of (G, p), then
uv uw
f supp(vw)
for each edge vw E(G).
Proof Let vw E(G) and suppose f is a support functional for pv pw. Applying
Lemma 1 with x0 = pv pw and y = uv uw, we have
x0 + t y x0
t
Since u is an infinitesimal flex of (G, p), limt0 1t ( x0 + t y x0 ) = 0 and so
f (y) = 0.
Let P be a polyhedral norm on Rd . For each facet F of P, denote by F the
linear functional
Lemma 3 Let P be a polyhedral norm on Rd , let F be a facet of P and let
x0 Rd . Then x0 cone(F ) if and only if the linear functional
is a support functional for x0.
F,x0 (x ) x0 P for each x P, and it follows that F,x0 xi0s a2Ps.uBppyo(r1t )f,uwncetihoanvael
Proof If x0 cone(F ), then by formula (4) F,x0 (x0) =
for x0. Conversely, if x0 / cone(F ), then by (4) F,x0 (x0) < x0 2P and so F,x0 is
not a support functional for x0.
For each oriented edge vw E (G), we denote by supp (vw) the set of all linear
pv pw .
functionals F which are support functionals for pv pw P
Proposition 4 Let (G, p) be a finite bar-joint framework in (Rd , P ). If a mapping
u : V (G) Rd satisfies
uv uw
v : (, ) Rd , v(t ) = pv + t uv,
is a finite flex of (G, p).
Proof Let vw E (G) and write x0 = pv pw and u0 = uv uw. If F is a support
x0 , then by the hypothesis F (u0) = 0. By Lemma 3, x0 is contained
functional for x0 P
in the conical hull of the facet F . Applying formulas (3) and (4),
x0 P =
x0 y = x0 F .
x0 + t u0 P =
= (x0 + t u0) F
Since G is a finite graph, the result holds with = minvwE(G) vw > 0.
The following is a characterisation of the space of infinitesimal flexes of a general
bar-joint framework in (Rd , P ).
Theorem 5 Let (G, p) be a (finite or infinite) bar-joint framework in (Rd , P ).
Then a mapping u : V (G) Rd is an infinitesimal flex of (G, p) if and only if
uv uw
for each edge vw E (G).
Proof If u is an infinitesimal flex of (G, p), then the result follows from Proposition 2.
For the converse, let vw E (G) and write x0 = pv pw and u0 = uv uw.
Applying the argument in the proof of Proposition 4, there exists vw > 0 with
x0 + t u0 P = x0 P for all |t | < vw. Hence u is an infinitesimal flex of (G, p).
3.2 Equivalence of Finite and Infinitesimal Rigidity
A placement of a simple graph G in Rd is a map p : V (G) Rd for which pv = pw
whenever vw E (G). A placement p : V (G) Rd is well-positioned with respect
to a polyhedral norm on Rd if pv pw is contained in the conical hull of exactly one
facet of the unit ball P for each edge vw E (G). We denote this unique facet by Fvw.
In the following discussion, G is a finite graph and each placement is identified with
a point p = ( pv)vV (G) in the product space vV (G) Rd which we regard as having
the usual topology. The set of all well-positioned placements of G in (Rd , P ) is an
open and dense subset of this product space. The configuration space for a bar-joint
framework (G, p) is defined as
V (G, p) =
x
Rd : xv xw P =
pv pw P
for all vw E (G) .
vV (G)
Proposition 6 Let (G, p) be a finite and well-positioned bar-joint framework in
(Rd , P ) with pv pw cone(Fvw) for each vw E (G). Then there exists
a neighbourhood U of p in vV (G) Rd such that
(i) if x U , then xv xw cone(Fvw) for each edge vw E (G),
(ii) (G, x ) is a well-positioned bar-joint framework for each x U and
(iii) V (G, p) U = {x U : Fvw (xv xw) = Fvw ( pv pw) for all vw E (G)}.
In particular, V (G, p) U = ( p + F (G, p)) U .
Proof Let vw E (G) be an oriented edge and consider the continuous map
Rd , (xv )v V (G) xv xw.
v V (G)
Since (G, p) is well-positioned, pv pw is an interior point of the conical hull of a
unique facet Fvw of P. The preimage Tvw1(cone(Fvw)) is an open neighbourhood of
p. Since G is a finite graph, the intersection
U =
Tvw1(cone(Fvw))
vwE(G)
is an open neighbourhood of p which satisfies (i), (ii) and (iii).
Since (G, p) is well-positioned, by Lemma 3, there is exactly one support functional
in supp (vw) for each edge vw and this functional is given by Fvw . If x U ,
then define u = (uv)vV (G) by setting uv = xv pv for each v V (G). By (iii),
x V (G, p) U if and only if x U and
Fvw (uv uw) = Fvw (xv xw) Fvw ( pv pw) = 0
for each edge vw E (G). By Theorem 5, the latter identity is equivalent to the
condition that u is an infinitesimal flex of (G, p). Thus x V (G, p) U if and only
if x U and x p F (G, p).
We now prove the equivalence of continuous rigidity and infinitesimal rigidity for
finite well-positioned bar-joint frameworks.
Theorem 7 Let (G, p) be a finite well-positioned bar-joint framework in a normed
space (Rd , P ), where P is a polyhedral norm. Then the following statements
are equivalent:
(i) (G, p) is continuously rigid.
(ii) (G, p) is infinitesimally rigid.
Proof (i) (ii). If u = (uv)vV (G) F (G, p) is an infinitesimal flex of (G, p),
then by Theorem 5 and Proposition 4, the family
v : (, ) Rd , v(t ) = pv + t uv, v V (G),
is a finite flex of (G, p) for some > 0. Since (G, p) is continuously rigid, this finite
flex must be trivial. Thus there exist > 0 and a continuous path c : (, ) Rd
such that v(t ) = pv + c(t ) for all |t | < and all v V (G). Now uv = v(0) = c (0)
for all v V (G) and so u is a constant, and hence trivial, infinitesimal flex of (G, p).
We conclude that (G, p) is infinitesimally rigid.
(ii) (i). If (G, p) has a finite flex given by the family
then consider the continuous path
: (, ) V (G, p), t (v(t ))vV (G).
By Proposition 6, V (G, p) U = ( p + F (G, p)) U for some neighbourhood U of
p. Since (0) = p, there exists > 0 such that (t ) V (G, p) U for all |t | < .
Choose t0 (, ) and define
u : V (G) Rd , uv = v(t0) pv.
Then u = (t0) p F (G, p) is an infinitesimal flex of (G, p). Since (G, p) is
infinitesimally rigid, u must be a trivial infinitesimal flex. Hence uv = c(t0) for all
v V (G) and some c(t0) Rd . Apply the same argument to show that for each
|t | < there exists c(t ) such that v(t ) = pv + c(t ) for all v V (G). Note that
c : (, ) Rd is continuous and so {v : v V (G)} is a trivial finite flex of
(G, p). We conclude that (G, p) is continuously rigid.
The non-equivalence of finite and infinitesimal rigidity for general finite bar-joint
frameworks in (Rd , P ) is demonstrated in Example 9.
3.3 The Rigidity Matrix
W(Red ,definePt)heasrfigoildloitwysm:FaitxrixanRoPr(dGer,inpg) ofofrthae fivneirtteicbesarV-jo(iGnt) farnadmeedwgoersk E( G(G, )p)anidn
choose an orientation on the edges of G. For each vertex v, assign d columns in
the rigidity matrix and label these columns pv,1, . . . , pv,d . For each directed edge
vw E (G) and each facet F with pv pw cone(F ), assign a row in the rigidity
matrix and label this row by (vw, F ). The entries for the row (vw, F ) are given by
0 0 F1 Fd 0 0 F1 Fd 0 0 ,
where pv pw cone(F ) and F = (F1, . . . , Fd ) Rd . If (G, p) is well-positioned,
then the rigidity matrix has size |E (G)| d|V (G)|.
Proposition 8 Let (G, p) be a finite bar-joint framework in (Rd , P ). Then
(i) F (G, p) = ker RP (G, p).
(ii) (G, p) is infinitesimally rigid if and only if rank RP (G, p) = d|V (G)| d.
Proof The system of equations in Theorem 5 is expressed by the matrix
equation RP (G, p)uT = 0 where we identify u : V (G) Rd with a row vector
(uv1 , . . . , uvn ) Rd|V (G)|. Thus F (G, p) = ker RP (G, p). The space of trivial
infinitesimal flexes of (G, p) has dimension d and so in general we have
rank RP (G, p) d|V (G)| d
with equality if and only if (G, p) is infinitesimally rigid.
If F is a facet of P and y1, y2, . . . , yd ext(P) are extreme points of P which
are contained in F , then for each column vector yk we compute [1 1] A1 yk = 1,
where A = [y1 yd ] M dd (R). Hence,
F = [1 1] A1.
Moreover, if y1, y2, . . . , yd are pairwise orthogonal, then
A1 =
F =
j=1
Example 9 Let P be a crosspolytope in Rd with 2d many extreme points ext(P) =
{ek : k = 1, . . . , d}, where e1, e2, . . . , ed is the usual basis in Rd . Then each facet F
contains d pairwise orthogonal extreme points y1, y2, . . . , yd each of Euclidean norm
1. By (8), F = dj=1 y j and the resulting polyhedral norm is the 1-norm
x P =
x y =
|xi | =
i=1
Consider for example the placements of the complete graph K2 in (R2, 1) illustrated
in Fig. 1. The polytope P is indicated on the left with facets labelled F1 and F2. The
extreme points of the polar set P which correspond to these facets are F1 = e1 +e2 =
(1, 1) and F2 = e1 e2 = (1, 1). The first placement is well-positioned with respect
to P and the rigidity matrix is
Evidently, this bar-joint framework has a non-trivial infinitesimal flex. The second
placement is not well-positioned and the rigidity matrix is
As the rigidity matrix has rank 2, this bar-joint framework is infinitesimally rigid in
(R2, 1), but continuously flexible.
1
1
Fig. 1 An infinitesimally flexible and an infinitesimally rigid placement of K2 in (R2, 1)
4 Edge-Labellings and Monochrome Subgraphs
In this section, we describe an edge-labelling on G which depends on the placement of
the bar-joint framework (G, p) in (Rd , P ) relative to the facets of P. We provide
methods for identifying infinitesimally flexible frameworks and subframeworks which
are relatively infinitesimally rigid. We then characterise infinitesimal rigidity for
barjoint frameworks with d framework colours in terms of the monochrome subgraphs
induced by this edge-labelling.
4.1 Edge-Labellings
Let (G, p) be a general bar-joint framework in (Rd , P ) (i.e. it is not assumed
here that (G, p) is finite or well-positioned). Since P is symmetric in Rd , if F is a
facet of P then F is also a facet of P. Denote by (P) the collection of all pairs
[F ] = {F, F }. For each edge vw E (G), define
(vw) = {[F ] (P) : pv pw cone(F ) cone(F )} .
We refer to the elements of (vw) as the framework colours of the edge vw. For
example, if pv pw lies in the conical hull of exactly one facet of P, then the edge
vw has just one framework colour. If pv pw lies along a ray through an extreme
point of P, then vw has at least d distinct framework colours. By Lemma 3, [F ]
is a framework colour for an edge vw if and only if either F or F is a support
pv pw .
functional for pv pw P
For each vertex v0 V (G), denote by (v0) the collection of framework colours
of all edges which are incident with v0:
v0wE(G)
Proposition 10 If a (finite or infinite) bar-joint framework (G, p) is infinitesimally
rigid in (Rd , P ), then |(v)| d for each vertex v V (G).
By Theorem 5, if u : V (G) Rd is defined by
then u is a non-trivial infinitesimal flex of (G, p).
uv =
We now consider the subgraphs of G which are spanned by edges possessing a
particular framework colour. For each facet F of P, define
E F (G, p) = {vw E (G) : [F ] (vw)}
and let G F be the subgraph of G spanned by E F (G, p). We refer to G F as a
monochrome subgraph of G.
Denote by (G, p) the collection of all framework colours of edges of G:
vwE(G)
We refer to the elements of (G, p) as the framework colours of the bar-joint
framework (G, p).
Proposition 11 Let (G, p) be a (finite or infinite) bar-joint framework which is
infinitesimally rigid in (Rd , P ). If C is a collection of framework colours of (G, p)
with |(G, p)\C | < d, then
[F]C
contains a spanning tree of G.
Proof Suppose that [F]C G F does not contain a spanning tree of G. Then there
exists a partition V (G) = V1 V2 for which there is no edge v1v2 E (G) with
framework colour contained in C satisfying v1 V1 and v2 V2. Since |(G, p)\C | < d,
there exists non-zero
By Theorem 5, if u : V (G) Rd is defined by
x
uv =
then u is a non-trivial infinitesimal flex of (G, p). We conclude that
contains a spanning tree of G.
[F]C G F
It is possible to construct examples which show that the converse to Proposition 11
does not hold in general. In Theorem 13, we show that a converse statement does hold
under the additional assumption that |(G, p)| = d.
4.2 Edge-Labelled Paths and Relative Infinitesimal Rigidity
Let (G, p) be a finite bar-joint framework in (Rd , P ) and, for each edge vw
E (G), let Xvw be the vector subspace of Rd :
Xvw =
ker F =
For each pair of vertices v, w V (G), denote by G (v, w) the set of all paths in G
from v to w.
A subframework of (G, p) is a bar-joint framework (H, p) obtained by restricting
p to the vertex set of a subgraph H . We say that (H, p) is relatively infinitesimally
rigid in (G, p) if the restriction of every infinitesimal flex of (G, p) to (H, p) is trivial.
Proposition 12 Let (G, p) be a finite bar-joint framework in (Rd , P ) and let
(H, p) be a subframework of (G, p). If for each pair of vertices v, w V (H )
then (H, p) is relatively infinitesimally rigid in (G, p).
Proof Let u F (G, p) be an infinitesimal flex of (G, p) and let v, w V (H ).
Suppose G (v, w), where = {v1v2, . . . , vn1vn} is a path in G with v = v1
and w = vn. Then by Theorem 5,
uv uw = (uv1 uv2 ) + (uv2 uv3 ) + + (uvn1 uvn ) X .
Since this holds for all paths in G (v, w), the hypothesis implies that uv = uw.
Applying this argument to every pair of vertices in H , we see that the restriction of
u to V (H ) is constant and hence a trivial infinitesimal flex of (H, p). Thus (H, p) is
relatively infinitesimally rigid in (G, p).
4.3 Monochrome Spanning Subgraphs
Applying the results of the previous sections, we can now characterise the
infinitesimally rigid bar-joint frameworks in (Rd , P ) which use exactly d framework
colours.
Theorem 13 Let (G, p) be a (finite or infinite) bar-joint framework in (Rd , P )
and suppose that |(G, p)| = d. Then the following statements are equivalent:
(i) (G, p) is infinitesimally rigid.
(ii) G F contains a spanning tree of G for each [F ] (G, p).
Proof The implication (i) (ii) follows from Proposition 11. To prove (ii) (i),
let u F (G, p). If v, w V (G), then for each framework colour [F ] (G, p)
there exists a path in G F from v to w. Hence
ker F = {0}
and, by Proposition 12, uv = uw. Applying this argument to all pairs v, w V (G),
we see that u is a trivial infinitesimal flex and so (G, p) is infinitesimally rigid.
A bar-joint framework (G, p) is minimally infinitesimally rigid in (Rd , P ) if it
is infinitesimally rigid and every subframework obtained by removing a single edge
from G is infinitesimally flexible.
Corollary 14 Let (G, p) be a (finite or infinite) bar-joint framework in (Rd , P ) and
suppose that |(G, p)| = d. If G F is a spanning tree in G for each [F ] (G, p),
then (G, p) is minimally infinitesimally rigid.
Proof By Theorem 13, (G, p) is infinitesimally rigid. If any edge vw is removed from
G, then G F is no longer a spanning tree for some [F ] (G, p). By Theorem 13,
the subframework (G\{vw}, p) is not infinitesimally rigid and so we conclude that
(G, p) is minimally infinitesimally rigid.
There exist bar-joint frameworks which show that the converse statement to
Corollary 14 does not hold in full generality. In the following corollary, the converse is
established for bar-joint frameworks that are well-positioned.
Corollary 15 Let (G, p) be a (finite or infinite) well-positioned bar-joint framework
in (Rd , P ) and suppose that |(G, p)| = d. Then the following statements are
equivalent:
(i) (G, p) is minimally infinitesimally rigid.
(ii) G F is a spanning tree in G for each [F ] (G, p).
Proof (i) (ii). Let [F ] (G, p). If (G, p) is minimally infinitesimally rigid,
then by Theorem 13 the monochrome subgraph G F contains a spanning tree of G.
Suppose vw is an edge of G which is contained in G F . Since (G, p) is minimally
infinitesimally rigid, (G\{vw}, p) is infinitesimally flexible. Since (G, p) is
wellpositioned, vw is contained in exactly one monochrome subgraph of G and so G F is
the only monochrome subgraph which is altered by removing the edge vw from G.
By Theorem 13, G F \{vw} does not contain a spanning tree of G. We conclude that
G F is a spanning tree of G. The implication (ii) (i) is proved in Corollary 14.
5 An Analogue of Lamans Theorem
In this section, we address the problem of whether there exists a combinatorial
description of the class of graphs for which a minimally infinitesimally rigid placement exists
in (Rd , P ). We restrict our attention to finite bar-joint frameworks and prove that in
two dimensions such a characterisation exists (Theorem 23). This result is analogous
to Lamans theorem [14] for bar-joint frameworks in the Euclidean plane and extends
[12, Thm. 4.6] which holds in the case where P is a quadrilateral.
5.1 Regular Placements
Let (G, Rd , P) denote the set of all well-positioned placements of a finite simple
graph G in (Rd , P ). A bar-joint framework (G, p) is regular in (Rd , P ) if the
function
(G, Rd , P) {1, 2, . . . , d|V (G)| d}, x rank RP (G, x )
achieves its maximum value at p.
Lemma 16 Let G be a finite simple graph.
vV (G)
(i) The set of placements of G in (Rd , P ) which are both well-positioned and
regular is an open set in vV (G) Rd .
(ii) The set of placements of G in (Rd , P ) which are well-positioned and not
regular is an open set in Rd .
Proof Let p be a well-positioned placement of G and let U be an open neighbourhood
of p as in the statement of Proposition 6. The matrix-valued function x RP (G, x )
is constant on U and so either (G, x ) is regular for all x U or (G, x ) is not regular
for all x U .
A finite simple graph G is (minimally) rigid in (Rd , P ) if there exists a
wellpositioned placement of G which is (minimally) infinitesimally rigid.
Example 17 The complete graph K4 is minimally rigid in (R2, P ) for every
polyhedral norm P . To see this, let F1, F2, . . . , Fn be the facets of P and let
x0 ext(P) be any extreme point of P. Then x0 is contained in exactly two facets,
F1 and F2 say. Choose a point x1 in the relative interior of F1 and a point x2 in the
relative interior of F2. Then by formulas (3) and (4),
Since (x0 F1) = (x0 F2) = x0 P = 1, if x1 and x2 are chosen to lie in a sufficiently
small neighbourhood of x0 then by continuity we may assume
We may also assume without loss of generality that
x1 F2 = mk=a1x (x1 Fk) > 0,
x2 F1 = mk=a2x (x2 Fk) > 0.
x1 F2 = x2 F1.
Define a placement p : V (K4) R2 by setting
pv0 = (0, 0), pv1 = x1, pv2 = (1 )x2, pv3 = x1 + (1 + )x2,
(v0v1) = [F1], (v0v2) = [F2], (v1v3) = [F2].
To determine the framework colours for the remaining edges, we will apply the above
identities together with formulas (3) and (4). Consider the edge v2v3. If k = 1 and
is sufficiently small, then applying (9)
Also by (9) and (12), we have
( pv3 pv2 ) F1 = (x1 F1) + 2 (x2 F1) = 1 + 2 (x2 F1) > 1.
We conclude that F1 is the unique facet of P for which pv3 pv2 P = ( pv3 pv2 ) F1
and so pv3 pv2 cone(F1). Thus (v2v3) = [F1]. Consider the edge v0v3.
Applying (10) and (11), for k = 1, 2 we have
( pv3 pv0 ) Fk = (x1 Fk) + (1 + ) (x2 Fk) < (x1 F2) + 1 + .
( pv3 pv0 ) F1 = (x1 F1) + (1 + )(x2 F1) < (x1 F2) + 1 +
( pv3 pv0 ) F2 = (x1 F2) + (1 + )(x2 F2) = (x1 F2) + 1 + .
Hence F2 is the unique facet of P for which pv3 pv0 P = ( pv3 pv0 ) F2. Thus
pv3 pv0 cone(F2) and so (v0v3) = [F2]. Finally, consider the edge v1v2.
Fig. 2 A framework colouring for an infinitesimally rigid placement of K4 in (R2, P )
Applying (13), we have
( pv2 pv1 ) F2 = (1 )(x2 F2) (x1 F2) = 1 (x2 F1)
We conclude that ( pv2 pv1 ) (F2) < pv2 pv1 P . Hence pv2 pv1 / cone(F2).
By making a small perturbation, we can assume that pv2 pv1 is contained in the
conical hull of exactly one facet of P and so (v1v2) = [Fk ] for some [Fk ] = [F2].
Thus (G, p) is well-positioned. This framework colouring is illustrated in Fig. 2 with
monochrome subgraphs G F1 and G F2 indicated in black and grey, respectively, and
G Fk indicated by the dotted line. Suppose u F (K4, p). To show that u is a trivial
infinitesimal flex, we apply the method of Proposition 12. The vertices v0 and v1
are joined by monochrome paths in both G F1 and G F2 and so uv0 = uv1 . Similarly,
uv2 = uv3 . The vertices v1 and v2 are joined by monochrome paths in G F2 and G Fk
and so uv1 = uv2 . Thus u is a constant and hence trivial infinitesimal flex of (K4, p).
We conclude that (K4, p) and all regular and well-positioned placements of K4 are
infinitesimally rigid.
5.2 Counting Conditions
The Maxwell counting conditions [17] state that a finite minimally infinitesimally rigid
bar-joint framework (G, p) in Euclidean space Rd must satisfy |E (G)| = d|V (G)|
d +21 with inequalities |E (H )| d|V (H )| d +21 for all subgraphs H containing at
least d vertices. The following analogous statement holds for polyhedral norms.
Proposition 18 Let (G, p) be a finite and well-positioned bar-joint framework in
(Rd , P ). If (G, p) is minimally infinitesimally rigid, then
(i) |E (G)| = d|V (G)| d and
(ii) |E (H )| d|V (H )| d for all subgraphs H of G.
Proof If (G, p) is minimally infinitesimally rigid, then by Proposition 8 the rigidity
matrix RP (G, p) is independent and
|E (G)| = rank RP (G, p) = d|V (G)| d.
The rigidity matrix for any subframework of (G, p) is also independent and so
|E (H )| = rank RP (H, p) d|V (H )| d
for all subgraphs H .
A graph G is (d, d)-tight if it satisfies the counting conditions in the above
proposition. The class of (2, 2)-tight graphs has the property that every member can be
constructed from a single vertex by applying a sequence of finitely many allowable
graph moves (see [18]). The allowable graph moves are:
A Henneberg 1-move G G adjoins a vertex v0 to G together with two edges v0v1 and v0v2 where v1, v2 V (G).
Proposition 19 The Henneberg 1-move preserves infinitesimal rigidity for
wellpositioned bar-joint frameworks in (R2, P ).
Proof Suppose (G, p) is well-positioned and infinitesimally rigid and let G G be a
Henneberg 1-move on the vertices v1, v2 V (G). Choose distinct [F1], [F2] (P)
and define a placement p of G by pv = pv for all v V (G) and
pv0 pv1 + cone (F1) cone (F1)
pv2 + cone (F2) cone (F2) .
Then (G , p ) is well-positioned and the edges v0v1 and v0v2 have framework colours
[F1] and [F2], respectively. If u F (G , p ), then the restriction of u to V (G) is
an infinitesimal flex of (G, p). This restriction must be trivial and hence constant. In
particular, uv1 = uv2 . By Theorem 5, F1 (uv0 uv1 ) = 0 and F2 (uv0 uv1 ) =
F2 (uv0 uv2 ) = 0 and so uv0 = uv1 . We conclude that (G , p ) is infinitesimally
rigid.
A Henneberg 2-move G G removes an edge v1v2 from G and adjoins a vertex
v0 together with three edges v0v1, v0v2 and v0v3.
Proposition 20 The Henneberg 2-move preserves infinitesimal rigidity for
wellpositioned bar-joint frameworks in (R2, P ).
Proof Suppose (G, p) is well-positioned and infinitesimally rigid and let G G be
a Henneberg 2-move on the vertices v1, v2, v3 V (G) and the edge v1v2 E (G). Let
[F1] be the unique framework colour for the edge v1v2 and choose any [F2] (P)
with [F2] = [F1]. Define a placement p of G by setting pv = pv for all v V (G)
and choosing pv0 to lie on the intersection of the line through pv1 and pv2 and the
double cone pv3 + (cone(F2) cone(F2)). (If pv1 , pv2 , pv3 are collinear, then
choose pv0 to lie in the intersection of this double cone and a small neighbourhood
of pv3 .) Then (G , p ) is well-positioned. Both edges v0v1 and v0v2 have framework
F1 (uv1 uv2 ) = F1 (uv1 uv0 ) + F1 (uv0 uv2 ) = 0.
Hence the restriction of u to V (G) is an infinitesimal flex of (G, p) and must be
trivial. In particular, uv1 = uv3 . Now F1 (uv0 uv1 ) = 0 and F2 (uv0 uv1 ) =
F2 (uv0 uv3 ) = 0 and so uv0 = uv1 . We conclude that u is a constant and hence
trivial infinitesimal flex of (G , p ).
Let v1v2 be an edge of G. An edge-to-K3 move G G (on the edge v1v2 and the
vertex v1) is obtained in two steps: Firstly, adjoin a new vertex v0 and two new edges
v0v1 and v0v2 to G (creating a copy of K3 with vertices v0, v1, v2). Secondly, each
edge v1w of G which is incident with v1 is either left unchanged or is removed and
replaced with the edge v0w.
Proposition 21 The edge-to-K3 move preserves infinitesimal rigidity for finite
wellpositioned bar-joint frameworks in (R2, P ).
Proof Suppose (G, p) is well-positioned and infinitesimally rigid and let G G be
an edge-to-K3 move on the vertex v1 V (G) and the edge v1v2 E (G). Let [F1] be
the unique framework colour for v1v2 and choose any [F2] (P) with [F2] = [F1].
Since v1 has finite valence, there exists an open ball B( pv1 , r ) such that if pv1 is
replaced with any point x B( pv1 , r ), then the induced framework colouring of G is
left unchanged. Define a placement p of G by setting pv = pv for all v V (G) and
choosing
pv0 ( pv1 + cone(F2)) B( pv1 , r ).
Then (G , p ) is well-positioned. Suppose u F (G , p ) is an infinitesimal flex
of (G , p ). The framework colours for the edges v0v1 and v0v2 are [F2] and [F1],
respectively. Thus there exists a path from v0 to v1 in the monochrome subgraph G F1
given by the edges v1v2, v2v0, and there exists a path from v0 to v1 in the monochrome
subgraph G F2 given by the edge v0v1. By the relative rigidity method of Proposition 12,
uv0 = uv1 . If an edge v1w in G has framework colour [F ] induced by (G, p) and
is replaced by v0w in G , then the framework colour is unchanged. Thus applying
Theorem 5,
F (uv1 uw) = F (uv1 uv0 ) + F (uv0 uw) = 0,
and so the restriction of u to V (G) is an infinitesimal flex of (G, p). This restriction
is constant since (G, p) is infinitesimally rigid and so u is a trivial infinitesimal flex
of (G , p ).
A vertex-to-K4 move G G replaces a vertex v0 V (G) with a copy of the
complete graph K4 by adjoining three new vertices v1, v2, v3 and six edges v0v1, v0v2,
v0v3, v1v2, v1v3, v2v3. Each edge v0w of G which is incident with v0 may be left
unchanged or replaced by one of v1w, v2w or v3w.
Proposition 22 The vertex-to-K4 move preserves infinitesimal rigidity for finite
wellpositioned bar-joint frameworks in (R2, P ).
Proof Suppose (G, p) is well-positioned and infinitesimally rigid and let G G
be a vertex-to-K4 move on the vertex v0 V (G) which introduces new vertices v1,
v2 and v3. Since v0 has finite valence, there exists an open ball B( pv0 , r ) such that if
pv0 is replaced with any point x B( pv0 , r ), then (G, x ) and (G, p) induce the same
framework colouring on G. Let (K4, p) be the well-positioned and infinitesimally
rigid placement of K4 constructed in Example 17. Define a well-positioned placement
p of G by setting pv = pv for all v V (G) and
where > 0 is chosen to be sufficiently small so that pv1 , pv2 and pv3 are all contained
in B( pv0 , r ). Suppose u F (G , p ). By the argument in Example 17, the restriction of
u to the vertices v0, v1, v2, v3 is constant. Thus if v0w is an edge of G with framework
colour [F ] which is replaced by vk w in G , then applying Theorem 5,
F (uv0 uw) = F (uv0 uvk ) + F (uvk uw) = 0,
and so the restriction of u to V (G) is an infinitesimal flex of (G, p). Since (G, p)
is infinitesimally rigid, this restriction is constant, and we conclude that u is a trivial
infinitesimal flex of (G , p ).
We now show that the class of finite graphs which have minimally infinitesimally
rigid well-positioned placements in (R2, P ) is precisely the class of (2, 2)-tight
graphs. In particular, the existence of such a placement does not depend on the choice
of polyhedral norm on R2.
Theorem 23 Let G be a finite simple graph and let
R2. The following statements are equivalent:
Proof (i) (ii). If G is minimally rigid, then there exists a placement p such that
(G, p) is minimally infinitesimally rigid in (R2, P ) and the result follows from
Proposition 18.
(ii) (i). If G is (2, 2)-tight, then there exists a finite sequence of allowable
graph moves, K1 G2 G3 G. Every placement of K1 is
certainly infinitesimally rigid. By Propositions 1922, for each graph in the sequence
there exists a well-positioned and infinitesimally rigid placement in (R2, P ). In
particular, (G, p) is infinitesimally rigid for some well-positioned placement p. If
a single edge is removed from G, then by Proposition 18 the resulting
subframework is infinitesimally flexible. Hence (G, p) is minimally infinitesimally rigid in
(R2, P ).
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