COMBINATORIAL AND GEOMETRIC RIGIDITY WITH SYMMETRY CONSTRAINTS BY BERND SCHULZE A DISSERTATION SUBMITTED TO THE FACULTY OF GRADUATE STUDIES IN PARTIAL FULFILMENT OF THE REQUIREMENTS FOR THE DEGREE OF DOCTOR OF PHILOSOPHY GRADUATE PROGRAM IN MATHEMATICS AND STATISTICS YORK UNIVERSITY, TORONTO, ONTARIO MAY 2009
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Combinatorial and Geometric Rigidity with Symmetry Constraints
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COMBINATORIAL AND GEOMETRIC RIGIDITY WITH
SYMMETRY CONSTRAINTS
BY BERND SCHULZE
A DISSERTATION SUBMITTED TO THE FACULTY OF GRADUATE
STUDIES IN PARTIAL FULFILMENT OF THE REQUIREMENTS
FOR THE DEGREE OF
DOCTOR OF PHILOSOPHY
GRADUATE PROGRAM IN MATHEMATICS AND STATISTICS
YORK UNIVERSITY,
TORONTO, ONTARIO
MAY 2009
Abstract
In this thesis, we investigate the rigidity and flexibility properties of frame-
works consisting of rigid bars and flexible joints that possess non-trivial sym-
metries.
Using techniques from group representation theory, we first prove that
the rigidity matrix of a symmetric framework can be transformed into a
block-diagonalized form. Based on this result, we prove a generalization of
the symmetry-extended version of Maxwell’s rule given in [25] which can
be applied to both injective and non-injective realizations in all dimensions.
We then use this rule to prove that a symmetric isostatic (i.e., minimal in-
finitesimally rigid) framework must obey some very simply stated restrictions
on the number of joints and bars that are ‘fixed’ by various symmetry op-
erations of the framework. In particular, it turns out that a 2-dimensional
isostatic framework must belong to one of only six possible point groups. For
3-dimensional isostatic frameworks, all point groups are possible, although
restrictions on the placement of structural components still apply.
For three of the five non-trivial symmetry groups in dimension 2 that
allow isostatic frameworks, namely for the groups C2 and C3 generated by
a half-turn and a 3-fold rotation, respectively, and for the group Cs gener-
ated by a reflection, we establish symmetric versions of Laman’s Theorem
([33, 46]). More precisely, we show that the necessary conditions derived from
the symmetry-extended version of Maxwell’s rule, together with the Laman
conditions, are also sufficient for a framework whose joints are positioned as
generically as possible subject to the given symmetry conditions to be iso-
static. Symmetric versions of Henneberg’s Theorem ([40, 33]) and Crapo’s
iv
Theorem ([20, 33, 67]) for the groups C2, C3, and Cs are also established. For
the remaining two non-trivial symmetry groups in dimension 2 that allow
isostatic frameworks, we offer some analogous conjectures.
Finally, we derive sufficient conditions for the existence of a finite flex of a
symmetric framework. Finite flexes detected with these results have the nice
property that they preserve all of the symmetries of the given framework.
v
To my parents
vi
Acknowledgements
First, I would like to thank my supervisor Prof. Walter Whiteley for his
invaluable advice, guidance, and support throughout my time as his student.
His care and enthusiasm for my work as well as his vast knowledge and
exceptional insights into mathematics have always been a great source of
motivation and inspiration for me. The countless enlightening conversations
I have had with Prof. Walter Whiteley over the last few years have not only
played a crucial role in the writing of this thesis, but they have also nourished
my intellectual maturity which I will benefit from for a long time to come. In
addition, Prof. Walter Whiteley has always been available for me whenever
I faced any sort of trouble or had a question about my research or writing. I
simply could not have wished for a better or friendlier supervisor.
Secondly, I would like to express many thanks to Prof. Asia Weiss and
Prof. Mike Zabrocki for taking the time to read my thesis and serve on my
supervisory committee in the midst of all their other activities.
Thanks also to Prof. Ada Chan, Prof. Andy Mirzaian, and Prof. Meera
Sitharam for agreeing to be members of my examining committee.
Further, I would like to thank the organizers and participants of the AIM
workshop in Palo Alto in December 2007 and the BIRS workshop in Banff
in July 2008, particularly Prof. Robert Connelly, Prof. Simon Guest, and
Prof. Brigitte Servatius, for all the interesting and fruitful discussions.
A special thanks goes to my mother, father, and sister for their continued
support and encouragement throughout my studies at home and abroad.
vii
They have always been there for me.
Last, but not least I would like to thank my partner Krishna Wu for all
The study of rigidity and flexibility has a rich history in what are cur-
rently a number of areas of engineering and mathematics, but historically
were connected in the work of many scientists who combined studies of en-
gineering, physics, and mathematics.
Early work which is now recognized as ‘rigidity theory’ included the con-
jecture of L. Euler about the rigidity of closed surfaces (e.g. polyhedra with
rigid faces) [23], the static and kinematic analysis of pin-jointed frameworks
for engineering structures (from the early 19th century) [49], and the large lit-
erature on linkages (frameworks with non-trivial - non-congruent - motions)
in the 19th century [52].
This wider range of work produced both counting rules (such as J.C.
Maxwell’s rules for built structures and M. Grubler’s counts for linkages) and
1
geometric analyses (such as reciprocal diagrams and various other tools for
analyzing the resolutions of forces in the structures, and analyzing possible
motions) [49, 52, 61, 62, 83].
By the end of the 19th century, there were detailed analyses of which
frameworks are ‘normally’ rigid (e.g. A.L. Cauchy’s famous theorem on the
uniqueness of convex triangulated polyhedra with fixed edge lengths [10, 22,
76]), as well as when certain frameworks that were ‘normally’ rigid became
flexible (e.g. R. Bricard’s work on flexible frameworks with the bars and
joints of an octahedron [9]).
The people working on engineering problems developed a number of prac-
tical methods for analyzing buildings (e.g. the Eiffel Tower) as well as numer-
ical and geometric rules of thumb for their design and construction. A whole
body of work, summarized by the engineer/mathematician L. Henneberg,
also developed explicit inductive techniques for generating rigid structures
[40]. Another stream, from A.F. Mobius through J. Plucker and F. Klein
to L. Cremona, investigated the projective geometry of static equilibria and
singular forms which lead to building failures, in fields with names such as
‘graphical statics’ [21, 83].
Pockets of work continued through the 20th century, including detailed
explorations of the design and analysis of mechanical linkages (mechanisms)
and some continuing geometric analyses of built structures such as built
trusses. However, much of the previous geometric and combinatorial the-
ory was submerged in the numerical analysis permitted by the developing
computers and the now ‘standard’ designs.
One notable exception to this decline in attention to the ’theory of rigid-
2
ity’ was the ongoing mathematical work on rigidity of polyhedral surfaces of
the Russian school of A.D. Alexandrov, N.V. Efimov, and A.V. Pogorelov,
and their analysis of the unique realizability of convex metrics [1].
In the past 40 years, there have been two major, parallel developments.
(A) A strong mathematical theory has flowered, refining old results
and techniques, with the splitting of key questions, techniques, and well-
developed results into combinatorial and geometric aspects.
These developments have brought in combinatorial methods from graph
theory, matroid theory, and associated combinatorial algorithms. The work
also refined geometric conditions that were sufficient for a shift in the rigidity
properties - projective conditions for the static/first-order kinematic theory,
and Euclidean and affine geometry for the theory of finite motions. The
static/first-order kinematic theory is expressed in the linear algebra of the
rigidity matrix, whose rank, row dependencies and column dependencies all
play key roles in the theory.
A highlight of these mathematical developments has been the basically
complete combinatorial theory of plane frameworks - most notably, the com-
binatorial characterization of rigid 2-dimensional ‘generic’ frameworks given
by G. Laman in 1970 [46]. Combined with strong results for certain classes
of structures in dimensions d ≥ 3, the work clarified the difficulties of char-
acterizing rigid ‘generic’ frameworks in 3- and higher-dimensional space.
During the 1970s, remarkable results have also been found in the theory
of flexible polyhedra: in 1975 H. Gluck was able to refine Euler’s famous
rigidity conjecture from 1766 [30], and in 1977 R. Connelly finally settled the
3
conjecture with his celebrated counterexample, the flexible ‘Connelly sphere’
[11].
(B) An expanding web of connections to problems in other fields where the
mathematical theory of rigidity makes substantial contributions to clarifying
and resolving central questions has arisen.
Starting with the connections with mechanical and civil engineering (hu-
man built structures), there are connections to the general theory of geomet-
ric constraints built into Computer Aided Design (CAD) and to computa-
tions for robotic motions. Less obvious, but very real, are the connections
to computer vision/recognition of geometric objects from geometric data.
In general, a sequence of problems in computational geometry have found
‘rigidity-type’ results and methods contribute to their understanding and
their solutions (as well as sometimes clarifying that the problems are ‘hard’).
In turn, computational geometry has contributed new problems and algo-
rithmic insights to the general theory of rigidity.
One step removed are the connections to scaled down natural structures,
such as granular materials which are sometimes modeled with packings of
spheres, abstracted as bar and joint frameworks. Over the last two decades,
there has also been a strong interest in applying rigidity theory to rapid
predictions of the rigid and flexible regions of large biomolecules, such as
proteins. Such predictions, even based on incomplete theories, are imple-
mented on the web because of the importance of flexibility and rigidity to
the function of biomolecules and the design of drugs to alter their functioning
[82].
Overall, both the elegant and expanding mathematical theory and the
4
growing network of applications have made rigidity theory a rich area for
research, and a source of new questions and new insights into a wide array
of pure and applied mathematical theories.
Symmetry is another central idea in geometry - and appears widely in
both natural structures such as crystals and biomolecules, and in structures
built by humans. One source of such symmetry is the efficiency of forming
the shape using multiple copies of a few key components.
Because the appearance of symmetry is wide-spread, sometimes ubiqui-
tous, there have recently been a series of papers by engineers and chemists,
which present criteria for rigidity and flexibility of symmetric frameworks.
The first of these breakthrough papers is due to the engineers R. Kangwai
and S. Guest: in 2000 they observed that the rigidity matrix of a symmet-
ric framework can be put into a block-diagonalized form using techniques
from group representation theory [44]. Using this result, some engineers and
chemists were able to make some further interesting and useful observations
concerning the rigidity of symmetric frameworks (see [25, 43], for example).
Many of these observations, however, are incomplete from a mathematical
point of view, since they are not presented with a mathematically precise
formulation nor with a thorough mathematical foundation or proofs. So
while this work has resulted in some important heuristics for engineers and
chemists to gain further insight into the rigidity properties of a symmetric
framework, there has not been a rigorous mathematical investigation of how
symmetry impacts the rigidity of frameworks. While this thesis was being
written, the mathematicians J.C. Owen and S.C. Power have also been work-
ing on grounding aspects of this theory [53]. In Chapter 4 we will describe
5
the one key area of overlap between their work and this thesis.
Again, the focus on classes of frameworks with given symmetries also has
both a combinatorial (graph automorphism) level and a geometric aspect
(point group, spatial isometries). In our study of these connections, we will
see some clearly combinatorial conditions about fixed vertices, edges, etc.
in the graph automorphisms which must show up in the geometry of the
realization.
When does this necessary geometry of symmetry from the graph auto-
morphisms and the geometry of reflections, rotations, and other isometries
force the rank of the rigidity matrix to drop? When does the geometry
of symmetry overlap with the geometry of the singular positions which are
traditionally expressed in projective form? We will see that under many cir-
cumstances, the addition of symmetry does not change the rigidity predicted
for more general, asymmetric realizations. In other circumstances there are
very simply stated added conditions.
The simplest example is the point group C3 in dimension 2 (Z3 as an
abstract group) which describes 3-fold rotational symmetry. We will see that
the combinatorial condition, once there is a group of graph automorphisms
associated with the point group C3, is simply that no vertices are fixed by
the automorphism corresponding to the 3-fold rotation (geometrically, no
vertices are placed on the center of rotation). This is a necessary condition
for any independent and rigid realization of the graph as a framework with
C3 symmetry (Chapter 4). It is also a sufficient condition for the most gen-
eral ‘C3-symmetric realizations’ to be isostatic (Chapter 5). The result is
striking in its simplicity: to test a ‘generic’ framework with C3 symmetry for
6
isostaticity, we just need to check the number of fixed vertices, as well as the
standard conditions for rigidity without symmetry.
More generally, the techniques in Chapter 4 work with counts of vertices
and edges, and counts of vertices and edges fixed by various elements of the
group. So the necessary conditions will always be of this type: vertices or
edges fixed by the automorphisms. The examination of when these necessary
conditions are also sufficient is the larger theme of Chapter 5, for an array
of plane symmetry groups. In fact, a collection of rigidity theory methods
which can now be called ’classical’ are symmetrized in Chapter 5 to establish
both necessary and sufficient conditions for realizations which are generic
with the given symmetry to be isostatic.
For general frameworks, an undercount of constraints becomes a predic-
tion of flexibility. For symmetric frameworks, we will show that there is an
extension that not only predicts finite motions, but predicts motions which
preserve the symmetries throughout their path (Chapter 6).
The interactions of combinatorics, geometry, and symmetry are rich. It is
no surprise that at almost every turn, we find not only fascinating and appeal-
ing results, but also possible extensions to explore as well as new questions
and new generalizations which can be conjectured and anticipated. We in-
vite the interested reader to join us in the exploration of these landscapes of
possibilities.
1.2 Outline of thesis
The thesis is organized as follows.
7
In Chapter 2, we give a brief introduction to rigidity, its linearized versions
infinitesimal and static rigidity, as well as generic (or combinatorial) rigidity.
We also introduce suitable mathematical definitions for the relevant terms
relating to symmetric structures that are frequently used in the chemistry
and engineering literature. In particular, we give a detailed mathematical
description of the Schoenflies notation for point groups in dimensions 2 and
3. We will be using this notation for all the examples throughout this thesis.
In Chapter 3, we introduce a natural classification of symmetric frame-
works. This classification is fundamental to all the results of this thesis. We
then define a symmetry-adapted notion of a ‘generic’ framework with respect
to this classification. This symmetrized notion of generic has the two funda-
mental properties that ‘almost all’ realizations in a given symmetry class are
generic and all generic realizations in this class share the same infinitesimal
rigidity properties. This classification therefore not only lays the foundation
for symmetrizing results in rigidity, infinitesimal rigidity, and static rigidity,
but it also allows us to develop a symmetry-adapted version of generic rigid-
ity theory in Chapter 5.
In the last two sections of the third chapter, we carefully examine the dif-
ficulties that arise in applying techniques from group representation theory
to the analysis of symmetric frameworks with non-injective configurations.
More precisely, in Section 3.3, we show that a framework with a non-injective
configuration can belong to more than one symmetry class, and we examine
how many distinct symmetry classes a given framework can possibly belong
to. In Section 3.4, we investigate under what conditions techniques from
group representation theory can be applied to the frameworks in a given
8
symmetry class.
All the results of the third chapter can be found in the manuscript [55]
which has been submitted for review.
Chapter 4 concerns the application of techniques from group representa-
tion theory to the rigidity analysis of symmetric frameworks. In Section 4.1,
we first give a complete self-contained mathematical proof that the rigid-
ity matrix of a symmetric framework can be block-diagonalized as described
by R. Kangwai and S. Guest in [44]. In Section 4.2, we use this result to
give a detailed proof for the symmetry-extended version of Maxwell’s rule
given by P. Fowler and S. Guest in [25]. This rule provides further necessary
conditions (in addition to Maxwell’s original condition from 1864 [49]) for a
symmetric framework (G, p) to be isostatic. While the rule in [25] is only
applicable to 2- or 3-dimensional frameworks with injective configurations,
we establish a more general result in this thesis, namely a rule that can be
applied to both injective and non-injective realizations in all dimensions.
The results of Sections 4.1 and 4.2 will constitute the main part of [56].
An alternate proof for the rule given in [25], as well as various general-
izations of this rule to other types of geometric constraint systems, is given
by J.C. Owen and S.C. Power in [53].
In Section 4.3, we show that the symmetry-extended version of Maxwell’s
rule can be used to prove that a symmetric isostatic framework must obey
some very simply stated restrictions on the number of joints and bars that are
‘fixed’ by various symmetry operations of the framework. In particular, these
restrictions imply that the symmetries of a 2-dimensional isostatic framework
must belong to one of only six possible point groups. For 3-dimensional iso-
9
static frameworks, all point groups are possible, although restrictions on the
placement of structural components still apply.
The main part of Section 4.3 is a mathematically explicit derivation of
the results presented without proof in [15]. This paper is joint work with R.
Connelly, P. Fowler, S. Guest, and W. Whiteley.
Finally, in Section 4.4, we use the results of the previous sections to also
establish necessary conditions for a symmetric framework to be independent
or infinitesimally rigid.
In Chapter 5, we present symmetric versions of some famous results
in generic rigidity theory. Given a graph G, Laman’s Theorem says that
Maxwell’s condition in 2D, i.e., |E(G)| = 2|V (G)| − 3, together with the
counts |E(H)| ≤ 2|V (H)| − 3 for all non-trivial subgraphs H of G, are nec-
essary and sufficient for all generic 2-dimensional realizations of G to be
isostatic. There are well known difficulties in extending this result to higher
dimensions (see [32, 33, 46], for example). Using the symmetry-adapted no-
tion of ‘generic’ introduced in Chapter 3, we establish symmetric versions of
Laman’s Theorem for three of the five non-trivial symmetry groups in dimen-
sion 2 that allow isostatic frameworks, namely for the groups C2 and C3 of
order 2 and 3 generated by a half-turn and a 3-fold rotation, respectively, and
for the group Cs of order 2 generated by a reflection. More precisely, we show
that for each of these groups, the conditions derived from the symmetry-
extended version of Maxwell’s rule, together with the Laman conditions, are
necessary and sufficient for realizations of G that are ‘generic’ within the
given symmetry class to be isostatic. These results were conjectured in [15].
Henneberg’s Theorem and Crapo’s Tree Covering Theorem are also fa-
10
mous combinatorial results that provide characterizations of generically 2-
isostatic graphs [20, 40, 33, 67, 68]. We show that for each of the symmetry
groups C2, C3 and Cs, there exist symmetric versions of these results as well.
The other two non-trivial symmetry groups in dimension 2 that allow iso-
static frameworks are the dihedral groups of order 4 and 6. For these groups,
we offer some analogous conjectures. To prove these conjectures with tech-
niques similar to the ones used for the results above, one has to consider an
unreasonably large number of cases.
In the final section of Chapter 5, we briefly discuss ‘symmetric-generically’
isostatic graphs in dimension 3.
The key results of the fifth chapter will be summarized in [57].
In Chapter 6, we study finite flexes of symmetric frameworks, i.e., flexes
that move the joints of a given framework on differentiable displacement
paths while holding the lengths of all bars fixed and changing the distance
between two unconnected joints. We prove that if a framework (G, p) is
‘generic’ within a given symmetry class and there exists a ‘fully-symmetric’
infinitesimal flex of (G, p) (i.e., the velocity vectors of the infinitesimal flex
remain unaltered under all symmetry operations of (G, p)), then (G, p) also
possesses a ‘symmetry-preserving’ finite flex, i.e., a flex which displaces the
joints of (G, p) in such a way that all the resulting frameworks have the same
symmetry as (G, p) (or possibly higher symmetry). This and other related
results are obtained by symmetrizing techniques described by L. Asimov and
B. Roth in [3] and by using the fact that the rigidity matrix of a symmetric
framework can be transformed into a block-diagonalized form as shown in
Chapter 4. As corollaries of these results, one obtains the results stated (but
11
not rigorously proven) in [35] and Proposition 1 in [43].
The finite flexes that can be detected with these symmetry-based methods
can in general not be found with the analogous non-symmetric methods.
The work of Chapter 6 will also be presented in [58].
Finally, in Chapter 7, we outline how the methods developed in this thesis
can be extended to analyze the rigidity and flexibility properties of various
other types of symmetric structures.
Several additional promising directions for future work are also presented
in this final chapter.
12
Chapter 2
Definitions and preliminaries
2.1 Graph theory terminology
We begin by establishing the graph theory vocabulary and notation we
will be using throughout this thesis.
Definition 2.1.1 A graph G is a finite nonempty set of objects called ver-
tices together with a (possibly empty) set of unordered pairs of distinct
vertices of G called edges. The vertex set of G is denoted by V (G) and the
edge set of G is denoted by E(G).
Definition 2.1.2 Two vertices u 6= v of a graph G are adjacent if u, v ∈E(G), and independent otherwise. A set S of vertices of G is independent if
every two vertices of S are independent.
Definition 2.1.3 Let G be a graph. The neighborhood NG(v) of a vertex
v ∈ V (G) is the set of all vertices that are adjacent to v and the elements of
13
NG(v) are called the neighbors of v.
Definition 2.1.4 Let G be a graph and e = u, v be an edge of G. Then
we say that u and e are incident, as are v and e. The valence valG(v) of
a vertex v ∈ V (G) is the number of edges of G that are incident with v.
Equivalently, valG(v) = |NG(v)|.
Definition 2.1.5 A graph is called complete if every two of its vertices are
adjacent. We write Kn for the complete graph on n vertices.
A graph G is called bipartite if the vertex set V (G) can be partitioned into
two sets X and Y (called partite sets) such that for every edge x, y ∈ E(G)
we have x ∈ X and y ∈ Y . A bipartite graph G with partite sets X and Y
is complete if x, y ∈ E(G) for all x ∈ X and y ∈ Y . We write Km,n for the
complete bipartite graph whose partite sets have cardinality m and n.
Definition 2.1.6 A graph H is a subgraph of a graph G if V (H) ⊆ V (G)
and E(H) ⊆ E(G), in which case we write H ⊆ G. A subgraph H of a graph
G is called spanning if |V (H)| = |V (G)|.
The simplest type of subgraph of a graph G is that obtained by deleting
a vertex or an edge from G. Let v be a vertex and e be an edge of G. Then
we write G − v for the subgraph of G that has V (G) \ v as its vertex
set and whose edges are those of G that are not incident with v. Similarly,
we write G− e for the subgraph of G that has V (G) as its vertex set and
E(G) \ e as its edge set. The deletion of a set of vertices or a set of edges
from G is defined and denoted analogously.
If u and v are independent vertices of G, then we write G +u, v for
14
the graph that has V (G) as its vertex set and E(G) ∪ u, v as its edge
set. The addition of a set of edges is again defined and denoted analogously.
Definition 2.1.7 Let G be a graph and U be a nonempty subset of V (G).
Then the subgraph 〈U〉 of G induced by U is the graph having vertex set U
and whose edges are those of G that are incident with two elements of U .
Definition 2.1.8 Let G1 and G2 be two graphs. The intersection G =
G1∩G2 is the graph with V (G) = V (G1)∩V (G2) and E(G) = E(G1)∩E(G2).
Similarly, the union G = G1 ∪G2 is the graph with V (G) = V (G1) ∪ V (G2)
and E(G) = E(G1) ∪ E(G2).
Definition 2.1.9 An automorphism of a graph G is a permutation α of
V (G) such that u, v ∈ E(G) if and only if α(u), α(v) ∈ E(G).
The automorphisms of a graph G form a group under composition which
is denoted by Aut(G).
Definition 2.1.10 Let H be a subgraph of a graph G and α ∈ Aut(G).
We define α(H) to be the subgraph of G that has α(V (H)
)as its vertex
set and α(E(H)
)as its edge set, where u, v ∈ α
(E(H)
)if and only if
α−1(u, v) = α−1(u), α−1(v) ∈ E(H).
We say that H is invariant under α if α(V (H)
)= V (H) and α
(E(H)
)=
E(H), in which case we write α(H) = H.
Example 2.1.1 The graph G in Figure 2.1 (a) has the automorphism
α = (v1 v2 v3)(v4 v5 v6). The subgraph H1 of G is invariant under α, but
the subgraph H2 of G is not, because α(E(H2)
) 6= E(H2).
15
...v3..v1
..v2
..v4
..v5
..v6
.G:
.(a)
...v3..v1
..v2
.H1:
.(b)
...v3..v1
..v2
.H2:
.(c)
Figure 2.1: An invariant (b) and a non-invariant subgraph (c) of the graph
G under α = (v1 v2 v3)(v4 v5 v6) ∈ Aut(G).
Definition 2.1.11 Let u and v be two (not necessarily distinct) vertices
of a graph G. A u-v path in G is a finite alternating sequence u =
u0, e1, u1, e2, . . . , uk−1, ek, uk = v of vertices and edges of G in which no vertex
is repeated and ei = ui−1, ui for i = 1, 2, . . . , k. A u-v path is called a cycle
if k ≥ 3 and u = v.
Let a u-v path P in G be given by u = u0, e1, u1, e2, . . . , uk−1, ek, uk = v
and let α ∈ Aut(G). Then we denote α(P ) to be the α(u)-α(v) path
α(u) = α(u0), α(e1), α(u1), α(e2), . . . , α(uk−1), α(ek), α(uk) = α(v) in G.
A vertex u is said to be connected to a vertex v in G if there exists a u−v
path in G. A graph G is connected if every two vertices of G are connected.
A graph with no cycles is called a forest and a connected forest is called
a tree.
A connected subgraph H of a graph G is a component of G if H = H ′
whenever H ′ is a connected subgraph of G containing H.
16
2.2 Introduction to rigidity theory
We now give a brief introduction to rigidity, its linearized versions in-
finitesimal and static rigidity, as well as generic rigidity, as we shall sym-
metrize results from each of these theories. The definitions and results listed
in this introduction are widely used in the rigidity theory literature so that
we will omit the proofs and leave more detailed explanations and illustrations
to be found in the references provided.
2.2.1 Rigidity
Definition 2.2.1 [32, 33, 81, 83] A framework (in Rd) is a pair (G, p), where
G is a graph and p : V (G) → Rd is a map with the property that p(u) 6= p(v)
for all u, v ∈ E(G). We also say that (G, p) is a d-dimensional realization
of the underlying graph G.
Given the vertex set V (G) = v1, . . . , vn of a graph G and a map p :
V (G) → Rd, it is often useful to identify p with a vector in Rdn by using the
order on V (G). In this case we also refer to p as a configuration of n points
in Rd.
Throughout this thesis we will simplify our notation by not differentiating
between an abstract vector and its coordinate vector relative to the canonical
basis.
Definition 2.2.2 Let (G, p) be a framework in Rd. A joint of (G, p) is
an ordered pair(v, p(v)
), where v ∈ V (G). A bar of (G, p) is an unordered
pair(
u, p(u)),(v, p(v)
)of joints of (G, p), where u, v ∈ E(G). We define
17
‖p(u)−p(v)‖ to be the length of the bar(
u, p(u)),(v, p(v)
), where ‖p(u)−
p(v)‖ is defined by the canonical inner product on Rd.
Note that we allow the map p of a framework (G, p) to be non-injective,
that is, two distinct joints(u, p(u)
)and
(v, p(v)
)of (G, p) may be located
at the same point p(u) = p(v) in Rd, provided that u and v are independent
vertices of G. However, if u, v ∈ E(G), then p(u) 6= p(v), and hence every
bar(
u, p(u)),(v, p(v)
)of (G, p) has a strictly positive length.
Definition 2.2.3 [32] Let (G, p) be a framework in Rd with V (G) =
v1, v2, . . . , vn. A motion of (G, p) is an indexed family of functions
Pi : [0, 1] → Rd, i = 1, 2, . . . , n, so that
(i) Pi(0) = p(vi) for all i;
(ii) Pi(t) is differentiable on [0, 1] for all i;
(iii) ‖Pi(t)− Pj(t)‖ = ‖p(vi)− p(vj)‖ for all t ∈ [0, 1] and vi, vj ∈ E(G).
A motion of a framework (G, p) displaces the joints of (G, p) on differen-
tiable displacement paths while preserving the lengths of all bars of (G, p).
Every framework has some trivial motions, namely those that correspond to
rigid motions of space (i.e., translations, rotations and their combinations).
Definition 2.2.4 A motion Pi of a framework (G, p) with V (G) =
v1, v2, . . . , vn is called a rigid motion if it preserves the distances between
every pair of joints of (G, p), that is, if ‖Pi(t)− Pj(t)‖ = ‖p(vi)− p(vj)‖ for
all t ∈ [0, 1] and all 1 ≤ i < j ≤ n.
18
Pi is called a flex if the distance between at least one pair of joints of
(G, p) is changed by Pi, that is, if ‖Pi(t)− Pj(t)‖ 6= ‖p(vi)− p(vj)‖ for all
t ∈ (0, 1] and some vi, vj /∈ E(G).
Definition 2.2.5 A framework (G, p) is called rigid if every motion of (G, p)
is a rigid motion. Otherwise (G, p) is called flexible.
.. .
.
.(a)
.. .
..
.(b)
.. .
.. . .
.(c)
.. .
. .
.(d)
Figure 2.2: A rigid (a) and a flexible (b) framework in the plane. The flex
shown in (c) takes the framework in (b) to the framework in (d).
Some alternate definitions of a rigid framework are common in the litera-
ture [3, 83] all of which are equivalent to Definition 2.2.5. We will introduce
some of these definitions in Chapter 6, where we examine the motions of
symmetric frameworks.
2.2.2 Infinitesimal rigidity
It is in general very difficult to determine whether a given framework is
rigid or not since it requires solving a system of quadratic equations. It is
therefore common to linearize this problem by differentiating the equations
in Definition 2.2.3 (iii). This gives rise to
Definition 2.2.6 [32, 33, 81, 83] Let (G, p) be a framework in Rd with
V (G) = v1, v2, . . . , vn. An infinitesimal motion of (G, p) is a function
19
u : V (G) → Rd such that
(p(vi)− p(vj)
) · (u(vi)− u(vj))
= 0 for all vi, vj ∈ E(G). (2.1)
An infinitesimal motion of a framework (G, p) is a set of displacement
vectors u(vi), one at each joint, that neither stretch nor compress the bars
of (G, p) at first order. More precisely, condition (2.1) says that for every
edge vi, vj ∈ E(G), the projections of u(vi) and u(vj) onto the line through
p(vi) and p(vj) have the same direction and the same length (see also Figure
2.3).
Definition 2.2.7 An infinitesimal motion u of a framework (G, p) with
V (G) = v1, v2, . . . , vn is called an infinitesimal rigid motion if there ex-
ists a rigid motion Pi of (G, p) such that for i = 1, 2, . . . , n, the vector
u(vi) is the derivative (at t = 0) of Pi. Otherwise, u is called an infinitesimal
flex of (G, p).
Remark 2.2.1 Let G be a graph with V (G) = v1, v2, . . . , vn and let u
be an infinitesimal motion of a d-dimensional realization (G, p) of G. If(p(vi) − p(vj)
) · (u(vi) − u(vj)) 6= 0 for some vi, vj /∈ E(G), then u is
an infinitesimal flex of (G, p). If the points p(v1), . . . , p(vn) span all of Rd
(in an affine sense), then the converse also holds, i.e., in this case, u is an
infinitesimal flex of (G, p) if and only if(p(vi)− p(vj)
) · (u(vi)− u(vj)) 6= 0
for some vi, vj /∈ E(G) or equivalently, u is an infinitesimal rigid motion of
(G, p) if and only if(p(vi)− p(vj)
) · (u(vi)− u(vj))
= 0 for all 1 ≤ i < j ≤ n
[32, 33, 81].
From now on, when we say that a set of points spans a space, then this
will always be in the affine sense.
20
...p1
..p2
.u1
.u2
.(a)
...p1
..p2
..p3
.u3.u1 = 0 .u2 = 0
.(b)
...p6
..p1
..p2
..p3
. .p4
. .p5
.u6
.u1
.u2
.u3.u4
.u5
.(c)
Figure 2.3: The arrows indicate the non-zero displacement vectors of an in-
finitesimal rigid motion (a) and infinitesimal flexes (b, c) of frameworks in
R2.
Definition 2.2.8 [32, 33, 81, 83] A framework (G, p) is infinitesimally rigid
if every infinitesimal motion of (G, p) is an infinitesimal rigid motion. Oth-
erwise (G, p) is said to be infinitesimally flexible.
The following theorem gives the main connection between rigidity and
infinitesimal rigidity. A proof of this result can be found in [3], [17] or [30],
for example.
Theorem 2.2.1 If a framework (G, p) is infinitesimally rigid, then (G, p) is
rigid.
Under certain conditions, rigidity and infinitesimal rigidity are equivalent.
We will give the relevant results in the end of Section 2.2.5 after we have
established the necessary definitions.
For a framework (G, p) whose underlying graph G has a vertex set that is
indexed from 1 to n, say V (G) = v1, v2, . . . , vn, we will frequently denote
21
p(vi) by pi for i = 1, 2, . . . , n. Similarly, for an infinitesimal motion u of
(G, p), we will frequently denote u(vi) by ui for all i. The kth component of
a vector x is denoted by (x)k.
The equations stated in Definition 2.2.6 form a system of linear equations
whose corresponding matrix is called the rigidity matrix. This matrix is
fundamental in the study of both infinitesimal and static rigidity.
Definition 2.2.9 [32, 33, 81, 83] Let G be a graph with V (G) =
v1, v2, . . . , vn and let p : V (G) → Rd. The rigidity matrix of (G, p) is
the |E(G)| × dn matrix
..R(G, p) =
...
0 . . . 0 pi − pj 0 . . . 0 pj − pi 0 . . . 0
...
.edge vi, vj,
.vi .vj.v1 .vn
that is, for each edge vi, vj ∈ E(G), R(G, p) has the row with (pi −pj)1, . . . , (pi−pj)d in the columns d(i−1)+1, . . . , di, (pj−pi)1, . . . , (pj−pi)d
in the columns d(j − 1) + 1, . . . , dj, and 0 elsewhere.
Remark 2.2.2 The rigidity matrix is defined for arbitrary pairs (G, p),
where G is a graph and p : V (G) → Rd is a map. If (G, p) is not a frame-
work, then there exists a pair of adjacent vertices of G that are mapped to
the same point in Rd under p and every such edge of G gives rise to a zero-row
in R(G, p).
If we identify an infinitesimal motion of a d-dimensional framework (G, p)
with a column vector in Rd|V (G)| (by using the order on V (G)), then the
22
kernel of R(G, p) is the space of infinitesimal motions of (G, p). It is well
known that the infinitesimal rigid motions arising from d translations and(
d2
)rotations of Rd form a basis for the space of infinitesimal rigid motions
of (G, p), provided that the points p1, . . . , pn span an affine subspace of Rd
of dimension at least d − 1 [33, 81]. Thus, for such a framework (G, p), we
have nullity(R(G, p)
) ≥ d +(
d2
)=
(d+12
)and (G, p) is infinitesimally rigid
if and only if nullity(R(G, p)
)=
(d+12
)or equivalently, rank
(R(G, p)
)=
d|V (G)| − (d+12
).
Theorem 2.2.2 [3, 30] A framework (G, p) in Rd is infinitesimally rigid if
and only if either rank(R(G, p)
)= d|V (G)|−(
d+12
)or G is a complete graph
Kn and the points p(v), v ∈ V (G), are affinely independent.
Remark 2.2.3 Let 1 ≤ m ≤ d and let (G, p) be a framework in Rd. If
(G, p) has at least m + 1 joints and the points p(v), v ∈ V (G), span an
affine subspace of Rd of dimension less than m, then (G, p) is infinitesimally
flexible (recall Figure 2.3 (b)). In particular, if (G, p) is infinitesimally rigid
and |V (G)| ≥ d, then the points p(v), v ∈ V (G), span an affine subspace of
Rd of dimension at least d− 1.
2.2.3 Static rigidity
We now also give a brief introduction to the static approach to rigidity.
The intuitive test for static rigidity of a framework (G, p) is to apply an
external load to (G, p) (i.e., a set of forces, one to each joint) and investigate
whether there exists a set of tensions and compressions in the bars of (G, p)
that reach an equilibrium with this load at the joints (see also Figure 2.4).
23
Of course only loads which do not correspond to a translation or rotation of
space can possibly be resolved in this way.
Definition 2.2.10 [21, 68, 76, 81] Let (G, p) be a framework in Rd with
V (G) = v1, v2, . . . , vn. A load on (G, p) is a function l : V (G) → Rd, where
for i = 1, 2, . . . , n, the vector l(vi) represents a force applied to the joint(vi, pi
)of (G, p).
A load l on (G, p) is called an equilibrium load if l satisfies
(i)∑n
i=1 li = 0;
(ii)∑n
i=1
((li)j(pi)k − (li)k(pi)j
)= 0 for all 1 ≤ j < k ≤ d,
where li denotes the vector l(vi) for each i.
The physical intuition for conditions (i) and (ii) in Definition 2.2.10 is
the following: condition (i) rules out loads that would produce a translation
of (G, p) and (ii) says that there is no net rotational twist of (G, p).
Definition 2.2.11 [21, 68, 76, 81] Let l be an equilibrium load on a frame-
work (G, p) in Rd with V (G) = v1, v2, . . . , vn. A resolution of l by (G, p)
is a function ω : E(G) → R such that at each joint(vi, pi
)of (G, p) we have
∑
j with vi,vj∈E(G)
ωij(pi − pj) + li = 0,
where ωij denotes ω(vi, vj) for all vi, vj ∈ E(G).
The scalars ωij represent tensions (ωij < 0) and compressions (ωij > 0)
in the bars of (G, p), so that the bar forces reach an equilibrium with li at
each joint(vi, pi
).
24
.. .
.(a)
. .
.(b)
.. .
.
.
.(c)
.. .
.
.
.(d)
.. ..
.
.(e)
Figure 2.4: (a), (b) The arrows indicate a tension (a) and a compression (b)
in a bar. (c) An equilibrium load on a non-degenerate triangle. This load can
be resolved by the triangle as shown in (d). (e) An unresolvable equilibrium
load on a degenerate triangle: for any joint of this framework, tensions or
compressions in the bars cannot reach an equilibrium with the load vector at
this joint.
Definition 2.2.12 [21, 68, 76, 81] A framework (G, p) is statically rigid if
every equilibrium load on (G, p) has a resolution by (G, p).
Note that if we identify l and ω with a column vector in Rdn and R|E(G)|,
respectively, then (after changing the sign of l) the equations in Definition
2.2.11 can be written in a compact form in terms of the rigidity matrix
R(G, p) as
R(G, p)T ω = l.
Let (vh, ph) and (vk, pk) be two joints of (G, p). Then it is easy to see that
Corollary 3.3.2 Let G be a graph, S be a symmetry group, Φ be a map from
S to Aut(G), and (G, p) ∈ R(G,S,Φ). Then for every Ψ : S → Aut(G) distinct
from Φ, we have (G, p) /∈ R(G,S,Ψ) if and only if Aut(G, p) = id.
71
Proof. It follows directly from Theorem 3.3.1 that Aut(G, p) = id if and
only if for every x ∈ S, the automorphism Φ(x) is the only automorphism of
G that satisfies x(p(v)
)= p
(Φ(x)(v)
)for all v ∈ V (G). ¤
Corollary 3.3.2 asserts that the type Φ : S → Aut(G) of a framework
(G, p) ∈ R(G,S) is unique if and only if Aut(G, p) only contains the identity
automorphism of G. In particular, we have the following result.
Corollary 3.3.3 Let G be a graph, S be a symmetry group, and Φ be a map
from S to Aut(G). If the map p of a framework (G, p) ∈ R(G,S,Φ) is injective,
then (G, p) /∈ R(G,S,Ψ) for every Ψ : S → Aut(G) distinct from Φ.
Proof. Let α be an element of Aut(G, p). Then we have p(v) = p(α(v)
)
for all v ∈ V (G), and since p is injective it follows that v = α(v) for all
v ∈ V (G). Thus, α is the identity automorphism of G and the result follows
from Corollary 3.3.2. ¤
The following examples show that the converse of Corollary 3.3.3 does
not hold, that is, a framework (G, p) ∈ R(G,S) that is of a unique type
Φ : S → Aut(G) can possibly have a non-injective map p.
Example 3.3.3 The framework (G, p) in Figure 3.8 (a) is a non-injective
realization of (G, C2) (since p5 = p6) with Aut(G, p) = id. So, (G, p) ∈R(G,C2) is of the unique type Φ : C2 → Aut(G), where Φ(Id) = id and
Φ(C2) = (v1 v2)(v3 v4)(v5 v6).
Example 3.3.4 The framework (G, p) in Figure 3.8 (b) is a non-injective
realization of (G, C3) (since p4 = p5 = p6) with Aut(G, p) = id. So,
72
(G, p) ∈ R(G,C3) is of the unique type Φ : C3 → Aut(G), where Φ is the
homomorphism defined by Φ(C3) = (v1 v2 v3)(v4 v5 v6).
.
..p6
..p5
..p3 . .p4
..p1
. .p2
.(a)
...p5..p6
..p4
..p3
..p1 . .p2
.(b)
Figure 3.8: Non-injective realizations with Aut(G, p) = id.
Remark 3.3.1 Let (G, p) ∈ R(G,S,Φ) be a framework with Aut(G, p) = idand let (G, q) ∈ R(G,S,Φ) be an (S, Φ)-generic framework. It follows immedi-
ately from the definition of (S, Φ)-generic (Definition 3.2.2) that two joints
(vi, qi) and (vj, qj) of (G, q) can only satisfy qi = qj if pi = pj. This says that
(G, q) also satisfies Aut(G, q) = id. Therefore, by Corollary 3.3.2, being of
a unique type is an (S, Φ)-generic property.
Remark 3.3.2 If a framework (G, p) ∈ R(G,S) is of distinct types Φ1, . . . Φk,
where k ≥ 2, then (G, p) is not (S, Φt)-generic for some t ∈ 1, . . . , k, as the
following argument shows.
Suppose to the contrary that (G, p) is (S, Φi)-generic for all i = 1, . . . , k
and let l ∈ 1, . . . , k. Since Aut(G, p) 6= id, there exist vertices v 6= w
of G such that p(v) = p(w) and α(v) = w for some α ∈ Aut(G, p). Since
(G, p) is (S, Φl)-generic, there must exist non-trivial symmetry operations
x, y ∈ S such that Φl(x)(v) = v and Φl(y)(w) = w, and the symmetry
73
elements corresponding to x and y must be the origin 0 = p(v) = p(w). If
for each x ∈ S with Φl(x)(v) = v, we replace Φl(x) by α Φl(x), then we
obtain a map Φt, t 6= l, with the property that for all x ∈ S, Φt(x)(v) 6= v.
Thus, (G, p) is not (S, Φt)-generic, a contradiction.
As an example, consider the framework (Gt, p) in Figure 3.7 (a). (Gt, p)
is (C2, Θa)-generic, but not (C2, Θb)-generic, because p3 = p4 and Θb(v3) = v4
(see Example 3.3.1).
The framework in Figure 3.7 (b) is a realization of (Gbp, Cs) of type Ξa
and Ξb which is neither (Cs, Ξa)-generic nor (Cs, Ξb)-generic, because p4 = p5
(see Example 3.3.2).
3.4 When is a type Φ of a framework a ho-
momorphism?
We will see in the next chapter that in order to use techniques from
group representation theory to analyze the rigidity properties of a symmetric
framework (G, p) ∈ R(G,S,Φ), we need Φ to be a homomorphism. In this
section, we therefore investigate the natural question of whether a type Φ :
S → Aut(G) of a given framework (G, p) ∈ R(G,S) is in fact a homomorphism
(rather than just a map).
Theorem 3.4.1 Let S be a symmetry group and (G, p) be a framework in
R(G,S) with Aut(G, p) = id. Then the unique map Φ : S → Aut(G) for
which (G, p) ∈ R(G,S,Φ) is a homomorphism.
74
Proof. Let x and y be any two elements of S. Then Φ(y) Φ(x) ∈ Aut(G)
satisfies
(y x
)(p(v)
)= y
(p(Φ(x)(v)
))= p
((Φ(y) Φ(x)
)(v)
)for all v ∈ V (G)
and, by Corollary 3.3.2, Φ(y)Φ(x) is the only automorphism of G with this
property. Thus, Φ(y x) = Φ(y) Φ(x). ¤
In particular, it follows from Corollary 3.3.3 and Theorem 3.4.1 that if
the map p of (G, p) ∈ R(G,S) is injective, then the unique type Φ of (G, p) is
a group homomorphism.
Theorem 3.4.2 Let S be a symmetry group, Φ : S → Aut(G) be a map,
and (G, p) be a framework in R(G,S,Φ).
(i) If Φ is a homomorphism, then Φ(S) is a subgroup of Aut(G);
(ii) if Φ(S) is a subgroup of Aut(G) and Φ(x) = Φ(y) whenever Φ(y) ∈Φ(x)Aut(G, p), then Φ is a homomorphism.
Proof. (i) It is a standard result in algebra that the homomorphic image of
a group is again a group.
(ii) Let x and y be any two elements of S. By the same argument as in
the proof of Theorem 3.4.1, we have
(y x
)(p(v)
)= p
((Φ(y) Φ(x)
)(v)
)for all v ∈ V (G).
It follows from Theorem 3.3.1 that Φ(y x) ∈ (Φ(y) Φ(x)
)Aut(G, p).
By assumption, Φ(S) contains at most one element of each of the cosets
of Aut(G, p). Since Φ(S) is a group, the element of the coset(Φ(y)
75
Φ(x))Aut(G, p) that lies in Φ(S) must be Φ(y) Φ(x). It follows that
Φ(y x) = Φ(y) Φ(x) and the proof is complete. ¤
For a framework (G, p) ∈ R(G,S) with Aut(G, p) 6= id, there does not
necessarily exist any homomorphism Φ : S → Aut(G) for which (G, p) ∈R(G,S,Φ), as the following examples illustrate.
.
. . . .
.
.
....
.
.
.
.
.
.
.v2 .v3
.v4.v1
.(a)
.
.
.
.
.p2, p4
.p1, p3
.(b)
Figure 3.9: A graph G (a) and a realization (G, p) ∈ R(G,Cs) (b) for which
there does not exist a homomorphism Φ : Cs → Aut(G) so that (G, p) is of
type Φ.
Example 3.4.1 Consider the graph G and the 2-dimensional realization
(G, p) of G shown in Figure 3.9 (a) and (b), respectively. Let s be the
reflection whose mirror line is shown in Figure 3.9 (b). All vertices of G
that are illustrated with the same color in Figure 3.9 have the same image
under p. Observe that the ‘14-turn-automorphism’ σ of G that permutes
the vertices v1, v2, v3, and v4 according to the cycle (v1 v2 v3 v4) satisfies
s(p(vi)
)= p
(σ(vi)
)for all vi ∈ V (G). Thus, s is a symmetry operation
of (G, p), and hence (G, p) is an element of R(G,Cs), where Cs = Id, s.Note that Aut(G, p) = id, σ2. Therefore, by Theorem 3.3.1, id and σ
76
are the two automorphisms of G that can turn Id ∈ Cs into a symmetry oper-
ation of (G, p). Similarly, either one of the elements of σAut(G, p) = σ, σ3can turn s ∈ Cs into a symmetry operation of (G, p). It now follows from The-
orem 3.4.2 (i) that there does not exist any homomorphism Φ : Cs → Aut(G)
such that (G, p) ∈ R(G,Cs) is of type Φ, because we cannot choose two ele-
ments, one from each of the cosets Aut(G, p) and σAut(G, p), that form a
subgroup of Aut(G).
.
..v1
..v2
..v3
..v4 .
.v5
..v6
..v7
. .v8
..v9
.(a)
.
. .
.p1, p4, p7
.p2, p5, p8 .p3, p6, p9
.(b)
Figure 3.10: A graph G (a) and a realization (G, p) ∈ R(G,C3) (b) for which
there does not exist a homomorphism Φ : C3 → Aut(G) so that (G, p) is of
type Φ.
Example 3.4.2 Consider the graph G and the 2-dimensional realization
(G, p) of G shown in Figure 3.10 (a) and (b), respectively. As in the previous
example, all vertices of G that are illustrated with the same color in Figure
3.10 have the same image under p. Note that (G, p) is an element of R(G,C3),
where C3 = Id, C3, C23 is a symmetry group in dimension 2, because the
automorphism γ = (v1 v2 . . . v9) of G satisfies C3
(p(vi)
)= p
(γ(vi)
)for all
vi ∈ V (G) and γ2 satisfies C23
(p(vi)
)= p
(γ2(vi)
)for all vi ∈ V (G).
We have Aut(G, p) = id, γ3, γ6, and hence γAut(G, p) = γ, γ4, γ7 and
77
γ2Aut(G, p) = γ2, γ5, γ8. Since C3 ∈ C3 has order 3 and each element in
γAut(G, p) has order 9 it follows that there does not exist any homomorphism
Φ : C3 → Aut(G) such that (G, p) is of type Φ.
Note that Examples 3.4.1 and 3.4.2 can easily be extended to obtain
further examples of frameworks (G, p) and symmetry groups S with the
property that there exists no homomorphism Φ : S → Aut(G) for which
(G, p) ∈ R(G,S,Φ).
78
Chapter 4
Using group representation
theory to analyze symmetric
frameworks
It is a common method in engineering, physics, and chemistry to apply
techniques from group representation theory to the analysis of symmetric
structures (see, for example, [26, 27, 34, 35, 44, 45]). In particular, some
recent papers have used these techniques to gain insight into the rigidity
properties of symmetric frameworks consisting of rigid bars and flexible joints
[15, 25, 43, 44, 53].
One of the fundamental observations resulting from this approach for
studying the rigidity of symmetric frameworks is due to R. Kangwai and S.
Guest ([44]): given a symmetric framework (G, p) and a non-trivial subgroup
S of its point group, there are techniques to block-diagonalize the rigidity
matrix of (G, p) into submatrix blocks in such a way that each block corre-
79
sponds to an irreducible representation of S. A number of interesting and
useful results concerning the rigidity of symmetric frameworks are based on
this block-diagonalization of the rigidity matrix [15, 25, 43]. However, since
the main focus of the work in [44], as well as in [15], [25], and [43], lies
on applications in engineering and chemistry, many of these results are not
presented with a mathematically precise formulation nor with a complete
mathematical verification.
In this chapter, we establish several major results. First, in Section 4.1,
we use the mathematical foundation we established in the previous chapter
to give a complete proof for the fact that the rigidity matrix of a symmetric
framework can be block-diagonalized in the way described above. Fundamen-
tal to this proof are our mathematically explicit definitions for the ‘external’
and ‘internal’ representation which were introduced in [25] and [44] only by
means of an example, and Lemma 4.1.1 which establishes the key connection
between these two representations.
Secondly, in Section 4.2, we apply the results of Section 4.1 to give a de-
tailed mathematical proof for the symmetry-extended version of Maxwell’s
rule given in [25]. This rule provides further necessary conditions (in ad-
dition to Maxwell’s original condition given in Theorem 2.2.7) for a sym-
metric framework to be isostatic. While the symmetry-extended version of
Maxwell’s rule, as formulated in [25], is only applicable to 2- or 3-dimensional
frameworks with injective configurations, we establish a more general result
in Section 4.2, namely a rule that can be applied to both injective and non-
injective realizations in all dimensions. The proof of this result is based on
Theorem 4.2.2 which in turn relies on the fact that the rigidity matrix of a
80
symmetric framework can be block-diagonalized as described in Section 4.1.
An alternate approach to proving the symmetry-extended version of
Maxwell’s rule in [25], as well as various generalizations of this rule to other
types of geometric constraint systems, is presented in [53] (see also Chapter
7).
In order to apply the symmetry-extended version of Maxwell’s rule to a
given framework (G, p), it is necessary to determine the dimensions of the
subspaces of infinitesimal rigid motions of (G, p) that are invariant under
the external representation. While in [25], the question of how to find the
dimensions of these subspaces is only briefly addressed and not answered
completely from a mathematical point of view (in particular, for all frame-
works in dimensions higher than 3, this question is not addressed at all), in
Section 4.2, we describe in detail how to determine the dimensions of these
subspaces for an arbitrary-dimensional framework.
The results of Sections 4.1 and 4.2 will also be presented in the paper
[56].
Since in [25] and [44], the rigidity properties of a symmetric framework
are studied from both the kinematic and static point of view simultaneously,
we develop the corresponding mathematical theory in this chapter in the
same manner.
In Section 4.3, we use the symmetry-extended version of Maxwell’s rule
to show that a symmetric isostatic framework in 2D or 3D must obey some
very simply stated restrictions on the number of structural elements that
are ‘fixed’ by various symmetry operations of the framework. In particular,
it turns out that a 2-dimensional isostatic framework must belong to one
81
of only six point groups. For 3-dimensional isostatic frameworks, all point
groups are possible. However, there still exist restrictions on the placement
of structural components. While analogous restrictions on the number of
‘fixed’ structural components can be established for symmetric frameworks
in an arbitrary dimension using the results of Section 4.2, we focus our atten-
tion on frameworks in dimensions 2 and 3, since they are of special interest
for current applications.
Most of the results in Section 4.3 appeared in the joint paper [15]. The
derivations also appeared there, and Sections 4.1 and 4.2 now provide a proof
that these methods are correct.
Finally, in Section 4.4 we use the results of Sections 4.1 and 4.2 to estab-
lish necessary conditions for a symmetric framework to be independent or
infinitesimally rigid.
4.1 Block-diagonalization of the rigidity ma-
trix
4.1.1 Basic definitions in group representation theory
We need the following notions from group representation theory.
Definition 4.1.1 Let S be a group and V be an n-dimensional vector space
over the field F . A linear representation of S with representation space V
is a group homomorphism H from S to GL(V ), where GL(V ) denotes the
group of all automorphisms of V . The dimension n of V is called the degree
of H.
82
Two linear representations H1 : S → GL(V1) and H2 : S → GL(V2) are
said to be equivalent if there exists an isomorphism h : V1 → V2 such that
h H1(x) h−1 = H2(x) for all x ∈ S.
Definition 4.1.2 Let S be a group, V be a vector space over the field F
and H : S → GL(V ) be a linear representation of S. A subspace U of V is
said to be H-invariant (or simply invariant if H is clear from the context)
if H(x)(U) ⊆ U for all x ∈ S. H is called irreducible if V and 0 are the
only H-invariant subspaces of V .
Note that the property of irreducibility depends on the field F . Since
we only consider frameworks in the real vector space Rd, the representation
space of any linear representation in this thesis is assumed to be a real vector
space.
In the examples throughout this thesis we use the Mulliken symbols (see
Appendix A or [6, 19, 37]) to denote the irreducible representations of a given
group. This is one of the standard notations in group representation theory
and its applications.
Definition 4.1.3 A linear representation H : S → GL(V ) is said to be
unitary with respect to a given inner product 〈v, w〉 if
〈H(x)(v), H(x)(w)〉 = 〈v, w〉 for all v, w ∈ V and all x ∈ S.
Remark 4.1.1 A unitary representation has the property that the orthog-
onal complement of an invariant subspace is again invariant [60].
Definition 4.1.4 Let H : S → GL(V ) be a linear representation of a group
S and let U be an invariant subspace of V . If for all x ∈ S, we restrict the
83
automorphism H(x) of V to the subspace U , then we obtain a new linear
representation H(U) of S with representation space U . H(U) is said to be a
subrepresentation of H.
Definition 4.1.5 Let H1 : S → GL(V1) and H2 : S → GL(V2) be two
linear representations of a group S. Then H1 ⊕ H2 : S → GL(V1 ⊕ V2)
is the representation of S which sends x ∈ S to H1 ⊕ H2(x), where H1 ⊕H2(x)
((v1, v2)
)=
(H1(x)(v1), H2(x)(v2)
)for all v1 ∈ V1 and v2 ∈ V2.
Definition 4.1.6 Let S be a group and F be a field. A matrix representation
of S is a homomorphism H from S to GL(n, F ), where GL(n, F ) denotes the
group of all invertible n× n matrices with entries in F .
Two matrix representations H1 : S → GL(n, F ) and H2 : S → GL(n, F )
are said to be equivalent if there exists an invertible matrix M such that
MH1(x)M−1 = H2(x) for all x ∈ S, in which case we write H1 w H2.
Let S be a group, V be an n-dimensional vector space over the field F ,
and H : S → GL(V ) be a linear representation of S. Given a basis B of
V , we may associate a matrix representation HB : S → GL(n, F ) to H by
defining HB(x) to be the matrix that represents the automorphism H(x)
with respect to the basis B for all x ∈ S. HB is then said to correspond
to H with respect to B. Note that two matrix representations H1 and H2
correspond to equivalent linear representations if and only if H1 w H2.
84
4.1.2 The internal and external representation
Given a graph G, a symmetry group S, and a homomorphism Φ : S →Aut(G), we define two particular matrix representations of S, the external
and the internal representation, both of which depend on G and Φ. These
two representations play the key role in a symmetry-based rigidity analysis
of a framework (G, p) ∈ R(G,S,Φ). Note that our definitions of these represen-
tations are mathematically explicit definitions of the external and internal
representation introduced in [25] and [44].
Definition 4.1.7 Let G be a graph with V (G) = v1, v2, . . . , vn and
E(G) = e1, e2, . . . , em, S be a symmetry group in dimension d, and Φ
be a homomorphism from S to Aut(G). For x ∈ S, let Mx denote the or-
thogonal d× d matrix which represents x with respect to the canonical basis
of Rd.
The external representation of S (with respect to G and Φ) is the matrix
representation He : S → GL(dn,R) that sends x ∈ S to the matrix He(x)
which is obtained from the transpose of the n×n permutation matrix corre-
sponding to Φ(x) (with respect to the enumeration V (G) = v1, v2, . . . , vn)by replacing each 1 with the matrix Mx and each 0 with a d× d zero-matrix.
The internal representation of S (with respect to G and Φ) is the matrix
representation Hi : S → GL(m,R) that sends x ∈ S to the transpose of the
permutation matrix corresponding to the permutation of E(G) (with respect
to the enumeration E(G) = e1, e2, . . . , em) which is induced by Φ(x).
Remark 4.1.2 It is easy to verify that both the external representation He
and the internal representation Hi of S (with respect to G and Φ) are in fact
85
matrix representations of the group S, provided that Φ is a homomorphism.
If, however, Φ is not a homomorphism, then He and Hi are also not homo-
morphisms, in which case neither He nor Hi is a matrix representation of the
group S.
Example 4.1.1 To illustrate the previous definition, let K3 be the com-
plete graph with V (K3) = v1, v2, v3 and E(K3) = e1, e2, e3, where
e1 = v1, v2, e2 = v1, v3 and e3 = v2, v3. Further, let Cs = Id, sbe the symmetry group in dimension 2 with
MId =
1 0
0 1
and Ms =
−1 0
0 1
,
and let Φ : Cs → Aut(K3) be the homomorphism defined by Φ(s) =
(v1 v2)(v3). Then we have
He(Id) =
1 0 0 0 0 0
0 1 0 0 0 0
0 0 1 0 0 0
0 0 0 1 0 0
0 0 0 0 1 0
0 0 0 0 0 1
, He(s) =
0 0 −1 0 0 0
0 0 0 1 0 0
−1 0 0 0 0 0
0 1 0 0 0 0
0 0 0 0 −1 0
0 0 0 0 0 1
,
Hi(Id) =
1 0 0
0 1 0
0 0 1
, Hi(s) =
1 0 0
0 0 1
0 1 0
.
For further examples, see [44] or [45].
86
...p1
..p2
..p3
.e1
.e3.e2
Figure 4.1: A framework (K3, p) ∈ R(K3,Cs,Φ).
4.1.3 The block-diagonalization
In this section, we use the mathematically explicit definitions of the ex-
ternal and internal representation from the previous section to prove that the
rigidity matrix of a symmetric framework can be transformed into a block-
diagonalized form. Basic to this proof is Lemma 4.1.1 which discloses the
essential mathematical connection between the external and internal repre-
sentation.
Recall from Section 2.2 that in the study of infinitesimal rigidity, we
consider the equation
R(G, p)u = z,
where R(G, p) is the rigidity matrix of a framework (G, p), u ∈ Rd|V (G)| is a
column vector that represents an assignment of d-dimensional displacement
vectors to the joints of (G, p), and z ∈ R|E(G)| is the column vector that
represents the distortions in the bars of (G, p) that are induced by u. The
component of z that corresponds to the edge vi, vj of G is also known as
the strain induced on the bar (vi, pi), (vj, pj) by u.
87
Similarly, in the study of static rigidity, we consider the equation
R(G, p)T ω = l,
where the column vector ω ∈ R|E(G)| is a stress of (G, p) and the column
vector l ∈ Rd|V (G)| is the load on (G, p) which is resolved by ω.
Now, suppose (G, p) is a symmetric framework in the set R(G,S,Φ), where S
is a symmetry group in dimension d and Φ : S → Aut(G) is a homomorphism.
Then, using the notation of Definition 4.1.7, and assuming that the ith row
of the rigidity matrix R(G, p) of (G, p) corresponds to the edge ei of G,
we have the following fundamental property of the external and internal
representation of S (with respect to G and Φ).
Lemma 4.1.1 Let G be a graph, S be a symmetry group, Φ be a homomor-
phism from S to Aut(G), and p ∈ ⋂x∈S Lx,Φ.
(i) If R(G, p)u = z, then for all x ∈ S, we have R(G, p)He(x)u = Hi(x)z;
(ii) if R(G, p)T ω = l, then for all x ∈ S, we have R(G, p)T Hi(x)ω =
He(x)l.
Proof. (i) Suppose R(G, p)u = z. Fix x ∈ S and let Mx be the orthogonal
matrix representing x with respect to the canonical basis of Rd. Also, let
Φ(x)(vi) = vk and Φ(x)(vj) = vl, and let ef = vi, vj and eh = vk, vl.Then, since p ∈ ⋂
x∈S Lx,Φ, we have
Mxpi = pk and Mxpj = pl.
By the definition of Hi(x), we have
(Hi(x)z
)h
= (z)f .
88
...pi
..pj
. .Mxpi = pk
. .Mxpj = pl
.ef .eh
.Mx
.Mx
.
.
.(z)h
.uk
.ul
.Hi(x)
.He(x)
.He(x)
.
.
.(z)f
.Mxui
.Mxuj
Figure 4.2: Illustration of the proof of Lemma 4.1.1 (i).
Similarly, it follows from the definition of He(x) that if u ∈ Rdn is replaced
by He(x)u, then uk ∈ Rd is replaced by Mxui and ul ∈ Rd by Mxuj. By the
definition of R(G, p), we have
(R(G, p)u
)h
= (z)h = (pk − pl) · uk + (pl − pk) · ul.
Therefore,
(R(G, p)He(x)u
)h
= (pk − pl) ·Mxui + (pl − pk) ·Mxuj
=(Mxpi −Mxpj
) ·Mxui +(Mxpj −Mxpi
) ·Mxuj
=(Mx(pi − pj)
) ·Mxui +(Mx(pj − pi)
) ·Mxuj
= (pi − pj) · ui + (pj − pi) · uj
= (z)f .
The penultimate equality sign is valid because the canonical inner product
on Rd is invariant under the orthogonal transformation x ∈ S. This proves
(i).
(ii) Suppose R(G, p)T ω = l. Fix x ∈ S and let Φ(x)(vi) = vk. Then,
since p ∈ ⋂x∈S Lx,Φ, we have
Mxpi = pk.
Let vi1 , vi2 , . . . , vij be the vertices in V (G) that are adjacent to vi, and let
eft = vi, vit for t = 1, 2, . . . , j. Further, choose an enumeration of the j
89
...pi
..pj
..Mxpi.= pk
..Mxpj.= pl
.(z)f .(z)h
.ui
.uj
.uk
.ul
.x
..pi
..pj
..pk
. .pl
.(z)h .(z)f
.Mxuk.Mxui
.Mxul
.Mxuj
.x
Figure 4.3: Illustration of the proof of Lemma 4.1.1 (i) in the case where x
is a reflection.
vertices that are adjacent to vk in such a way that
Mxpit = pkt ,
and let eht = vk, vkt for t = 1, 2, . . . , j. For the vertex vk, the equation
R(G, p)T ω = l yields the vector-equation
(pk − pk1)(ω)h1 + . . . + (pk − pkj)(ω)hj
= lk. (4.1)
...pi . .Mxpi = pk
..pi1
..pij
.ef1
.efj
..Mxpi1 = pk1
..Mxpij = pkj
.eh1
.ehj
.Mx
.Mx
.Mx
..lk
.(ω)h1
.(ω)hj
. .Mxli
.(ω)f1
.(ω)fj
.He(x)
.Hi(x)
.Hi(x)
Figure 4.4: Illustration of the proof of Lemma 4.1.1 (ii).
If l ∈ Rdn is replaced by He(x)l, then on the right-hand side of equation
(4.1), lk ∈ Rd is replaced by Mxli and if ω is replaced by Hi(x)ω, then the
90
left-hand side of equation (4.1) is replaced by
(pk − pk1)(ω)f1 + . . . + (pk − pkj)(ω)fj
=(Mxpi −Mxpi1
)(ω)f1 + . . . +
(Mxpi −Mxpij
)(ω)fj
= Mx
((pi − pi1)(ω)f1 + . . . + (pi − pij)(ω)fj
)
= Mxli.
This completes the proof. ¤
In the following, we again let G be a graph with V (G) = v1, v2, . . . , vnand E(G) = e1, e2, . . . , em, S be a symmetry group in dimension d, and Φ
be a homomorphism from S to Aut(G).
Let He be the external and Hi be the internal representation of S (with
respect to G and Φ). Then we let H ′e : S → GL(Rdn) be the linear represen-
tation of S that sends x ∈ S to the automorphism H ′e(x) which is represented
by the matrix He(x) with respect to the canonical basis of the R-vector space
Rdn. Similarly, we let H ′i : S → GL(Rm) be the linear representation of S
that sends x ∈ S to the automorphism H ′i(x) which is represented by the
matrix Hi(x) with respect to the canonical basis of the R-vector space Rm.
So, the external representation He corresponds to the linear representation
H ′e with respect to the canonical basis of Rdn and the internal representation
Hi corresponds to the linear representation H ′i with respect to the canonical
basis of Rm.
From group representation theory we know that every finite group has, up
to equivalency, only finitely many irreducible linear representations and that
every linear representation of such a group can be written uniquely, up to
equivalency of the direct summands, as a direct sum of the irreducible linear
91
representations of this group [42, 60]. So, let S have r pairwise non-equivalent
irreducible linear representations I1, I2, . . . , Ir and let
H ′e = λ1I1 ⊕ . . .⊕ λrIr, where λ1, . . . , λr ∈ N ∪ 0. (4.2)
For each t = 1, . . . , r, there exist λt subspaces(V
(It)e
)1, . . . ,
(V
(It)e
)λt
of the
R-vector space Rdn which correspond to the λt direct summands in (4.2), so
that
Rdn = V (I1)e ⊕ . . .⊕ V (Ir)
e , (4.3)
where
V (It)e =
(V (It)
e
)1⊕ . . .⊕ (
V (It)e
)λt
. (4.4)
Let(B
(It)e
)1, . . . ,
(B
(It)e
)λt
be bases of the subspaces in (4.4). Then
B(It)e =
(B(It)
e
)1∪ . . . ∪ (
B(It)e
)λt
is a basis of V(It)e and
Be = B(I1)e ∪ . . . ∪B(Ir)
e (4.5)
is a basis of the R-vector space Rdn.
Consider now the matrix representation He that corresponds to the linear
representation H ′e with respect to the basis Be. For x ∈ S, we have
He(x) = T−1e He(x)Te,
where the ith column of Te is the coordinate vector of the ith basis vector
of Be relative to the canonical basis, that is, Te is the matrix of the basis
transformation from the canonical basis of the R-vector space Rdn to the
basis Be. The column vectors of He(x) are the coordinates of the images of
92
the basis vectors in Be under H ′e(x) relative to the basis Be. So, for each
x ∈ S, the matrix He(x) has the same block form, namely
He(x) =
(A
(I1)e
)1(x)
. . . 0(A
(I1)e
)λ1
(x)
. . .(A
(Ir)e
)1(x)
0. . .
(A
(Ir)e
)λr
(x)
.
The block-matrix(A
(It)e
)j(x) represents the restriction of the linear trans-
formation H ′e(x) to the subspace
(V
(It)e
)j
with respect to the basis(B
(It)e
)j.
Since for a given t, each of the subspaces(V
(It)e
)j, j = 1, . . . , λt, corresponds
to the same irreducible linear representation It, we can choose the bases of
the subspaces(V
(It)e
)j
in such a way that
(A(It)
e
)1(x) = . . . =
(A(It)
e
)λt
(x) =: A(It)e (x).
In the following we assume that the basis Be is chosen in this way.
The above observations about the linear representation H ′e of S can be
transferred analogously to the linear representation H ′i of S. Let the direct
sum decomposition of H ′i be given by
H ′i = µ1I1 ⊕ . . .⊕ µrIr, where µ1, . . . , µr ∈ N ∪ 0. (4.6)
For each t = 1, . . . , r, there exist µt subspaces(V
(It)i
)1, . . . ,
(V
(It)i
)µt
of the
R-vector space Rm which correspond to the µt direct summands in (4.6), so
that
Rm = V(I1)i ⊕ . . .⊕ V
(Ir)i , (4.7)
93
where
V(It)i =
(V
(It)i
)1⊕ . . .⊕ (
V(It)i
)µt
. (4.8)
Let(B
(It)i
)1, . . . ,
(B
(It)i
)µt
be bases of the subspaces in (4.8). Then
B(It)i =
(B
(It)i
)1∪ . . . ∪ (
B(It)i
)µt
is a basis of V(It)i and
Bi = B(I1)i ∪ . . . ∪B
(Ir)i
is a basis of the R-vector space Rm.
Consider now the matrix representation Hi that corresponds to the linear
representation H ′i with respect to the basis Bi. Let Ti be the matrix of the
basis transformation from the canonical basis of the R-vector space Rm to
the basis Bi. Then for x ∈ S, we have
Hi(x) = T−1i Hi(x)Ti.
So, the matrix Hi(x) has the same block form for each x ∈ S, namely
Hi(x) =
(A
(I1)i
)1(x)
. . . 0(A
(I1)i
)µ1
(x)
. . .(A
(Ir)i
)1(x)
0. . .
(A
(Ir)i
)µr
(x)
,
and for each t = 1, 2, . . . , r, we can choose the bases of the subspaces(V
(It)i
)j
in such a way that
(A
(It)i
)1(x) = . . . =
(A
(It)i
)µt
(x) =: A(It)i (x) = A(It)
e (x).
94
In the following we assume that Bi is chosen in this way.
Definition 4.1.8 With the notation above, we say that a vector v ∈ Rdn
is symmetric with respect to the irreducible linear representation It of S if
v ∈ V(It)e . Similarly, we say that a vector w ∈ Rm is symmetric with respect
to the irreducible linear representation It of S if w ∈ V(It)i .
We are now in the position to state the fundamental theorem for analyzing
the rigidity properties of a symmetric framework using group representation
theory.
Theorem 4.1.2 Let G be a graph, S be a symmetry group with pairwise non-
equivalent irreducible linear representations I1, . . . , Ir, Φ be a homomorphism
from S to Aut(G), and p ∈ ⋂x∈S Lx,Φ.
(i) If R(G, p)u = z and u is symmetric with respect to It, then z is also
symmetric with respect to It;
(ii) if R(G, p)T ω = l and ω is symmetric with respect to It, then l is also
symmetric with respect to It.
Proof. (i) Suppose S is a symmetry group in dimension d and G is a graph
with n vertices. Let u ∈ (V
(It)e
)j. By the direct sum decomposition of V
(It)e
in (4.4), the result follows if we can show that z = R(G, p)u ∈ V(It)i . By
the decomposition of R|E(G)| into direct summands in (4.8), z has a unique
decomposition of the form
z =r∑
α=1
µα∑
β=1
zα,β, where zα,β ∈(V
(Iα)i
)β.
95
We now interpret R(G, p) : Rdn → R|E(G)| as a linear transformation and
for given m and k, we define the projection map Rm,k corresponding to
R(G, p)|(V
(It)e
)j
by
Rm,k :
(V
(It)e
)j→ (
V(Im)i
)k
u 7→ zm,k
.
We need to show that for all m 6= t, Rm,k is the zero map. So, let m 6= t.
Clearly, Rm,k is a linear transformation.
The image of Rm,k is an H ′i-invariant subspace of
(V
(Im)i
)k, as the fol-
lowing argument shows. Fix x ∈ S and let z′ be in the image of Rm,k, say
z′ = Rm,k(u′). Then, by assumption, H ′
e(x)(u′) ∈ (V
(It)e
)j
and, by Lemma
4.1.1 (i), H ′i(x)(z′) is the image of H ′
e(x)(u′) under Rm,k.
Since Im is an irreducible linear representation of S,(V
(Im)i
)k
and 0are the only H ′
i-invariant subspaces of(V
(Im)i
)k. If the image of Rm,k is the
null-space, then we are done, otherwise Rm,k is surjective.
Next, we show that the kernel of Rm,k is an H ′e-invariant subspace of
(V
(It)e
)j. Fix x ∈ S and let u′ be in the kernel of Rm,k, that is, Rm,k(u
′) = 0.
Then, again by Lemma 4.1.1 (i), the image of H ′e(x)(u′) under Rm,k is
H ′i(x)(0) = 0, and hence H ′
e(x)(u′) is also in the kernel of Rm,k.
Since It is an irreducible linear representation of S, we either have
ker (Rm,k) =(V
(It)e
)j, in which case we are done, or ker (Rm,k) = 0,
in which case Rm,k is injective.
So, assume Rm,k is bijective. Let the matrix that represents Rm,k with
respect to the bases(B
(It)e
)jand
(B
(Im)i
)k
be denoted by Rm,k. Then Rm,k is
an invertible matrix. Let u be the coordinate vector of an element in(V
(It)e
)j
relative to the basis(B
(It)e
)j
and let z be the coordinate vector of the image
96
of u under Rm,k relative to the basis(B
(Im)i
)k. Then, by Lemma 4.1.1 (i),
for any x ∈ S, we have
Rm,k
(A(It)
e
)j(x)u =
(A
(Im)i
)k(x)z =
(A
(Im)i
)k(x)Rm,ku,
and hence also
Rm,k
(A(It)
e
)j(x) =
(A
(Im)i
)k(x)Rm,k.
Therefore,
Rm,k
(A(It)
e
)j(x)R−1
m,k =(A
(Im)i
)k(x) =
(A(Im)
e
)k(x) for all x ∈ S,
which says that It and Im are equivalent representations, a contradiction.
This completes the proof of part (i).
With the help of Lemma 4.1.1 (ii), part (ii) can be proved completely
analogously to part (i). ¤
Theorem 4.1.2 (i) says that if u ∈ Rdn is an assignment of displacement
vectors to the joints of a framework (G, p) ∈ R(G,S,Φ) and u is symmetric
with respect to It, then the strains induced on the bars of (G, p) by u must
also be symmetric with respect to It. Similarly, Theorem 4.1.2 (ii) says that
if ω is a resolution of an equilibrium load l on (G, p) ∈ R(G,S,Φ) and ω is
symmetric with respect to It, then l must also be symmetric with respect to
It.
An immediate consequence of Theorem 4.1.2 is that the matrices R(G, p)
and R(G, p)T can be block-diagonalized in such a way that the original rigid-
ity problems R(G, p)u = z and R(G, p)T ω = l are decomposed into subprob-
lems, where each subproblem considers, respectively, the relationship between
vectors u and z and vectors ω and l that are symmetric with respect to the
same irreducible linear representation It. This is specified in
97
Corollary 4.1.3 Let G be a graph, S be a symmetry group with pair-
wise non-equivalent irreducible linear representations I1, . . . , Ir, Φ be a ho-
momorphism from S to Aut(G), and p ∈ ⋂x∈S Lx,Φ. Then the matrices
T−1i R(G, p)Te and T−1
e R(G, p)T Ti are block-diagonalized in such a way that
there exists (at most) one submatrix block for each irreducible linear repre-
sentation It of S.
Proof. Suppose R(G, p)u = z, and let u be the coordinate vector of u rela-
tive to the basis Be and z be the coordinate vector of z relative to the basis Bi.
Further, let R(G, p) be the matrix that represents the linear transformation
R(G, p) with respect to the bases Be and Bi, that is,
R(G, p) = T−1i R(G, p)Te.
Then, by changing coordinates relative to the canonical bases of Rdn and Rm
into coordinates relative to the bases Be and Bi, the equation
R(G, p)u = z
is converted into the equation
R(G, p)u = z.
By Theorem 4.1.2 (i), the matrix R(G, p) is block-diagonalized in such a way
that there exists (at most) one submatrix block for each irreducible linear
representation It of S and the submatrix block corresponding to It is a matrix
of the size dim(V
(It)i
) × dim(V
(It)e
). In particular, a submatrix block can
possibly be an ‘empty matrix’ which has rows but no columns or alternatively
columns but no rows.
98
Similarly, if we denote ω to be the coordinate vector of ω relative to the
basis Bi, l to be the coordinate vector of l relative to the basis Be, and
R(G, p)T = T−1e R(G, p)T Ti,
then we may carry out the same changes of coordinates as above to convert
the equation
R(G, p)T ω = l
into the equation
R(G, p)T ω = l.
By Theorem 4.1.2 (ii), the matrix R(G, p)T is again block-diagonalized in
such a way that there exists (at most) one block for each It. ¤
Remark 4.1.3 Note that the matrix R(G, p)T is equal to the transpose of
the matrix R(G, p) if and only if both of the matrices Te and Ti are orthogonal
matrices (i.e., T−1e = T T
e and T−1i = T T
i ) if and only if both Be and Bi
are orthonormal bases. Since the external and internal representation are
both unitary representations (for all x ∈ S, He(x) and Hi(x) are orthogonal
matrices), the invariant subspaces in (4.3) and (4.7) are mutually orthogonal
(see [24, 60], for example). Thus, Be and Bi can always be chosen to be
orthonormal.
Example 4.1.2 Let K3, Cs = Id, s, and Φ be as in Example 4.1.1 and
consider the framework (K3, p) ∈ R(K3,Cs,Φ) shown in Figures 4.1 and 4.5,
where
p1 =
−1
0
, p2 =
1
0
, and p3 =
0
2
.
99
The rigidity matrix of (K3, p) is given by
R(K3, p) =
(p1 − p2)1 (p1 − p2)2 (p2 − p1)1 (p2 − p1)2 0 0
(p1 − p3)1 (p1 − p3)2 0 0 (p3 − p1)1 (p3 − p1)2
0 0 (p2 − p3)1 (p2 − p3)2 (p3 − p2)1 (p3 − p2)2
=
−2 0 2 0 0 0
−1 −2 0 0 1 2
0 0 1 −2 −1 2
.
The symmetry group Cs has two non-equivalent irreducible linear repre-
sentations both of which are of degree 1. In the Mulliken notation, they are
denoted by A′ and A′′. A′ maps both Id and s to the identity transforma-
tion, whereas A′′ maps Id to the identity transformation and s to the linear
transformation A′′(s) which is defined by A′′(s)(x) = −x for all x ∈ R. We
have
R6 = V (A′)e ⊕ V (A′′)
e
and
R3 = V(A′)i ⊕ V
(A′′)i .
It is easy to see that the elements of the subspace V(A′)e of R6 are of the form
u1
u2
−u1
u2
0
u3
, where u1, u2, u3 ∈ R,
100
(see Figure 4.5 (a)), so that an orthonormal basis B(A′)e of V
(A′)e is given by
B(A′)e =
1√2
0
− 1√2
0
0
0
,
0
1√2
0
1√2
0
0
,
0
0
0
0
0
1
.
Similarly, the elements of the subspace V(A′′)e of R6 are of the form
u1
u2
u1
−u2
u3
0
, where u1, u2, u3 ∈ R,
(see Figure 4.5 (b)), so that an orthonormal basis B(A′′)e of V
(A′′)e is given by
B(A′′)e =
1√2
0
1√2
0
0
0
,
0
1√2
0
− 1√2
0
0
,
0
0
0
0
1
0
.
Orthonormal bases B(A′)i and B
(A′′)i for the subspaces V
(A′)i and V
(A′′)i of R3
101
can be found analogously (see Figure 4.5 (c), (d)). We let
B(A′)i =
1
0
0
,
0
1√2
1√2
.
and
B(A′′)i =
0
1√2
− 1√2
.
Therefore, we have
...p1
..p2
..p3
.e1
.e3.e2
.
(u1
u2
).
(−u1
u2
)
.
(0u3
)
.(a)
..p1
..p2
..p3
.e1
.e3.e2
.z2
.z1
.z2
.(c)
...p1
..p2
. .p3
.e1
.e3.e2
.
(u1
u2
)
.
(u1
−u2
)
.
(u3
0
)
.(b)
..p1
..p2
..p3
.e1
.e3.e2
.z1
.0
.−z1
.(d)
Figure 4.5: (a, b) Vectors of the H ′e-invariant subspaces V
(A′)e (a) and V
(A′′)e
(b) of R6; (c, d) vectors of the H ′i-invariant subspaces V
(A′)i (c) and V
(A′′)i
(d) of R3.
102
Te =
1√2
0 0 1√2
0 0
0 1√2
0 0 1√2
0
− 1√2
0 0 1√2
0 0
0 1√2
0 0 − 1√2
0
0 0 0 0 0 10 0 1 0 0 0
and
Ti =
1 0 00 1√
21√2
0 1√2− 1√
2
.
Thus,
R(K3, p) = T−1i R(K3, p)Te =
−2√
2 0 0 0 0 0
−1 −2 2√
2 0 0 0
0 0 0 −1 −2√
2
and
R(K3, p)T = T−1e R(K3, p)T Ti =
−2√
2 −1 00 −2 0
0 2√
2 00 0 −10 0 −2
0 0√
2
.
Remark 4.1.4 In the previous example, we were able to find the invariant
subspaces V(A′)e , V
(A′′)e of R6 and V
(A′)i , V
(A′′)i of R3 by inspection because Cs is
a small symmetry group with only two elements. This is of course generally
not possible. There are, however, some standard methods and algorithms
for finding the symmetry adapted bases Be and Bi for any given symmetry
group. Good sources for these methods are [24, 50], for example.
As we will see in Section 4.2, knowledge of only the sizes of the subma-
trix blocks that appear in the block-diagonalized rigidity matrices of a given
symmetric framework allows us to gain significant insight into the rigidity
properties of the framework. Since, with the aid of character theory, the
103
sizes of these submatrix blocks can be determined very easily without ex-
plicitly finding the bases Be and Bi, there exist a number of applications of
Corollary 4.1.3 (such as the symmetry-extended version of Maxwell’s rule we
will discuss in the following sections) that do not require finding the block-
diagonalized rigidity matrices explicitly.
Remark 4.1.5 The matrices R(G, p)TR(G, p) and R(G, p)R(G, p)T are also
of interest in some areas of rigidity theory [17, 44]. In structural engi-
neering, these matrices are called the stiffness matrix and the flexibility
matrix, respectively. It follows immediately from Corollary 4.1.3 that if
p ∈ ⋂x∈S Lx,Φ, then these matrices can also be block-diagonalized in such a
way that there exists (at most) one block for each irreducible representation
It of S. In fact, it is easy to see that the matrices T−1e R(G, p)TR(G, p)Te
and T−1i R(G, p)R(G, p)T Ti have the desired block-form.
In the following sections of this chapter, as well as in Chapter 6, we will
discuss some interesting applications of the fact that the rigidity matrix of
a symmetric framework can be block-diagonalized in the way described in
Corollary 4.1.3.
4.2 A symmetry-extended version of
Maxwell’s rule
Recall from Section 2.2.5 that Maxwell’s rule (Theorem 2.2.7) gives a
necessary condition for a d-dimensional framework (G, p) to be isostatic. If
104
(G, p) is a 2- or 3-dimensional symmetric framework with an injective con-
figuration, then the symmetry-extended version of Maxwell’s rule given in
[25] provides further necessary conditions for (G, p) to be isostatic. Though
the rule in [25] is a useful tool for engineers and chemists to analyze the
rigidity properties of symmetric structures in 2D and 3D, it is unsatisfactory
from a mathematical point of view since it cannot be applied to frameworks
in dimensions higher than 3 and since a complete mathematical proof of
this result has not been provided. In the following sections, we aim to give a
mathematical proof not only for the rule in [25], but also for an extended rule
that can be applied to a symmetric framework with a possibly non-injective
configuration in an arbitrary dimension.
In this section, we first develop all the necessary mathematical back-
ground that was omitted in [25]. This background consists of three major
parts. First, we show that the subspaces R and T of all rotational and trans-
lational infinitesimal rigid motions of a given symmetric framework (G, p)
are invariant under the external representation H ′e (see Lemma 4.2.1). This
allows us to define subrepresentations of H ′e for the subspaces R and T . We
then prove that the block-diagonalized form of the rigidity matrix of (G, p)
gives rise to additional necessary conditions for (G, p) to be isostatic (see
Theorem 4.2.2). The symmetry-extended version of Maxwell’s rule is based
on these conditions. Finally, we describe in detail how to determine the
dimensions of the H ′e-invariant subspaces of R and T . This is essential in
applying the symmetry-extended version of Maxwell’s rule to a given sym-
metric framework.
Using some basic techniques from character theory, all the results of this
105
section combined will allow us to formulate the symmetry-extended version
of Maxwell’s rule given in [25] (as well as its extension to higher dimensions)
as a mathematical theorem in Section 4.2.2.
For the remainder of this chapter, we will continue to use the notation of
the previous section.
4.2.1 The necessary conditions
As before, we let G be a graph, S be a symmetry group in dimension
d with pairwise non-equivalent irreducible linear representations I1, . . . , Ir,
Φ be a homomorphism from S to Aut(G), and (G, p) be a framework in
R(G,S,Φ).
In this section, we make the additional assumption that the points p(v),
v ∈ V (G), span all of Rd.
Recall from the previous section that we have the decomposition
Rdn = V (I1)e ⊕ . . .⊕ V (Ir)
e (4.9)
with
V (It)e =
(V (It)
e
)1⊕ . . .⊕ (
V (It)e
)λt
(4.10)
of Rdn into H ′e-invariant subspaces.
While the scalars λt (as well as the subspaces that appear as direct sum-
mands in (4.9)) are uniquely determined in this decomposition, the subspaces
that appear as direct summands in (4.10) are not [60]. In order to derive the
symmetry-extended version of Maxwell’s rule, the subspaces in (4.10) shall
now be chosen appropriately.
Since the points p(v), v ∈ V (G), span all of Rd, the subspace N =
106
ker(R(Kn, p)
)of Rdn, where Kn is the complete graph on V (G), is the
space consisting of all infinitesimal rigid motions of (G, p). This space can
be written as the direct sum
N = T ⊕R,
where T is the space of all translational and R is the space of all rotational
infinitesimal rigid motions of (G, p). More precisely, a basis of T is given by
Tj| j = 1, . . . , d, where for j = 1, . . . , d, Tj : V (G) → Rd is the map that
sends each v ∈ V (G) to the jth canonical basis vector ej of Rd, and a basis of
R is given by Rij| 1 ≤ i < j ≤ d, where for 1 ≤ i < j ≤ d, Rij : V (G) → Rd
is the map defined by Rij(vk) = (pk)iej − (pk)jei for all k = 1, . . . , n [81].
Each of the maps Tj and Rij is of course identified with a vector in Rdn (by
using the order on V (G)).
Note that in the context of static rigidity, T is the space of all translational
loads and R is the space of all rotational loads on (G, p).
Using the notation of the previous paragraph we have the following result.
Lemma 4.2.1 For every dimension d, the subspaces T , R, and N of Rdn
are H ′e-invariant.
Proof. Fix a dimension d. We show first that N = ker(R(Kn, p)
)is
H ′e-invariant. Since p ∈ ⋂
x∈S Lx,Φ, it follows from Lemma 4.1.1 that if
R(Kn, p)u = z, then for all x ∈ S, we have
R(Kn, p)He(x)u = Hi(x)z, (4.11)
where Hi is the internal representation of S with respect to Kn and Φ. Let
u ∈ N , i.e., R(Kn, p)u = 0. Then for any x ∈ S, we have
Hi(x)R(Kn, p)u = Hi(x)0 = 0.
107
By (4.11), we have Hi(x)R(Kn, p)u = R(Kn, p)He(x)u, and hence
R(Kn, p)He(x)u = 0.
Thus, for all x ∈ S, He(x)u ∈ ker(R(Kn, p)
), which says that N is H ′
e-
invariant.
Next, we show that T is also H ′e-invariant. Let x ∈ S and let, as usual, Mx
denote the orthogonal matrix that represents x with respect to the canonical
basis of Rd. Then for j = 1, . . . , d, we have
He(x)Tj =
Mxej
...
Mxej
= (Mx)1jT1 + . . . + (Mx)djTd.
It follows that T is H ′e-invariant.
It remains to show that R is H ′e-invariant. Since for all x ∈ S, He(x)
is an orthogonal matrix, H ′e is a unitary representation (with respect to the
canonical inner product on Rdn). Therefore, the subrepresentation H′(N)e of
H ′e with representation space N is also unitary (with respect to the inner
product obtained by restricting the canonical inner product on Rdn to N).
So, by Remark 4.1.1, it suffices to show that R is the orthogonal complement
of T in N .
Let t be any element of T and r be any element of R. Then
t =
w
...
w
for some w ∈ Rd
108
and
r =
V p1
...
V pn
for some skew-symmetric matrix V .
Since the point∑n
i=1 pi must be fixed by every symmetry operation x ∈ S,
we may wlog define an origin so that∑n
i=1 pi = 0. Then the inner product
of t and r is given by
t · r =n∑
i=1
wT V pi
= wT V
n∑i=1
pi = 0.
This gives the result. ¤
Since, by Lemma 4.2.1, N is an H ′e-invariant subspace of Rdn, it fol-
lows from Maschke’s Theorem (see [42, 51, 60], for example) that N has an
H ′e-invariant complement Q in Rdn. We may therefore form the subrepresen-
tation H′(Q)e of H ′
e with representation space Q. Since H′(Q)e is a direct sum
of irreducible linear representations of S, say
H ′(Q)e = κ1I1 ⊕ . . .⊕ κrIr, where κ1, . . . , κr ∈ N ∪ 0, (4.12)
we obtain, analogously to (4.10), a decomposition of Q of the form
Q = V(I1)Q ⊕ . . .⊕ V
(Ir)Q ,
where
V(It)Q =
(V
(It)Q
)1⊕ . . .⊕ (
V(It)Q
)κt
. (4.13)
Similarly, since both T and R are also H ′e-invariant subspaces of Rdn, we
may form the subrepresentations H′(T )e and H
′(R)e of H ′
e with respective rep-
resentation spaces T and R. This gives rise to a decomposition of T of the
109
form
T = V(I1)T ⊕ . . .⊕ V
(Ir)T ,
where
V(It)T =
(V
(It)T
)1⊕ . . .⊕ (
V(It)T
)θt
,
and to a decomposition of R of the form
R = V(I1)R ⊕ . . .⊕ V
(Ir)R ,
where
V(It)R =
(V
(It)R
)1⊕ . . .⊕ (
V(It)R
)ρt
.
We can now choose the decomposition in (4.10) in such a way that
V (It)e = V
(It)Q ⊕ V
(It)T ⊕ V
(It)R . (4.14)
In the following we assume that the subspaces(V
(It)e
)jare chosen in this way.
We are now in the position to derive the necessary conditions for
(G, p) ∈ R(G,S,Φ) to be isostatic upon which the symmetry-extended version
of Maxwell’s rule is based.
Theorem 4.2.2 Let G be a graph, S be a symmetry group in dimension d
with pairwise non-equivalent irreducible linear representations I1, . . . , Ir, and
Φ : S → Aut(G) be a homomorphism. If (G, p) is an isostatic framework
in R(G,S,Φ) with the property that the points p(v), v ∈ V (G), span all of Rd,
then for t = 1, 2, . . . , r, we have
dim(V
(It)Q
)= dim
(V
(It)i
). (4.15)
110
Proof. Suppose first that dim(V
(It)Q
)> dim
(V
(It)i
)for some t. In this
case we give two separate arguments to show that (G, p) is not isostatic, one
that is based on infinitesimal rigidity and another one that is based on static
rigidity. This will later allow us to obtain information about both kinematic
and static rigidity properties of symmetric frameworks with the symmetry-
extended version of Maxwell’s rule.
Since dim(V
(It)Q
)> dim
(V
(It)i
), it follows from Corollary 4.1.3 that there
exists an element u 6= 0 in V(It)Q that lies in the kernel of the linear trans-
formation which is represented by the matrix R(G, p) with respect to the
bases Be and Bi. In other words, u is an infinitesimal flex of (G, p) (which is
symmetric with respect to It), and hence (G, p) is not isostatic.
Alternatively, it follows from Corollary 4.1.3 that there exists an element
l in V(It)Q that does not lie in the image of the linear transformation which
is represented by the matrix R(G, p)T with respect to the bases Be and Bi.
This says that l is an unresolvable equilibrium load on (G, p) (which is sym-
metric with respect to It), so that we may again conclude that (G, p) is not
isostatic.
Suppose now that dim(V
(It)Q
)< dim
(V
(It)i
)for some t. Then, analo-
gously as above, there exists an element ω 6= 0 in V(It)i that lies in the kernel
of the linear transformation which is represented by the matrix R(G, p)T with
respect to the bases Be and Bi. This says that ω is a non-zero self-stress of
(G, p) (which is symmetric with respect to It). So, it again follows that (G, p)
is not isostatic. ¤
Example 4.2.1 Recall from Example 4.1.2 that for the framework (K3, p) ∈
111
R(K3,Cs,Φ) shown in Figure 4.6, we have
dim(V (A′)
e
)= 3
dim(V
(A′)i
)= 2
dim(V (A′′)
e
)= 3
dim(V
(A′′)i
)= 1.
It is easy to see that the 2-dimensional space T of all translational infinites-
imal rigid motions of (K3, p) can be written as the direct sum
T = V(A′)T ⊕ V
(A′′)T ,
where V(A′)T is the space of dimension 1 generated by the infinitesimal rigid
motion shown in Figure 4.6 (a), and V(A′′)T is the space of dimension 1 gen-
erated by the infinitesimal rigid motion shown in Figure 4.6 (b). Moreover,
the 1-dimensional space R of rotational infinitesimal rigid motions of (K3, p)
is clearly generated by the infinitesimal rigid motion shown in Figure 4.6 (c),
so that R = V(A′′)R and dim
(V
(A′)R
)= 0. It follows from equation (4.14) that
...p1
..p2
..p3
.(a)
...p1
..p2
..p3
.(b)
...p1
..p2
..p3
.(c)
Figure 4.6: (a) A basis for the subspace V(A′)T ; (b) a basis for the subspace
V(A′′)T ; (c) a basis for the subspace R = V
(A′′)R .
112
dim(V
(A′)Q
)= dim
(V (A′)
e
)− dim(V
(A′)T
)− dim(V
(A′)R
)= dim
(V
(A′)i
)= 2
and
dim(V
(A′′)Q
)= dim
(V (A′′)
e
)−dim(V
(A′′)T
)−dim(V
(A′′)R
)= dim
(V
(A′′)i
)= 1,
so that the conditions (4.15) in Theorem 4.2.2 are satisfied for the isostatic
framework (K3, p).
In general, finding the dimensions of the subspaces V(It)Q and V
(It)i by
inspection is not as easy as in the previous example. In the following, we
therefore describe a systematic method, based on techniques from character
theory, for determining the dimensions of these subspaces, so that we can
apply Theorem 4.2.2 to a symmetric framework with an arbitrary point group
in any dimension. We begin by introducing the necessary vocabulary.
Definition 4.2.1 Let A = (aij) be an n× n square matrix. The trace of A
is defined to be Tr(A) =∑n
i=1 aii.
It is an important and well-known fact that the trace of a matrix is
invariant under a similarity transformation [19, 37]. This gives rise to
Definition 4.2.2 Let H : S → GL(V ) be a linear representation of a group
S, B be a basis of V , and HB be the matrix representation that corresponds to
H with respect to B. The character χ(H) of H is the function χ(H) : S → R
that sends x ∈ S to Tr(HB(x)
).
For a fixed enumeration x1, . . . , xk of the elements of the group S, we
will frequently also refer to the vector(Tr
(HB(x1)
), . . . , T r
(HB(xk)
))as
the character of H.
113
In the following we need some well-known results from character theory
which we summarize in
Theorem 4.2.3 [19, 37, 42, 60] Let S be a group with r pairwise non-
equivalent irreducible linear representations I1, . . . , Ir and let H : S →GL(V ) be a linear representation of S with H = α1I1 ⊕ . . . ⊕ αrIr, where
αt ∈ N ∪ 0 for all t = 1, . . . , r.
(i) If H = H1 ⊕H2 for some linear representations H1 and H2 of S, then
χ(H) = χ(H1) + χ(H2);
(ii) χ(H) can be written uniquely as a linear combination of the characters
χ(I1), . . . , χ(Ir) as
χ(H) = α1χ(I1) + . . . + αrχ(Ir);
(iii) For every t = 1, . . . , r, we have
αt =1
‖χ(It)‖2
(χ(H) · χ(It)
).
We first explain how we can determine the dimensions of the subspaces
V(It)i for all t = 1, . . . , r.
It follows from the direct sum decomposition of H ′i in (4.6) that for t =
1, . . . , r, the dimension of V(It)i is the degree of It multiplied by µt. Since
the degree of each irreducible linear representation It can be read off from
the character tables in Appendix A (or, if a more complete list of character
tables is required, from the tables in [2, 4, 37]), we only need to determine
the values of the µt. This can easily be done by means of the formula given in
Theorem 4.2.3 (iii), because the characters of the irreducible representations
114
It can simply be read off from the above-mentioned character tables and
the character of H ′i can be found by setting up the internal representation
matrices Hi(x), x ∈ S.
Finding the dimensions of the subspaces V(It)Q for all t = 1, . . . , r requires
a little more work. It follows from (4.14) that for t = 1, . . . , r, we have
dim(V
(It)Q
)= dim
(V (It)
e
)− dim(V
(It)T
)− dim(V
(It)R
).
The dimensions of the subspaces V(It)e can be determined in the analogous
way as the dimensions of the subspaces V(It)i : for t = 1, . . . , r, the dimension
of the subspace V(It)e is equal to the degree of It multiplied by λt. Note that
the values of the λt in (4.10) can again easily be computed with the help of
Theorem 4.2.3 (iii) since the character of H ′e can be found by setting up the
external representation matrices He(x), x ∈ S.
For t = 1, . . . , r, the dimension of the subspace V(It)T is the degree of It
multiplied by θt and the dimension of the subspace V(It)R is the degree of It
multiplied by ρt. So, in order to determine the dimensions of the subspaces
V(It)T and V
(It)R with the formula in Theorem 4.2.3 (iii), it only remains to
determine the characters χ(H′(T )e ) and χ(H
′(R)e ).
We first show how to compute the character χ(H′(T )e ). It follows directly
from the proof of Lemma 4.2.1 that if S is a symmetry group in dimension
d and x ∈ S, then the matrix that represents the linear transformation
H′(T )e (x) with respect to the basis T1, . . . , Td is the orthogonal matrix Mx
that represents x with respect to the canonical basis of Rd. This says that
for a fixed enumeration x1, . . . , xk of the elements of S, we have
χ(H ′(T )e ) =
(Tr(Mx1), . . . , T r(Mxk
)).
115
For example, if S is a symmetry group in dimension 2, then the compo-
nent of χ(H′(T )e ) that corresponds to the identity Id ∈ S is equal to 2, each
component of χ(H′(T )e ) that corresponds to a rotational symmetry operation
Cm ∈ S is equal to 2 cos(
2πm
), and each component of χ(H
′(T )e ) that corre-
sponds to a reflection s ∈ S is equal to 0 (see also Table 4.1 in Section 4.3.2).
For a symmetry group S in dimension 3, the explicit values of the com-
ponents of χ(H′(T )e ) are summarized in Table 4.2 in Section 4.3.3.
The character χ(H′(R)e ) can be determined similarly. As an example, we
compute the character χ(H′(R)e ) in the case where S is a symmetry group in
dimension 2. Every element of S is then either the identity Id, a rotation
Cm about the origin by an angle of 2πm
, or a reflection s in a line through the
origin. Note that R is a one-dimensional subspace of R2n a basis of which is
given by R12. Let Cm be a rotational symmetry operation of (G, p) with
MCm =
cos
(2πm
) − sin(
2πm
)
sin(
2πm
)cos
(2πm
)
.
Then, by using the definition of the external representation He of S (with
respect to G and Φ) and the fact that (G, p) ∈ R(G,S,Φ), it is easy to verify
that
He(Cm)R12 = R12.
Similarly, if s is a reflectional symmetry operation of (G, p) with
Ms =
cos (θ) sin (θ)
sin (θ) − cos (θ)
,
then it is again easy to verify that
He(s)R12 = −R12.
116
It follows that the matrices which represent the linear transformations
H′(R)e (Id), H
′(R)e (Cm), and H
′(R)e (s) with respect to the basis R12 are the
1 × 1 matrices (i.e., scalars) 1, 1, and −1, respectively. Therefore, if d = 2,
the character χ(H′(R)e ) is the vector defined as follows: each component of
χ(H′(R)e ) that corresponds to the identity Id ∈ S or a rotational symmetry
operation Cm ∈ S is equal to 1, and each component of χ(H′(R)e ) that cor-
responds to a reflection s ∈ S is equal to −1 (see also Table 4.1 in Section
4.3.2).
Note that analogous calculations as above can easily be carried out for
any symmetry group in dimension d > 2 as well. For a symmetry group S in
dimension 3, the values of the components of χ(H′(R)e ) are again summarized
in Table 4.2 in Section 4.3.3.
Example 4.2.2 Let us apply the methods described above to the framework
(K3, p) ∈ R(K3,Cs,Φ) from Example 4.2.1. From the representation matrices in
Example 4.1.1 we immediately deduce that χ(H ′e) = (6, 0) and χ(H ′
i) = (3, 1).
Therefore, if we let
H ′e = λ1A
′ ⊕ λ2A′′
H ′i = µ1A
′ ⊕ µ2A′′,
then, by the formula in Theorem 4.2.3 (iii), we have
λ1 =1
2
(6 · 1 + 0 · 1) = 3
λ2 =1
2
(6 · 1 + 0 · (−1)
)= 3
µ1 =1
2
(3 · 1 + 1 · 1) = 2
µ2 =1
2
(3 · 1 + 1 · (−1)
)= 1.
117
Further, for the characters χ(H′(T )e ) and χ(H
′(R)e ), we have, as shown above,
χ(H′(T )e ) = (2, 0) and χ(H
′(R)e ) = (1,−1). So, if we let
H ′(T )e = θ1A
′ ⊕ θ2A′′
H ′(R)e = ρ1A
′ ⊕ ρ2A′′,
then, again by the formula in Theorem 4.2.3 (iii), we obtain θ1 = 1, θ2 = 1,
ρ1 = 0, and ρ2 = 1. Since both A′ and A′′ are linear representations of degree
1, it follows that
dim(V
(A′)Q
)= dim
(V (A′)
e
)− dim(V
(A′)T
)− dim(V
(A′)R
)
= 3− 1− 0
= 2
dim(V
(A′′)Q
)= dim
(V (A′′)
e
)− dim(V
(A′′)T
)− dim(V
(A′′)R
)
= 3− 1− 1
= 1
and
dim(V
(A′)i
)= 2
dim(V
(A′′)i
)= 1.
4.2.2 The rule
Using the mathematical background established in the previous section,
we can now prove a symmetry-extended version of Maxwell’s rule that can
be applied to both injective and non-injective symmetric realizations in any
dimension. Note that for dimensions 2 and 3, Theorem 4.2.4 is a rigorous
118
mathematical formulation of the rule given in [25].
The condition (4.16) in Theorem 4.2.4 is obtained by combining all of
the conditions in (4.15) into a single equation using some basic techniques
from character theory. This enables us to check the conditions in (4.15) with
very little computational effort, so that the symmetry-extended version of
Maxwell’s rule is in the same spirit as Maxwell’s original rule in the sense
that it only requires a simple count of joints and bars that are ‘fixed’ by
various symmetry operations.
From now on we will simplify our notation of the previous section by
denoting the characters χ(H ′e), χ(H ′
i), χ(H′(Q)e ), χ(H
′(T )e ), and χ(H
′(R)e ) by
Xe, Xi, XQ, XT , and XR, respectively.
Theorem 4.2.4 (Symmetry-extended version of Maxwell’s rule)
Let G be a graph, S be a symmetry group in dimension d with pairwise non-
equivalent irreducible linear representations I1, . . . , Ir, and Φ : S → Aut(G)
be a homomorphism. If (G, p) is an isostatic framework in R(G,S,Φ) with the
property that the points p(v), v ∈ V (G), span all of Rd, then we have
XQ = Xi. (4.16)
Proof. Suppose XQ 6= Xi. Then, by Theorem 4.2.3 (ii) and equations (4.6)
Since XQ 6= Xi, we may conclude that (K3,3, p) is not isostatic. Note that
Maxwell’s original rule would not have detected this because |E(K3,3)| =
2|V (K3,3)| − 3 = 9.
With the help of the formula in Theorem 4.2.3 (iii) we obtain
XQ = 3A1 + 2A2 + 2B1 + 2B2 and
Xi = 4A1 + 2A2 + 2B1 + B2,
which implies that (K3,3, p) has a non-zero self-stress which is symmetric with
respect to A1 and an infinitesimal flex (as well as an unresolvable equilibrium
load) which is symmetric with respect to B2 (see Figure 4.7).
Remark 4.2.2 Given a framework (G, p) ∈ R(G,S), we need to specify a
type Φ : S → Aut(G) in order to apply the symmetry-extended version of
Maxwell’s rule (Theorem 4.2.4) to (G, p) and S, because Φ determines the
characters Xe and Xi. Of course, we also need to make sure that Φ is a
homomorphism, for otherwise the external and internal representation (with
respect to G and Φ) are not matrix representations of S (see Remark 4.1.2).
124
.
..p5
..p3
. .p6
..p1 . .p2
..p4
.sv
.sh
.(a)
.
..p5 .
.p3
..p6
..p1 . .p2
..p4
.sv
.sh
.(b)
.
..p5 .
.p3
..p6
..p1 ..p2
..p4
.sv
.sh
.(c)
Figure 4.7: (a) An infinitesimal flex of (K3,3, p) ∈ R(K3,3,C2v ,Φ) which is sym-
metric with respect to B2 (the displacement vector at each joint of (K3,3, p)
remains unchanged under Id and sv and is reversed under C2 and sh). (b)
An unresolvable equilibrium load on (K3,3, p) which is symmetric with respect
to B2. (c) A self-stress of (K3,3, p) which is symmetric with respect to A1 (the
tensions and compressions in the bars of (K3,3, p) remain unchanged under
all symmetry operations in C2v).
125
Recall that if Aut(G, p) = id, then, by Corollary 3.3.2, (G, p) ∈ R(G,S)
is of a unique type Φ and, by Theorem 3.4.1, Φ is a homomorphism, so that
the external and internal representation are uniquely determined in this case
and Theorem 4.2.4 can be applied to (G, p) and S in a unique way. By Corol-
lary 3.3.3, this is in particular the case if (G, p) is a framework whose map
p is injective. In the following section, we will see that if p is injective, then
the characters Xe and Xi can be found in a particularly easy way (without
determining the type Φ) by simply examining the geometric positions of the
joints and bars of (G, p).
Since in [25] only injective realizations in R2 and R3 are considered, The-
orem 4.2.4 includes the symmetrized version of Maxwell’s rule given in [25]
as a special case.
If Aut(G, p) 6= id, then, by Theorem 3.3.1, (G, p) ∈ R(G,S) is not of
a unique type, and hence we may apply Theorem 4.2.4 to (G, p) and S by
using any homomorphism Φ for which (G, p) ∈ R(G,S,Φ).
Note that Examples 3.4.1 and 3.4.2 show that there exist frameworks
(G, p) ∈ R(G,S) for which we cannot apply the symmetry-extended version
of Maxwell’s rule to (G, p) and S at all, because there does not exist any
homomorphism Φ so that (G, p) ∈ R(G,S) is of type Φ.
Remark 4.2.3 Let G be a graph, S be a symmetry group in dimension d,
and Φ : S → Aut(G) be a homomorphism, so that the set R(G,S,Φ) contains a
framework (G, p) with the property that the points p(v), v ∈ V (G), span all
of Rd. Then it follows from Lemma 3.2.2 and Theorem 3.2.3 that the con-
dition (4.16) in the symmetry-extended version of Maxwell’s rule (Theorem
4.2.4) is also a necessary condition for G to be (S, Φ)-generically isostatic.
126
In particular, if the condition (4.16) does not hold (i.e., if XQ 6= Xi) then we
may conclude that every framework in the set R(G,S,Φ) is not isostatic.
Remark 4.2.4 There exist a number of further classical counting rules, sim-
ilar to Maxwell’s rule, that can predict the rigidity and flexibility properties
of various other types of structures, such as pinned frameworks (i.e., frame-
works that have some of their joints firmly anchored to the ground), body-
bar frameworks, and body-hinge frameworks, for example. For each of these
rules, symmetry extensions can be derived using techniques from group rep-
resentation theory (see, for example, [25, 34, 36, 53, 56]). We will discuss
some of these rules in Chapter 7.
4.3 Restrictions on the number of fixed joints
and bars of symmetric isostatic frame-
works
In this section, we show that the necessary conditions given in the
symmetry-extended version of Maxwell’s rule for a 2- or 3-dimensional sym-
metric framework (G, p) ∈ R(G,S,Φ) to be isostatic are equivalent to some
very simply stated restrictions on the number of joints and bars of (G, p)
that are fixed by various symmetry operations in S.
The basic results in this section are from the joint paper [15].
127
4.3.1 Fixed versus geometrically unshifted
We begin by summarizing some simple observations regarding the geo-
metric position of a joint or a bar of a framework (G, p) ∈ R(G,S,Φ) that is
fixed by an element in S (with respect to Φ).
Definition 4.3.1 Let x be a symmetry operation of a framework (G, p).
Then a joint(v, p(v)
)of (G, p) is said to be geometrically unshifted by x if
x(p(v)
)= p(v) or equivalently, if p(v) is contained in the symmetry element
Fx corresponding to x.
Similarly, a bar(
v, p(v)),(w, p(w)
)of (G, p) is said to be geometri-
cally unshifted by x if x(p(v), p(w)) = p(v), p(w) or equivalently, if the
undirected line segment p(v)p(w) is equal to the undirected line segment
x(p(v)
)x(p(w)
).
..
.
.(a)
. .
.
.
.s
.(b)
.C2
Figure 4.8: Geometrically unshifted bars in dimension 2: (a) a bar that is
geometrically unshifted by a half-turn C2; (b) possible placement of a bar that
is geometrically unshifted by a reflection s.
Theorem 4.3.1 Let G be a graph, S be a symmetry group, Φ : S → Aut(G)
be a map, (G, p) be a framework in R(G,S,Φ), and x be an element in S.
(i) If a joint j of (G, p) is fixed by x (with respect to Φ), then j is geomet-
rically unshifted by x;
128
(ii) if a bar b of (G, p) is fixed by x (with respect to Φ), then b is geometri-
cally unshifted by x;
(iii) if p is injective and a joint j of (G, p) is geometrically unshifted by x,
then j is fixed by x;
(iv) if p is injective and a bar b of (G, p) is geometrically unshifted by x,
then b is fixed by x.
Proof. (i) Let j =(v, p(v)
)be a joint of (G, p) that is fixed by x (with
respect to Φ). Then we have x(p(v)
)= p
(Φ(x)(v)
)= p(v), which says that
j is geometrically unshifted by x.
(ii) Let b =(
v, p(v)),(w, p(w)
)be a bar of (G, p) that is fixed by x
with respect to Φ. Then we have x(p(v), p(w)) =
x(p(v)
), x
(p(w)
)=
p(Φ(x)(v)
), p
(Φ(x)(w)
)= p(v), p(w), which says that b is geometrically
unshifted by x.
(iii) Let j =(v, p(v)
)be geometrically unshifted by x. Then we have
p(v) = x(p(v)
)= p
(Φ(x)(v)
). Since p is injective, it follows that v = Φ(x)(v).
Thus, j is fixed by x.
(iv) Let b =(
v, p(v)),(w, p(w)
)be geometrically unshifted by x.
Then we have p(v), p(w) = x(p(v), p(w)) =
x(p(v)
), x
(p(w)
)=
p(Φ(x)(v)
), p
(Φ(x)(w)
). Since p is injective, it follows that v, w =
Φ(x)(v), Φ(x)(w) = Φ(x)(v, w). Thus, b is fixed by x. ¤
It follows from Theorem 4.3.1 that if the map p of a framework (G, p) ∈R(G,S,Φ) is injective, then a joint or a bar of (G, p) is geometrically unshifted
by x ∈ S if and only if it is fixed by x. Therefore, if (G, p) is a framework
whose map p is injective, then the determination of whether a joint j or a
129
.
.. ...(b)
..
..
.Cm
.(a)
Figure 4.9: Possible placement of a bar that is geometrically unshifted by: (a)
any rotation Cm, m ≥ 2 (in dimension 3); (b) a half-turn C2 (in dimension
3) alone.
...
..
.(a)
..
..
.s.(b)
Figure 4.10: Possible placement of a bar that is geometrically unshifted by
a reflection s (in dimension 3): (a) lying in the reflection plane; (b) lying
perpendicular to the reflection plane.
130
.
..
...Sm
.(a)
.
..
...i = S2
.(b)
Figure 4.11: Possible placement of a bar that is geometrically unshifted by:
(a) any improper rotation Sm, m ≥ 2 (in dimension 3); (b) an inversion
i = S2 (in dimension 3) alone.
bar b of (G, p) is fixed by x ∈ S only requires a simple examination of the
geometric positions of j and b with respect to the symmetry element corre-
sponding to x. Figures 4.8, 4.9, 4.10 and 4.11 show the possible geometric
positions of a bar that is geometrically unshifted by the relevant symmetry
operations in dimensions 2 and 3.
If the map p of (G, p) ∈ R(G,S,Φ) is not injective, then a joint or a bar
that is geometrically unshifted by x ∈ S is not necessarily fixed by x (with
respect to Φ).
For example, the joints (v3, p3) and (v4, p4) of the framework (Gt, p) ∈R(Gt,C2,Θb) in Example 3.3.1 are both geometrically unshifted by C2, since
both p3 and p4 lie on the center of rotation of C2, but they are not fixed
by C2 with respect to Θb, since Θb(C2)(v3) = v4. Similarly, the bar(v3, p3), (v4, p4)
of the framework (Gbp, p) ∈ R(Gbp,Cs,Ξb) in Example 3.3.2
is geometrically unshifted by s, but not fixed by s with respect to Ξb, since
Ξb(v3, v4) = v3, v5 6= v3, v4.So, if the map p of a framework (G, p) ∈ R(G,S,Φ) is not injective, then
131
we can only find the joints and bars of (G, p) that are fixed by x ∈ S (with
respect to Φ) by considering the graph automorphism Φ(x).
4.3.2 Isostatic frameworks in dimension 2
Suppose (G, p) is an isostatic framework in R(G,S,Φ), where S is a sym-
metry group in dimension 2, Φ : S → Aut(G) is a homomorphism, and the
points p(v), v ∈ V (G), span all of R2.
Since S is a symmetry group in dimension 2, every element of S is of one
of the following three types: the identity Id, a rotation Cm, where m ≥ 2,
or a reflection s. This allows us to calculate the (2-dimensional) symmetry-
extended Maxwell’s equation (4.17) for (G, p) componentwise, as shown in
Table 4.1.
In this table we distinguish a half-turn C2 from a rotation Cm, where
m > 2, because there may exist bars of (G, p) that are fixed by a half-turn,
but there cannot be any bars of (G, p) that are fixed by a rotation Cm, where
m > 2.
By Table 4.1, the symmetry-extended Maxwell’s equation for the isostatic
framework (G, p) ∈ R(G,S,Φ) reduces to the following four equations:
Id: |E(G)| = 2|V (G)| − 3 (4.18)
C2: 2jΦ(C2) + bΦ(C2) = 1 (4.19)
Cm, m > 2: (jΦ(Cm) − 1) cos
(2π
m
)=
1
2(4.20)
s: bΦ(s) = 1, (4.21)
132
Id C2 Cm, m > 2 s
XJ |V (G)| jΦ(C2) jΦ(Cm) jΦ(s)
XT 2 −2 2 cos(
2πm
)0
XR 1 1 1 −1
XQ 2|V (G)| − 3 −2jΦ(C2) + 1 2(jΦ(Cm) − 1) cos(
2πm
)− 1 1
Xi |E(G)| bΦ(C2) 0 bΦ(s)
Table 4.1: Calculations of characters in the 2-dimensional symmetry-
extended Maxwell’s equation.
where a given equation applies when the corresponding symmetry operation
is present in S.
Some observations arising from this set of equations are:
(i) Since every symmetry group contains the identity Id, equation (4.18)
holds and simply restates the condition in Maxwell’s original rule (The-
orem 2.2.7).
(ii) If S contains a half-turn C2, then it follows from (4.19) and the fact that
both jΦ(C2) and bΦ(C2) must be non-negative integers that jΦ(C2) = 0
and bΦ(C2) = 1. In particular, since all bars of (G, p) (except the one
that is fixed by C2 with respect to Φ) and all joints of (G, p) occur in
pairs, it follows that |V (G)| is even and |E(G)| is odd.
(iii) If S contains a rotation Cm, m > 2, then it follows from (4.20) that
either jΦ(Cm) = 0 and m = 3 or jΦ(Cm) = 2 and m = 6. However,
if jΦ(Cm) = 2 and m = 6, then S also contains a half-turn C2 = C36
with jΦ(C2) = 2, contradicting (4.19). Therefore, S cannot contain a
133
rotational symmetry operation Cm with m > 3 and when either C2 or
C3 is present in S, then jΦ(C2) = 0 and jΦ(C3) = 0. Note that if S
contains a rotation C3, then all joints and bars of (G, p) occur in sets
of 3.
(iv) Finally, equation (4.21) says that if S contains a reflection s, then we
must have bΦ(s) = 1. However, (4.21) does not impose a restriction on
the number of joints that are fixed by s (with respect to Φ).
An immediate consequence of these observations is that the point group
of (G, p) must be one of the following six symmetry groups in dimension 2:
C1, C2, C3, Cs, C2v or C3v. Figure 5.48 depicts examples of small 2-dimensional
isostatic frameworks for each of these symmetry groups.
Group by group, the above results can be summarized as follows.
Theorem 4.3.2 Let G be a graph, S be a symmetry group in dimension 2,
Φ : S → Aut(G) be a homomorphism, and (G, p) be an isostatic framework
in R(G,S,Φ) with the property that the points p(v), v ∈ V (G), span all of R2.
(i) If S = C1, then |E(G)| = 2|V (G)| − 3;
(ii) if S = C2, then |E(G)| = 2|V (G)| − 3, jΦ(C2) = 0 and bΦ(C2) = 1;
(iii) if S = C3, then |E(G)| = 2|V (G)| − 3 and jΦ(C3) = 0;
(iv) if S = Cs, then |E(G)| = 2|V (G)| − 3 and bΦ(s) = 1;
(v) if S = C2v, then |E(G)| = 2|V (G)|−3, jΦ(C2) = 0 and bΦ(C2) = bΦ(s) = 1
for all reflections s ∈ C2v;
134
.. .
.
.(a)
. .
..
.(b)
. .
.
..
.
.(c)
.
.
.. .
. .
.
.(d.i)
.(d.ii)
. .
.
.
.(e)
...
.
..
.
.
. .
.(f.ii)
.(f.i)
. .
.
Figure 4.12: Examples, for each of the possible point groups, of small 2-
i.e., for each edge vi, vj ∈ E(G), R(G, p, q) has the row with(q(vi, vj)
)1
and(q(vi, vj)
)2
in the columns 2i − 1 and 2i, −(q(vi, vj)
)1
and
160
−(q(vi, vj)
)2
in the columns 2(j − 1) and 2j, and 0 elsewhere.
We say that (G, p, q) is independent if R(G, p, q) has linearly independent
rows.
Remark 5.1.1 If (G, p, q) is a frame with the property that p(vi) 6= p(vj)
whenever vi, vj ∈ E(G), then we obtain the rigidity matrix of the frame-
work (G, p) by multiplying each row of R(G, p, q) by its corresponding scalar
λij. Therefore, if (G, p, q) is independent, so is (G, p).
Lemma 5.1.3 Let (G, p, q) be an independent frame in R2 and let pt :
V (G) → R[t] × R[t] and qt : E(G) → R[t] × R[t] be such that (G, pa, qa)
is a frame in R2 for every a ∈ R. If (G, pa, qa) = (G, p, q) for a = 0, then
(G, pa, qa) is an independent frame in R2 for almost all a ∈ R.
Proof. Note that the rows of R(G, pt, qt) are linearly dependent (over the
quotient field of R[t]) if and only if the determinants of all the |E(G)|×|E(G)|submatrices of R(G, pt, qt) are identically zero. These determinants are poly-
nomials in t. Thus, the set of all a ∈ R with the property that R(G, pa, qa)
has a non-trivial row dependency is a variety F whose complement, if non-
empty, is a dense open set. Since a = 0 is in the complement of F we can
conclude that for almost all a, (G, pa, qa) is independent. ¤
Each time Lemma 5.1.3 is applied in this chapter, the polynomials in
R(G, pt, qt) are linear polynomials in t.
Since the characterizations of (S, Φ)-generically isostatic graphs can be
given in the most natural way if S = C3, followed by S = C2 and S = Cs, the
sections in this chapter are arranged according to this order.
161
5.2 Characterizations of (C3, Φ)-generically
isostatic graphs
5.2.1 Symmetrized Henneberg moves and 3Tree2 par-
titions for C3
We need the following inductive construction techniques to obtain a sym-
metrized Henneberg’s Theorem for C3.
Definition 5.2.1 Let G be a graph, C3 = Id, C3, C23 be a sym-
metry group in dimension 2, and Φ : C3 → Aut(G) be a homomor-
phism. Let v1, v2 be two distinct vertices of G and v, w, z /∈ V (G).
Then the graph G with V (G) = V (G) ∪ v, w, z and E(G) =
By the induction hypothesis, R(Gk−1,C3,Φk−1) 6= ∅ and Gk−1 is (C3, Φk−1)-
generically isostatic. Further, by the definition of the (C3, Φ) construction
sequence, we have jΦk−1(C3) = 0. So, if G is a (C3, Φk−1) vertex addition of
Gk−1, then the result follows from Lemma 5.2.6; if G is a (C3, Φk−1) edge split
of Gk−1, then the result follows from Lemma 5.2.7, and if G is a (C3, Φk−1)
∆ extension of Gk−1, then the result follows from Lemma 5.2.8. ¤
187
5.3 Characterizations of (C2, Φ)-generically
isostatic graphs
We will find that the symmetric versions of Laman’s Theorem for the
symmetry groups C2 and Cs cannot be proved as straightforwardly as for the
group C3. In the following, we demonstrate one of the additional difficulties
that arises in proving the Laman-type results for C2 and Cs by means of some
simple examples.
Let G be a graph that satisfies the conditions of Case 1 or Case 2 in
the proof of Lemma 5.2.3, i.e., G satisfies the Laman conditions, jγ = 0,
and G has a vertex v with NG(v) = v1, v2, v3 so that v, γ(v), γ2(v)is an independent subset of V (G). Then it follows from this proof that
if there exists a pair i, j ⊆ 1, 2, 3 such that every subgraph H of
G′ = G−v, γ(v), γ2(v) with vi, vj ∈ V (H) satisfies |E(H)| ≤ 2|V (H)| − 4,
then G = G′ +vi, vj, γ(vi), γ(vj), γ2(vi), γ
2(vj)
satisfies the Laman
conditions.
For the symmetry groups C2 and Cs, the situation can possibly be more
complicated, as the following examples illustrate.
Let G be the underlying graph of the frameworks in Figure 5.15 (a) and
(b), C2 = Id, C2 and Cs = Id, s be symmetry groups in dimension 2,
and Φ : C2 → Aut(G) and Ψ : Cs → Aut(G) be the homomorphisms that
map C2 and s to the automorphism (v w)(v1 v6)(v2 v7)(v3 v8)(v4 v9)(v5 v10)
of G, respectively. Observe that G satisfies the Laman conditions and
that the conditions in Theorem 4.3.2 also hold for C2 and Φ, as well as
for Cs and Ψ. Moreover, every subgraph H of G′ = G − v, w with
188
.
..p1
..p2
..p3.
.p4
..p5
..p6
..p7
. .p8
..p9
..p10
..p(v)
.
.p(w)
.(a)
.
..p1
..p2
..p3
..p4
..p5
..p6
..p7
..p8
..p9
..p10
..p(v) . .p(w)
.(b)
Figure 5.15: A realization of (G, C2) of type Φ (a) and a realization of (G, Cs)
of type Ψ (b).
v1, v2 ∈ V (H) satisfies |E(H)| ≤ 2|V (H)| − 4. However, G = G′ +v1, v2, v6, v7
does not satisfy the Laman conditions, since for the sub-
graph H = 〈v1, v2, v4, v5, v6, v7, v9, v10〉 of G, we have |E(H)| = 2|V (H)|−2.
5.3.1 Symmetrized Henneberg moves and 3Tree2 par-
titions for C2
We need the following inductive construction techniques to obtain a sym-
metrized Henneberg’s Theorem for C2.
.
.
..
..v1
.v2.γ(v1)
.γ(v2) .
.
..
.
. .
.v1
.v2 .γ(v1)
.γ(v2).v .w
Figure 5.16: A (C2, Φ) vertex addition of a graph G, where Φ(C2) = γ.
Definition 5.3.1 Let G be a graph, C2 = Id, C2 be the half-turn
symmetry group in dimension 2, and Φ : C2 → Aut(G) be a homo-
189
morphism. Let v1, v2 be two distinct vertices of G and v, w /∈ V (G).
Then the graph G with V (G) = V (G) ∪ v, w and E(G) = E(G) ∪v, v1, v, v2, w, Φ(C2)(v1), w, Φ(C2)(v2)
is called a (C2, Φ) vertex ad-
dition (by (v, w)) of G.
Definition 5.3.2 Let G be a graph, C2 = Id, C2 be the half-turn symme-
try group in dimension 2, and Φ : C2 → Aut(G) be a homomorphism. Let
v1, v2, v3 be three distinct vertices of G such that v1, v2 ∈ E(G) and v1, v2is not fixed by Φ(C2) and let v, w /∈ V (G). Then the graph G with V (G) =
V (G) ∪ v, w and E(G) =(E(G) \ v1, v2, Φ(C2)(v1), Φ(C2)(v2)
) ∪v, vi| i = 1, 2, 3
∪ w, Φ(C2)(vi)| i = 1, 2, 3
is called a (C2, Φ) edge
split (on (v1, v2, Φ(C2)(v1), Φ(C2)(v2)); (v, w)) of G.
.
.v1
.v2
.v3
.γ(v1)
.γ(v2)
.γ(v3)
.
.
.
.
.
. .
.v1
.v2
.v3
.γ(v1)
.γ(v2)
.γ(v3).v .w
.
.
.
.
.
.. .
Figure 5.17: A (C2, Φ) edge split of a graph G, where Φ(C2) = γ.
Remark 5.3.1 Each of the constructions in Definitions 5.3.1 and 5.3.2 has
the property that if the graph G satisfies the Laman conditions, then so does
G. This follows from Theorems 2.2.9 and 2.2.12 and the fact that we can
obtain a (C2, Φ) vertex addition of G by a sequence of two vertex 2-additions,
and a (C2, Φ) edge split of G by a sequence of two edge 2-splits.
In order to extend Crapo’s Theorem to C2 we need the following sym-
metrized definition of a 3Tree2 partition.
190
.
..v1
..v2
...γ(v1).γ(v2)
.
...v1
..v2
.. .γ(v1)
...v3 .γ(v2)
.γ(v3).
Figure 5.18: (C2, Φ) 3Tree2 partitions of graphs, where Φ(C2) = γ. The edges
in black color represent edges of the invariant trees.
Definition 5.3.3 Let G be a graph, C2 = Id, C2 be the half-turn sym-
metry group in dimension 2, and Φ : C2 → Aut(G) be a homomorphism. A
(C2, Φ) 3Tree2 partition of G is a 3Tree2 partition E(T0), E(T1), E(T2) of
G such that Φ(C2)(T1) = T2 and Φ(C2)(T0) = T0. The tree T0 is called the
invariant tree of E(T0), E(T1), E(T2).
5.3.2 The main result for C2
Theorem 5.3.1 Let G be a graph with |V (G)| ≥ 2, C2 = Id, C2 be the
half-turn symmetry group in dimension 2, and Φ : C2 → Aut(G) be a homo-
morphism. The following are equivalent:
(i) R(G,C2,Φ) 6= ∅ and G is (C2, Φ)-generically isostatic;
(ii) |E(G)| = 2|V (G)| − 3, |E(H)| ≤ 2|V (H)| − 3 for all H ⊆ G with
So, suppose there exist two distinct pairs in 1, 2, 3, say wlog 1, 2and 1, 3, such that every subgraph H of G′ with v1, v2 ∈ V (H) or
v1, v3 ∈ V (H) satisfies |E(H)| ≤ 2|V (H)| − 4. Then every subgraph
H of G′ with σ(v1), σ(v2) ∈ V (H) or σ(v1), σ(v3) ∈ V (H) also satisfies
|E(H)| ≤ 2|V (H)| − 4, because G′ is invariant under σ.
Suppose there exist subgraphs H1 and H2 of G′ with v1, v2, σ(v1), σ(v2) ∈V (H1) and v1, v3, σ(v1), σ(v3) ∈ V (H2) satisfying |E(Hi)| = 2|V (Hi)| − 4 for
i = 1, 2. Then there also exist subgraphs σ(H1) ⊆ G′ and σ(H2) ⊆ G′ with
v1, v2, σ(v1), σ(v2) ∈ V(σ(H1)
)and v1, v3, σ(v1), σ(v3) ∈ V
(σ(H2)
)satisfying
|E(σ(Hi)
)| = 2|V (σ(Hi)
)| − 4 for i = 1, 2. Let H ′i = Hi ∪ σ(Hi) for i = 1, 2.
221
Then
|E(H ′1)| = |E(H1)|+ |E(
σ(H1))| − |E(
H1 ∩ σ(H1))|
≥ 2|V (H1)| − 4 + 2|V (σ(H1)
)| − 4− (2|V (H1 ∩ σ(H1)
)| − 4)
= 2|V (H ′1)| − 4,
because H1 ∩ σ(H1) is a subgraph of G′ with v1, v2 ∈ V(H1 ∩ σ(H1)
). Since
H ′1 is also a subgraph of G′ with v1, v2 ∈ V (H ′
1) it follows that
|E(H ′1)| = 2|V (H ′
1)| − 4.
Similarly, we have
|E(H ′2)| = 2|V (H ′
2)| − 4.
So, both H ′1 and H ′
2 have an even number of edges. Moreover, both of these
graphs are invariant under σ, which says that neither E(H ′1) nor E(H ′
2)
contains the edge e of G that is fixed by σ. Note that H ′1 ∩H ′
2 is a subgraph
of G with v1, σ(v1) ∈ V (H ′1 ∩H ′
2), and hence satisfies the count
|E(H ′1 ∩H ′
2)| ≤ 2|V (H ′1 ∩H ′
2)| − 3,
because G satisfies the Laman conditions. Since H ′1 ∩ H ′
2 is also invariant
under σ and E(H ′1∩H ′
2) does not contain the edge e, |E(H ′1∩H ′
2)| is an even
number, and hence the above upper bound for |E(H ′1 ∩H ′
2)| can be lowered
to
|E(H ′1 ∩H ′
2)| ≤ 2|V (H ′1 ∩H ′
2)| − 4.
Thus, H ′ = H ′1 ∪H ′
2 satisfies
|E(H ′)| = |E(H ′1)|+ |E(H ′
2)| − |E(H ′1 ∩H ′
2)|
≥ 2|V (H ′1)| − 4 + 2|V (H ′
2)| − 4− (2|V (H ′1 ∩H ′
2)| − 4)
= 2|V (H ′)| − 4.
222
This is a contradiction, because H ′ is a subgraph of G′ with vi, σ(vi) ∈ V (H ′)
for all i = 1, 2, 3.
So, for i, j = 1, 2 or i, j = 1, 3, say wlog i, j = 1, 2, we
have that every subgraph H of G′ with vi, vj, σ(vi), σ(vj) ∈ V (H) satisfies
|E(H)| ≤ 2|V (H)| − 5.
Thus, G = G′ +v1, v2, σ(v1), σ(v2)
satisfies the Laman conditions
and if we define Φ by Φ(x) = Φ(x)|V (G) for all x ∈ Cs, then Φ(x) ∈ Aut(G)
for all x ∈ Cs and Φ : Cs → Aut(G) is a homomorphism. Since we also
have bΦ(s) = 1, it follows from the induction hypothesis that there exists a
which is a contradiction to the fact that G satisfies the Laman conditions.
Therefore, |N ′| ≥ k + 1. By assumption, every vertex in N ′ has valence at
least 5 in G, and since y, z /∈ N ′, every vertex in N ′ must even have valence
at least 6 in G. Therefore, the average valence in G is at least
2k · 3 + (k + 1) · 6 +(|V (G)| − (2k + k + 1)
) · 4|V (G)| = 4 +
2
|V (G)| ,
234
which again contradicts the fact that the average valence in G is 4 − 6|V (G)|
(see Lemma 5.1.1).
So, as claimed, there exists a vertex v ∈ V (G) with NG(v) = v1, v2, v3,σ(vi) = vi for all i = 1, 2, 3, and valG(vi) = 4 for some i ∈ 1, 2, 3, say wlog
valG(v1) = 4 with NG(v1) = v, σ(v), w, σ(w).
.
.v3 = σ(v3)
.v1 = σ(v1)
.v2 = σ(v2)
.σ(w) .w
.v .σ(v)
.
.
.. .
. .
Figure 5.32: If a graph G satisfies the conditions in Theorem 5.4.1 (ii), has no
vertex of valence two, no vertex of valence three that is fixed by σ, and every
3-valent vertex v of G (except possibly the vertices that are incident with the
edge that is fixed by σ) has the property that σ(u) = u for all u ∈ NG(v), then
there exists v ∈ V (G) with NG(v) = v1, v2, v3, σ(vi) = vi for all i = 1, 2, 3,
and valG(vi) = 4 for some i ∈ 1, 2, 3.
Let G′ = G − v1. We claim that G = G′ +v, w, σ(v), σ(w)
is also a v1 − σ(v1) path in T(k−1)0 distinct from P ′′.
Thus, T(k)0 is a tree and
E
(T
(k)0
), E
(T
(k)1
), E
(T
(k)2
)is a (Cs, Φ) ‖ 3Tree2
partition of G.
Case 4.2: Suppose v1, v2 /∈ E(T
(k−1)0
), say wlog v1, v2 ∈ E
(T
(k−1)1
).
Then we also have σ(v1), σ(v2) ∈ E(T
(k−1)2
).
245
Case 4.2a: If v3 ∈ V(T
(k−1)0
), then σ(v3) ∈ V
(T
(k−1)0
). In this case we
define T(k)0 to be the tree with
V(T
(k)0
)= V
(T
(k−1)0
) ∪ v, w
E(T
(k)0
)= E
(T
(k−1)0
) ∪ v, v3, w, σ(v3),
T(k)1 to be the tree with
V(T
(k)1
)= V
(T
(k−1)1
) ∪ v
E(T
(k)1
)=
(E
(T
(k−1)1
) \ v1, v2) ∪ v, v1, v, v2
,
and T(k)2 to be the tree with
V(T
(k)2
)= V
(T
(k−1)2
) ∪ w
E(T
(k)2
)=
(E
(T
(k−1)2
) \ σ(v1), σ(v2))
∪w, σ(v1), w, σ(v2).
ThenE
(T
(k)0
), E
(T
(k)1
), E
(T
(k)2
)is a (Cs, Φ) ⊥ or (Cs, Φ) ‖ 3Tree2 partition
of G.
Case 4.2b: If v3 /∈ V(T
(k−1)0
), then v3 ∈ V
(T
(k−1)i
)for i = 1, 2, and we
define
T(k)0 = T
(k−1)0 ,
T(k)1 to be the tree with
V(T
(k)1
)= V
(T
(k−1)1
) ∪ v, w
E(T
(k)1
)=
(E
(T
(k−1)1
) \ v1, v2)
∪v, v1, v, v2, w, σ(v3),
246
.
.v1 .σ(v1)
.v2 .σ(v2)
.v3 .σ(v3)
. .
. .
. .
.
.v1 .σ(v1)
.v2 .σ(v2)
.v3 .σ(v3)
. .
. .
. .
. ..v .w
.
.v1 .σ(v1)
.v2 .σ(v2)
.v3 .σ(v3)
. .
. .
. .
.
.v1 .σ(v1)
.v2 .σ(v2)
.v3 .σ(v3)
. .
. .
. .
. ..v .w
Figure 5.39: Construction of a (Cs, Φ) ‖ 3Tree2 partition of G in the case
where G is a (Cs, Φk−1) double edge split of Gk−1, v1, v2 ∈ E(T
(k−1)1
)and
σ(v1), σ(v2) ∈ E(T
(k−1)2
). The edges in black color represent edges of the
invariant tree.
and T(k)2 to be the tree with
V(T
(k)2
)= V
(T
(k−1)2
) ∪ v, w
E(T
(k)2
)=
(E
(T
(k−1)2
) \ σ(v1), σ(v2))
∪ w, σ(v1), w, σ(v2), v, v3.
ThenE
(T
(k)0
), E
(T
(k)1
), E
(T
(k)2
)is again a (Cs, Φ) ⊥ or (Cs, Φ) ‖ 3Tree2
partition of G.
Case 5: Finally, suppose that G is a (Cs, Φk−1) X-replacement by v of
Gk−1 with E(G) =(E(Gk−1)\
v1, v2, v3, v4)∪v, vi| i ∈ 1, 2, 3, 4
.
Then Φk−1(s)(v1, v2) = v3, v4. Wlog we assume Φk−1(s)(v1) = σ(v1) =
v3 and Φk−1(s)(v2) = σ(v2) = v4.
247
Case 5.1: Suppose v1, v2 /∈ E(T
(k−1)0
), say wlog v1, v2 ∈ E
(T
(k−1)1
).
Then v3, v4 ∈ E(T
(k−1)2
). So, if we define
T(k)0 = T
(k−1)0 ,
T(k)1 to be the tree with
V(T
(k)1
)= V
(T
(k−1)1
) ∪ v
E(T
(k)1
)=
(E
(T
(k−1)1
) \ v1, v2) ∪ v, v1, v, v2
,
and T(k)2 to be the tree with
V(T
(k)2
)= V
(T
(k−1)2
) ∪ v
E(T
(k)2
)=
(E
(T
(k−1)2
) \ v3, v4) ∪ v, v3, v, v4
,
thenE
(T
(k)0
), E
(T
(k)1
), E
(T
(k)2
)is a (Cs, Φ) ⊥ or (Cs, Φ) ‖ 3Tree2 partition
of G.
.
.v1 .v3 = σ(v1)
.v4 = σ(v2) .v2
. .
. .
.
.v1 .v3 = σ(v1)
.v4 = σ(v2) .v2
. .
. .
..v
Figure 5.40: Construction of a (Cs, Φ) ‖ 3Tree2 partition of G in the case
where G is a (Cs, Φk−1) X-replacement of Gk−1, v1, v2 ∈ E(T
(k−1)1
)and
v3, v4 ∈ E(T
(k−1)2
).
Case 5.2: Suppose v1, v2, v3, v4 ∈ E(T
(k−1)0
). Since T
(k−1)0 is a tree
and Φk−1(s)(T
(k−1)0
)= T
(k−1)0 , there either exists a v1 − v3 path that does
not contain the vertices v2 and v4 or a v2− v4 path that does not contain the
vertices v1 and v3 in T(k−1)0 . Suppose wlog that P is a v2 − v4 path in T
(k−1)0
248
that does not contain the vertices v1 and v3. Wlog we may also assume that
v2 ∈ V(T
(k−1)2
), and hence v4 ∈ V
(T
(k−1)1
). If all the vertices and edges of
P , as well as the edges v1, v2 and v3, v4, are deleted from T(k−1)0 , then
the resulting subgraph of T(k−1)0 has at least two components, namely the
components A with v1 ∈ V (A) and σ(A) = B with v3 ∈ V (B).
Case 5.2.1: Suppose V (A) = v1. Then we also have V (B) = v3.
Case 5.2.1a: If v1 ∈ V(T
(k−1)1
), then v3 ∈ V
(T
(k−1)2
). In this case we
define T(k)0 to be the tree with
V(T
(k)0
)= V
(T
(k−1)0
) \ v1, v3
E(T
(k)0
)= E
(T
(k−1)0
) \ v1, v2, v3, v4,
T(k)1 to be the tree with
V(T
(k)1
)= V
(T
(k−1)1
) ∪ v, v3
E(T
(k)1
)= E
(T
(k−1)1
) ∪ v, v3, v, v4,
and T(k)2 to be the tree with
V(T
(k)2
)= V
(T
(k−1)2
) ∪ v, v1
E(T
(k)2
)= E
(T
(k−1)2
) ∪ v, v1, v, v2.
Case 5.2.1b: If v1 ∈ V(T
(k−1)2
), then v3 ∈ V
(T
(k−1)1
). In this case we
define T(k)0 to be the tree with
V(T
(k)0
)= V
(T
(k−1)0
) \ v1, v3
E(T
(k)0
)= E
(T
(k−1)0
) \ v1, v2, v3, v4,
249
T(k)1 to be the tree with
V(T
(k)1
)= V
(T
(k−1)1
) ∪ v, v1
E(T
(k)1
)= E
(T
(k−1)1
) ∪ v, v1, v, v4,
and T(k)2 to be the tree with
V(T
(k)2
)= V
(T
(k−1)2
) ∪ v, v3
E(T
(k)2
)= E
(T
(k−1)2
) ∪ v, v2, v, v3.
In both cases,E
(T
(k)0
), E
(T
(k)1
), E
(T
(k)2
)is a (Cs, Φ) ⊥ or (Cs, Φ) ‖ 3Tree2
partition of G.
.
.v1 .v3 = σ(v1)
.v4 = σ(v2) .v2
. .
. .
.
.v1 .v3 = σ(v1)
.v4 = σ(v2) .v2
. .
. .
..v
.
.v1 .v3 = σ(v1)
.v4 = σ(v2) .v2
. .
. .
.
.v1 .v3 = σ(v1)
.v4 = σ(v2) .v2
. .
. .
..v
Figure 5.41: Construction of a (Cs, Φ) ‖ 3Tree2 partition of G in the case
where G is a (Cs, Φk−1) X-replacement of Gk−1 and v1, v2, v3, v4 ∈E
(T
(k−1)0
). The edges in black color represent edges of the invariant tree.
Case 5.2.2: Finally, suppose |V (A)| = |V (B)| = m ≥ 2. Then we first
carry out the same construction as in Case 5.2.1. Subsequently, we delete all
the edges of A and B from E(T
(k)0
), one edge from both A and B at a time,
and add them to either E(T
(k)1
)or E
(T
(k)2
)in the following way.
250
Let A be the subgraph of A that only contains the single vertex v1 and
let B be the subgraph of B that only contains the single vertex σ(v1) = v3.
Let v1, z be an edge of A. Then v3, σ(z) is an edge of B. By the
construction in Case 5.2.1, v1, v3 ∈ V(T
(k)i
)for i = 1, 2. Also, σ(z) 6= z and
z, σ(z) ∈ V(T
(k)0
), which says that either z ∈ V
(T
(k)1
)and σ(z) ∈ V
(T
(k)2
)
or z ∈ V(T
(k)2
)and σ(z) ∈ V
(T
(k)1
).
We now delete the edges v1, z and v3, σ(z) from E(T
(k)0
)and if z ∈
V(T
(k)1
), then we add v1, z to E
(T
(k)2
)and v3, σ(z) to E
(T
(k)1
), and
if z ∈ V(T
(k)2
), then we add v1, z to E
(T
(k)1
)and v3, σ(z) to E
(T
(k)2
).
Subsequently, we add the vertex z to V (A), the vertex σ(z) to V (B), the edge
v1, z to E(A), and the edge v3, σ(z) to E(B). If we then have A = A,
then B = B andE
(T
(k)0
), E
(T
(k)1
), E
(T
(k)2
)is a (Cs, Φ) ⊥ or (Cs, Φ) ‖
3Tree2 partition of G.
Otherwise, there exists an edge x, y in E(A) \ E(A) with x ∈ V (A)
and y ∈ V (A) \ V (A), and hence there also exists the edge σ(x), σ(y) in
E(B) \ E(B) with σ(x) ∈ V (B) and σ(y) ∈ V (B) \ V (B). Note that since
x ∈ V (A) and σ(x) ∈ V (B), we have x, σ(x) ∈ V(T
(k)i
)for i = 1, 2. So, we
can repeat the above construction step for the edges x, y and σ(x), σ(y).This process can be continued until A = A and B = B. ¤
Lemma 5.4.5 Let G be a graph with |V (G)| ≥ 2, Cs = Id, s be a symmetry
group in dimension 2, and Φ : Cs → Aut(G) be a homomorphism. If G has a
proper (Cs, Φ) 3Tree2 ⊥ partition or a proper (Cs, Φ) 3Tree2 ‖ partition, then
R(G,Cs,Φ) 6= ∅ and G is (Cs, Φ)-generically isostatic.
Proof. Case 1: Suppose G has a proper (Cs, Φ) ‖ 3Tree2 partition
E(T0), E(T1), E(T2). In the following, we again denote Φ(s) by σ. There
251
exists an edge e = w, z ∈ E(T1) such that σ(w) = w and σ(z) = z and,
by Remark 5.4.3, valT1(w) = 1, w ∈ E(T0), and no other vertex of G that is
fixed by σ is a vertex of T0.
Since G has a 3Tree2 partition, G satisfies the count |E(G)| = 2|V (G)|−3.
Therefore, by Theorems 2.2.5 and 3.2.3, it suffices to find some framework
(G, p) ∈ R(G,Cs,Φ) that is independent.
Let Vi be the set of vertices of G that are not in V (Ti) for i = 0, 1, 2 and
let (G, p, q) be the frame with p : V (G) → R2 and q : E(G) → R2 defined by
p(v) =
(0, 1) if v ∈ V0
(−1, 0) if v ∈ V1
(1, 0) if v ∈ V2 \ v(0, 0) if v = w
q(b) =
(1, 0) if b ∈ EV1,w or b ∈ EV2\w,w
(2, 0) if b ∈ EV1,V2\w
(−1, 1) if b ∈ E(T1) \w, z
(1, 1) if b ∈ E(T2)
(0, 1) if b = w, z
,
where for disjoint sets X,Y ∈ V (G), EX,Y denotes the set of edges of G
incident with a vertex in X and a vertex in Y .
We claim that the generalized rigidity matrix R(G, p, q) has linearly in-
dependent rows. To see this, we first rearrange the columns of R(G, p, q) in
such a way that we obtain the matrix R′(G, p, q) which has the (2i−1)st col-
umn of R(G, p, q) in its ith column and the (2i)th column of R(G, p, q) in its
(|V (G)|+ i)th column for i = 1, 2, . . . , |V (G)|. Let Fb denote the row vector
of R′(G, p, q) that corresponds to the edge b ∈ E(G). We then rearrange the
252
..w
.(−1, 0) .(1, 0).(0, 0)
.(0, 1)
.T1.T2
.V1 .V2 \ w
.V0
.T0
. .
.
.
Figure 5.42: The frame (G, p, q) in Case 1 of the proof of Lemma 5.4.5.
rows of R′(G, p, q) in such a way that we obtain the matrix R′′(G, p, q) which
has the vectors Fb with b ∈ E(T0) in the rows 1, 2, . . . , |E(T0)|, the vectors
Fb with b ∈ E(T1) \w, z in the following |E(T1)| − 1 rows, the vector
Fw,z in the next row, and the vectors Fb with b ∈ E(T2) in the last |E(T2)|rows. So R′′(G, p, q) is of the form
1 −1
... 0
2 −2
−1 1 1 −1
......
−1 1 1 −1
0 1 −1
1 −1 1 −1
......
1 −1 1 −1
.
Clearly, R(G, p, q) has a row dependency if and only if R′′(G, p, q) does.
Suppose R′′(G, p, q) has a row dependency of the form
∑
b∈E(G)
αbFb = 0,
253
where αb 6= 0 for some b ∈ E(T0). Since T0 is a tree, it follows that
∑
b∈E(T0)
αbFb 6= 0.
Thus, there exists a vertex vs ∈ V (T0), s ∈ 1, 2, . . . , |V (G)|, such that
∑
b∈E(T0)
αb(Fb)s = C 6= 0.
Since vs ∈ V (T0), vs belongs to either T1 or T2.
Suppose first that vs ∈ V (T2) and vs /∈ V (T1). Then (Fb)s = 0 and
(Fb)|V (G)|+s = 0 for all b ∈ E(T1) and we have
∑
b∈E(T2)
αb(Fb)s = −C.
This says that
∑
b∈E(T2)
αb(Fb)|V (G)|+s =∑
b∈E(G)
αb(Fb)|V (G)|+s = −C 6= 0,
a contradiction.
So, suppose that vs ∈ V (T1) and vs /∈ V (T2). Then (Fb)s = 0 and
(Fb)|V (G)|+s = 0 for all b ∈ E(T2) and we have
∑
b∈E(T1)
αb(Fb)s = −C.
Note that vs 6= w, because valT1(w) = 1 and (Fw,z)s = 0 for all s =
1, 2, . . . , |V (G)|. Also, vs 6= z, since z /∈ V (T0). Therefore,
∑
b∈E(T1)
αb(Fb)|V (G)|+s =∑
b∈E(G)
αb(Fb)|V (G)|+s = C 6= 0,
which is again a contradiction. So, if∑
b∈E(G) αbFb = 0 is a row dependency
of R′′(G, p, q), then αb = 0 for all b ∈ E(T0).
254
It is now only left to show that the matrix R(G, p, q) which is obtained
from R′′(G, p, q) by deleting those rows of R′′(G, p, q) that correspond to the
edges of T0 has linearly independent rows. Clearly, R(G, p, q) has linearly in-
dependent rows if and only if the matrix R(G, p, q) has linearly independent
rows, where R(G, p, q) is obtained by deleting the row Fw,z from R(G, p, q).
In order to show that R(G, p, q) has linearly independent rows we may mul-
tiply R(G, p, q) by appropriate matrices of basis transformation and then use
arguments analogous to above. So, as claimed, the frame (G, p, q) is inde-
pendent.
Now, if (G, p) is not a framework, then we need to symmetrically
pull apart those joints of (G, p, q) that have the same location in R2 and
whose vertices are adjacent. So suppose |V1| ≥ 2. Then it follows that
|V1| = |V2 \w| ≥ 2, because σ(V1) = V2 \w. Since E(T0), E(T1), E(T2)is proper, one of 〈V1〉 ∩ Ti, i = 0, 2, is not connected.
Suppose first that 〈V1〉 ∩ T0 is not connected. Then 〈V2 \ w〉 ∩ T0 is
..w.(−1, 0) .(1, 0).(0, 0)
.(0, 1)
.T1.T2
.V1 \ A .(V2 \ w) \ σ(A)
.A .σ(A)
.V0
.T0
. .
.
.
. .
. .
Figure 5.43: The frame (G, pt, qt) in the case where 〈V1〉∩T0 is not connected.
also not connected. Let A be the set of vertices in one of the components of
〈V1〉 ∩ T0 and σ(A) be the set of vertices in the corresponding component of
255
〈V2 \ w〉 ∩T0. For t ∈ R, we define pt : V (G) → R2 and qt : E(G) → R2 by
pt(v) =
(−1− t,−t) if v ∈ A
(1 + t,−t) if v ∈ σ(A)
p(v) otherwise
qt(b) =
(1 + t, t) if b ∈ EA,w
(2 + t, t) if b ∈ EA,(V2\w)\σ(A)
(1 + t,−t) if b ∈ Eσ(A),w
(2 + t,−t) if b ∈ Eσ(A),V1\A
q(b) otherwise
.
Suppose now that 〈V1〉 ∩ T2 is not connected. Then 〈V2 \ w〉 ∩ T1 is
also not connected. Let B and σ(B) be the vertex sets of components of
〈V1〉 ∩ T2 and 〈V2 \ w〉 ∩ T1, respectively. In this case, for t ∈ R, we define
pt : V (G) → R2 and qt : E(G) → R2 by
..w.(−1, 0) .(1, 0).(0, 0)
.(0, 1)
.V1 \B .(V2 \ w) \ σ(B)
.V0
.T0
. .
.
.. ...
.T2 .T1
.
. .
.B .σ(B)
Figure 5.44: The frame (G, pt, qt) in the case where 〈V1〉∩T2 is not connected.
pt(v) =
(−1− t, 0) if v ∈ B
(1 + t, 0) if v ∈ σ(B)
p(v) otherwise
256
qt(b) =
(1 + t, 1) if b ∈ EB,V0
(−1− t, 1) if b ∈ Eσ(B),V0
q(b) otherwise
.
In both cases, we have (G, pt, qt) = (G, p, q) if t = 0. Therefore, by
Lemma 5.1.3, there exists a t0 ∈ R, t0 6= 0, such that the frame (G, pt0 , qt0)
is independent. This process can be continued until we obtain an indepen-
dent frame (G, p, q) which has the property that if p(u) = p(v) for some
u, v ∈ E(G), then u, v ∈ V0.
Suppose (G, p) is still not a framework. Then |V0| ≥ 2 and since
E(T0), E(T1), E(T2) is proper, 〈V0〉 ∩ T1 or 〈V0〉 ∩ T2 is not connected.
In fact, since σ(〈V0〉 ∩ T1) = 〈V0〉 ∩ T2, both 〈V0〉 ∩ T1 and 〈V0〉 ∩ T2 are
not connected. Let A be the set of vertices in one of the components of
〈V0〉 ∩ T2 and σ(A) be the set of vertices in the corresponding component
of 〈V0〉 ∩ T1. We denote A ∩ σ(A) by D and A ∪ σ(A) by F . Clearly,
ED,V0\F = ∅, EA\D,V0\F ⊆ E(T1) and Eσ(A)\D,V0\F ⊆ E(T2). Further, we have
EA\D,σ(A)\D = ∅ as the following argument shows.
Suppose to the contrary that there exists x, y ∈ E(G) with x ∈ A \D
and y ∈ σ(A) \ D. Then x, y ∈ E(T1) or x, y ∈ E(T2), say wlog
x, y ∈ E(T2). Since x, y ∈ E(〈V0〉), it follows that x, y ∈ E(〈V0〉∩T2).
Therefore, since x ∈ A, y must also be a vertex of A, contradicting the fact
that y ∈ σ(A) \D.
Finally, note that EA\D,D ⊆ E(T2), because if x, y ∈ E(T1), where
x ∈ A \D and y ∈ D, then we must have x ∈ σ(A), contradicting x ∈ A \D.
Similarly, we have Eσ(A)\D,D ⊆ E(T1).
257
So, for t ∈ R, we define pt : V (G) → R2 and qt : E(G) → R2 by
pt(v) =
(−t, 1 + t) if v ∈ A \D
(t, 1 + t) if v ∈ σ(A) \D
(0, 1 + 2t) if v ∈ D
p(v) otherwise
qt(b) =
q(b) + (t, t) if b ∈ EV2\w,σ(A)\D
q(b) + (−t, t) if b ∈ EV1,A\D
q(b) + (0, 2t) if b ∈ EV2\w,D
q(b) + (0, 2t) if b ∈ EV1,D
q(b) otherwise
.
Then (G, pt, qt) = (G, p, q) if t = 0. Therefore, by Lemma 5.1.3, there exists
..w
.(−1, 0) .(1, 0).(0, 0)
.(0, 1)
.V1 .V2 \ w
.V0 \ F
.T0
. .
.
.
.
. .
.D
.A \D .σ(A) \D
.T2 .T1
.
. .
. .
.
Figure 5.45: The frame (G, pt, qt).
a t0 ∈ R, t0 6= 0, such that the frame (G, pt0 , qt0) is independent.
Now, if |A \ D| ≥ 2, then |σ(A) \ D| = |A \ D| ≥ 2. Since
E(T0), E(T1), E(T2) is proper, 〈A \ D〉 ∩ T1 or 〈A \ D〉 ∩ T2 is not con-
nected, say wlog 〈A \ D〉 ∩ T2 is not connected. Then 〈σ(A) \ D〉 ∩ T1 is
258
also not connected. Let B be the set of vertices in one of the components of
〈A \D〉 ∩T2 and σ(B) be the set of vertices in the corresponding component
of 〈σ(A) \ D〉 ∩ T1. Then, by using arguments analogous to above, we can
pull apart the vertices of B from (A \D) \ B in the direction of the vector
(−t, t) and the vertices of σ(B) from (σ(A) \ D) \ σ(B) in the direction of
the vector (t, t) in order to obtain a new independent frame.
This process can be continued until we obtain an independent frame
(G, p, q) with p(u) 6= p(v) for all u, v ∈ E(G). Then, by Remark 5.1.1,
(G, p) is an independent framework and, if necessary, an appropriate rota-
tion of (G, p) about the origin yields an independent framework in the set
R(G,Cs,Φ).
Case 2: Suppose G has a proper (Cs, Φ) ⊥ 3Tree2 partition
E(T0), E(T1), E(T2). Let Vi be the set of vertices of G that are not in
V (Ti) for i = 0, 1, 2 and let e0 = (0, 1), e1 = (−1, 0), and e2 = (1, 0). We let
(G, p, q) be the frame with p : V (G) → R2 and q : E(G) → R2 defined by
p(v) = ei if v ∈ Vi
q(b) =
(2, 0) if b ∈ E(T0)
(−1, 1) if b ∈ E(T1)
(1, 1) if b ∈ E(T2)
.
The proof that (G, p, q) is independent and that we can construct an
independent framework (G, p) ∈ R(G,Cs,Φ) is analogous to the proof of Case
1. ¤
Lemmas 5.4.2, 5.4.3, 5.4.4, and 5.4.5 provide a complete proof for Theo-
rem 5.4.1.
259
..e1 .e2
.e0
.T1.T2
.V1 .V2
.V0
.T0
. .
.
Figure 5.46: The frame (G, p, q) in Case 2 of the proof of Lemma 5.4.5.
Remark 5.4.5 By generalizing the geometric proofs of Lemmas 5.2.6 and
5.2.7, and by using the fact that the framework (G, p) which is obtained
from an isostatic framework (G, p) by performing an X-replacement on G
and placing the new vertex in G at the point of intersection of the two bars
that were removed from (G, p) is again isostatic (see the proof of Proposition
3.9 in [68], for example), it is straightforward to also give a direct geometric
proof that condition (iii) implies condition (i) in Theorem 5.4.1, i.e., that the
existence of a (Cs, Φ) construction sequence for G implies that R(G,Cs,Φ) 6= ∅and that G is (Cs, Φ)-generically isostatic.
Remark 5.4.6 Theorem 5.4.1 still holds if we omit (Cs, Φi) single edge splits
in condition (iii). However, all the other inductive construction techniques,
including the (Cs, Φi) X-replacement, are necessary to characterize all (Cs, Φ)-
generically isostatic graphs in terms of an inductive construction sequence.
260
.
.
.
.
.
.
.
.
.
.
. .
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
Figure 5.47: Two frameworks whose underlying graphs satisfy the conditions
of Case B.2.3 in the proof of Lemma 5.4.3 with respect to the types Φ that
are uniquely determined by the injective realizations. Any symmetrized Hen-
neberg’s sequence for any of these two graphs needs to include a (Cs, Φi)
X-replacement.
5.5 Conjectures, algorithms, and further re-
marks
5.5.1 Dimension 2
For the symmetry groups C2v and C3v in dimension 2, it was conjectured in
[15] that the following Laman-type characterizations of (C2v, Φ)- and (C3v, Φ)-
generically isostatic graphs hold.
Conjecture 5.5.1 Let G be a graph with |V (G)| ≥ 2, C2v be a symmetry
group in dimension 2, and Φ : C2v → Aut(G) be a homomorphism. Then
R(G,C2v ,Φ) 6= ∅ and G is (C2v, Φ)-generically isostatic if and only if
(i) |E(G)| = 2|V (G)| − 3 and |E(H)| ≤ 2|V (H)| − 3 for all H ⊆ G with
|V (H)| ≥ 2 (Laman conditions);
261
(ii) jΦ(C2) = 0 and bΦ(C2) = bΦ(s) = 1 for both reflections s ∈ C2v.
Conjecture 5.5.2 Let G be a graph with |V (G)| ≥ 3, C3v be a symmetry
group in dimension 2, and Φ : C3v → Aut(G) be a homomorphism. Then
R(G,C3v ,Φ) 6= ∅ and G is (C3v, Φ)-generically isostatic if and only if
(i) |E(G)| = 2|V (G)| − 3 and |E(H)| ≤ 2|V (H)| − 3 for all H ⊆ G with
|V (H)| ≥ 2 (Laman conditions);
(ii) jΦ(C3) = 0 and bΦ(s) = 1 for all reflections s ∈ C3v.
If one wants to prove these conjectures in the analogous way as the sym-
metrized Laman’s Theorems for C3, C2, and Cs, one has to consider two basic
cases: first, the case where the given graph G has a vertex of valence 2, and
secondly, the case where G has a vertex of valence 3 and no vertex of valence
2.
For each of the groups C2v and C3v, the first case can be treated in a
straightforward fashion by using appropriate symmetrized versions of a ver-
tex 2-addition.
We have seen in the previous section that for vertices of valence 3, the
presence of a single reflection s in the symmetry group S gives rise to a
large number of subcases that need to be treated separately, where each
subcase corresponds to a particular allocation of the 3-valent vertex and its
three neighbors to the permutation cycles of the graph automorphism Φ(s).
Since the symmetry groups C2v and C3v contain more than just one reflection
(namely two and three, respectively), the number of subcases that need to
be considered for these groups is even larger than it was in the case of Cs.
262
So, while we suspect that the above conjectures can be proven in this way,
the number of cases that need to be treated in these proofs is unreasonably
large for the scope of this thesis.
We conjecture that (C3v, Φ)-generically isostatic graphs can also be char-
acterized by means of symmetrized 3Tree2 partitions. We need the following
definitions.
Definition 5.5.1 Let G be a graph, C3v = 〈C3, s〉 be a symmetry group in
dimension 2, and Φ : C3v → Aut(G) be a homomorphism. A (C3v, Φ) 3Tree2
⊥ partition of G is a 3Tree2 partition E(T0), E(T1), E(T2) of G such that
(i) Φ(C3)(Ti) = Ti+1 for i = 0, 1, 2, where the indices are added modulo 3;
(ii) there exists i ∈ 0, 1, 2 such that Φ(s)(Ti) = Ti and Φ(s)(Ti+1) = Ti+2.
.
.γ2(v)
.v .γ(v) = σ(v)
.γ2(w)
.w
.γ(w).
. .
.
.
..T0
.T2
.T1
.(a)
.
.γ2(x)
.x .γ(x) = σ(x)
.γ2(w)
.w
.γ(w)
.γ2(v)
.v
.γ(v)
.
. .
.
.
..
.
..T0
.T1.T2
.(b)
Figure 5.48: A (C3v, Φ) 3Tree2 ⊥ partition of a graph (a) and a (C3v, Φ)
3Tree2 ‖ partition of a graph (b), where Φ(C3) = γ and Φ(s) = σ.
Definition 5.5.2 Let G be a graph, C3v = 〈C3, s〉 be a symmetry group in
dimension 2, and Φ : C3v → Aut(G) be a homomorphism such that there
263
exists an edge e = v, w ∈ E(G) with Φ(s)(v) = v and Φ(s)(w) = w. A
(C3v, Φ) 3Tree2 ‖ partition of G is a 3Tree2 partition E(T0), E(T1), E(T2)of G such that
(i) Φ(C3)(Ti) = Ti+1 for i = 0, 1, 2, where the indices are added modulo 3;
(ii) e ∈ E(T1), Φ(s)(T1 − v) = T2 − Φ(C3)(v) and Φ(s)(T0 −Φ(C2
3)(v)) = T0 − Φ(C23)(v).
Conjecture 5.5.3 Let G be a graph with |V (G)| ≥ 3, C3v be a symmetry
group in dimension 2, and Φ : C3v → Aut(G) be a homomorphism. Then
R(G,C3v ,Φ) 6= ∅ and G is (C3v, Φ)-generically isostatic if and only if G has a
proper (C3v, Φ) 3Tree2 ⊥ partition or a proper (C3v, Φ) 3Tree2 ‖ partition.
Due to the structure of the group C2v, there does not seem to exist an
analogous characterization of (C2v, Φ)-generically isostatic graphs in terms of
symmetrized 3Tree2 partitions.
Algorithms
An immediate consequence of the symmetrized Laman’s Theorems for C3,
C2, and Cs (and the above conjectures for C2v and C3v) is that there is (would
be) a polynomial time algorithm to determine whether a given graph G is
(S, Φ)-generically isostatic. In fact, although the Laman conditions involve
an exponential number of subgraphs of G, there are several algorithms that
determine whether they hold in c|V (G)||E(G)| steps, where c is a constant.
The pebble game ([39]) is an example for such an algorithm. The additional
264
symmetry conditions for the number of fixed structural elements can trivially
be checked in constant time, from the graph automorphisms.
5.5.2 Dimension 3
There is no known characterization of generically 3-isostatic graphs,
although we have the necessary conditions identified in Theorem 2.2.8:
|E(G)| = 3|V (G)| − 6 and |E(H)| ≤ 3|V (H)| − 6 for all subgraphs H of
G with |V (H)| ≥ 3. Recall from Section 2.2.5, however, that there are
a number of inductive construction techniques which are known to preserve
the generic rigidity properties of a graph. Suppose we are given such a generi-
cally 3-isostatic graph G with a construction sequence. Further, suppose S is
a symmetry group in dimension 3 and Φ : S → Aut(G) is a homomorphism.
Then the graph G can only be (S, Φ)-generically isostatic if G satisfies all the
conditions given in Theorem 4.3.3 for the symmetry operations in S (with Φ
as the underlying type) as well as the corresponding conditions for all sub-
graphs H with the full count |E(H)| = 3|V (H)|−6 and either reflectional or
half-turn symmetry. One may ask whether all of these conditions combined
are also sufficient for G to be (S, Φ)-generically isostatic. The following ex-
ample shows that this is in general not the case.
Consider the (Cs, Φ)-generic realization (K4,6, p) of the complete bipar-
tite graph K4,6 with partite sets X = v1, v2, v3, v4, and Y = v5, . . . , v10shown in Figure 5.49. The graph K4,6 is generically 3-isostatic, and for the
reflection s ∈ Cs, we have jΦ(s) = bΦ(s) = 0, so that all the symmetry con-
ditions given in Section 4.3.3 are satisfied. However, the mirror symmetry
forces the four points p1, p2, p3, p4 corresponding to the vertices in X to be
265
.
....
....
..
....
..
.....p1
.p5
.p6
.p7
.p8
.p9
.p10
.p2
.p3
.p4
Figure 5.49: A (Cs, Φ)-generic realization of the complete bipartite graph K4,6.
coplanar, so that it follows from the results in [71] or [75] that (K4,6, p) is
infinitesimally flexible.
Further examples and a more detailed investigation of how ‘flatness’
caused by symmetry gives rise to additional necessary conditions for a graph
to be (S, Φ)-generically isostatic, where S is a symmetry group in dimension
3, will be presented in [59]. This builds on [70].
What these ‘failures from flatness’ show is that symmetry in dimension
d > 2 induces extra conditions for a graph G to be (S, Φ)-generically isostatic
beyond those of
(a) G being generically d-isostatic and
(b) the symmetry-extended Maxwell’s rules for G and all subgraphs of G
with the full Maxwell count.
We conjecture that flatness is the only additional concern, and that it can be
made into a finite set of added combinatorial conditions, for each symmetry
group.
266
5.5.3 Independence and infinitesimal rigidity
In Section 4.4, we described how to derive necessary conditions for a d-
dimensional symmetric framework (G, p) ∈ R(G,S,Φ) to be independent and
infinitesimally rigid, respectively, provided that the type Φ : S → Aut(G) of
(G, p) is a group homomorphism and that the points p(v), v ∈ V (G), span
all of Rd.
Given a graph G, it follows immediately from Lemma 3.2.2 and Theorem
3.2.3 that if there exists a framework (G, p) ∈ R(G,S,Φ) so that p(v), v ∈ V (G),
span all of Rd, then the conditions derived in Section 4.4 are also necessary
conditions for G to be (S, Φ)-generically independent and (S, Φ)-generically
infinitesimally rigid, respectively.
In this section, we investigate whether sufficient conditions for a graph
to be (S, Φ)-generically independent or (S, Φ)-generically infinitesimally rigid
can also be established.
In our exploration of these questions we restrict our attention to the
symmetry group C3 in dimension 2, since for this group we have the most
natural characterizations of (S, Φ)-generically isostatic graphs, as we have
seen in the previous sections. Similar explorations, however, can of course
also be carried out for other symmetry groups, with analogous corollaries for
C2 and Cs, and analogous conjectures for these groups and others.
Using the techniques described in Section 4.4, we obtain the following
result.
Theorem 5.5.4 Let G be a graph, C3 = Id, C3, C23 be a symmetry group
in dimension 2, and Φ : C3 → Aut(G) be a homomorphism, so that the set
267
R(G,C3,Φ) contains a framework (G, p) with the property that the points p(v),
v ∈ V (G), span all of R2.
(i) If G is (C3, Φ)-generically independent, then |E(G)| ≤ 2|V (G)| − 3 −2jΦ(C3);
(ii) if G is (C3, Φ)-generically infinitesimally rigid, then |E(G)| ≥ 2|V (G)|−3 + jΦ(C3).
Independence
We suppose first that jΦ(C3) = 0. In this case, we claim that for each of
the characterizations of (C3, Φ)-generically isostatic graphs given in Theorem
5.2.1, there also exists an analogous characterization of (C3, Φ)-generically
independent graphs. We need the following definitions.
Definition 5.5.3 Let G be a graph, C3 = Id, C3, C23 be a symmetry group
in dimension 2, Φ : C3 → Aut(G) be a homomorphism, and v0 ∈ V (G). Then
(a) the graph G with V (G) = V (G)∪v, w, z and E(G) = E(G) is called
a (C3, Φ) partial vertex addition of order 0 (by (v, w, z)) of G .
(b) the graph G with V (G) = V (G) ∪ v, w, z and E(G) = E(G) ∪v, v0, w, Φ(C3)(v0), z, Φ(C2
3)(v0)
is called a (C3, Φ) partial ver-
tex addition of order 1 (by (v, w, z)) of G.
(c) the graph G with V (G) = V (G) ∪ v, w, z and E(G) = E(G) ∪v, w, w, z, z, v is called a (C3, Φ) ∆ addition (by (v, w, z)) of
G.
268
. ..
. .
.z
.v .w
.
.. . .
.. .
.
. .
.z
.v .w
..(a) .(b) .
. .
.
. .
.z
.v .w
..(c) .
Figure 5.50: A (C3, Φ) partial vertex addition of order 0 of a graph G (a), a
(C3, Φ) partial vertex addition of order 1 of a graph G (b), and a (C3, Φ) ∆
addition of a graph G (c).
Definition 5.5.4 Let G be a graph, C3 = Id, C3, C23 be a symmetry group
in dimension 2, and Φ : C3 → Aut(G) be a homomorphism. A (C3, Φ)
3Forest2 partition of G is a partition of E(G) into the edge sets of three edge
disjoint forests F0, F1, F2 such that each vertex of G belongs to exactly two of
the forests and Φ(C3)(Fi) = Fi+1 for i = 0, 1, 2, where the indices are added
modulo 3.
A 3Forest2 partition is called proper if no non-trivial subtrees of distinct
forests Fi have the same span.
The proof of Theorem 5.2.1 can be extended directly to a proof of the
following result.
Corollary 5.5.5 Let G be a graph with |V (G)| ≥ 3, C3 = Id, C3, C23 be a
symmetry group in dimension 2, and Φ : C3 → Aut(G) be a homomorphism
so that jΦ(C3) = 0. The following are equivalent:
269
(i) R(G,C3,Φ) 6= ∅ and G is (C3, Φ)-generically independent;
(ii) |E(G)| = 2|V (G)| − 3 and |E(H)| ≤ 2|V (H)| − 3 for all H ⊆ G with
|V (H)| ≥ 2 (Laman conditions);
(iii) there exists a sequence
(G0, Φ0), (G1, Φ1), . . . , (Gk, Φk) = (G, Φ)
such that
(a) G0 = K3 or G0 is the graph with three vertices and no edges, Gi+1
is a (C3, Φi) partial vertex addition of order 0 or 1, a (C3, Φi) vertex
addition, a (C3, Φi) edge split, a (C3, Φi) ∆ addition, or a (C3, Φi)
∆ extension of Gi with V (Gi+1) = V (Gi) ∪ vi+1, wi+1, zi+1 for
all i = 0, 1, . . . , k − 1;
(b) Φ0 : C3 → Aut(G0) is a non-trivial homomorphism and for
all i = 0, 1, . . . , k − 1, Φi+1 : C3 → Aut(Gi+1) is the homo-
morphism defined by Φi+1(x)|V (Gi) = Φi(x) for all x ∈ C3 and
Φi+1(C3)|vi+1,wi+1,zi+1 = (vi+1 wi+1 zi+1);
(iv) G has a proper (C3, Φ) 3Forest2 partition.
Corollary 5.5.6 Let G be a graph with |V (G)| ≥ 3, C3 = Id, C3, C23 be a
symmetry group in dimension 2, and Φ : C3 → Aut(G) be a homomorphism
so that jΦ(C3) = 0. Then G is (C3, Φ)-generically independent if and only if G
is a subgraph of a (C3, Φ′)-generically isostatic graph G′ with V (G) = V (G′),
where Φ′ : C3 → Aut(G′) is defined by Φ′(x) = Φ(x) for all x ∈ C3.
270
Proof. If G is a subgraph of a (C3, Φ′)-generically isostatic graph G′ with
V (G) = V (G′), where Φ′ : C3 → Aut(G′) is defined by Φ′(x) = Φ(x) for all
x ∈ C3, then G is clearly (C3, Φ)-generically independent.
The converse follows immediately from Corollary 5.5.5 (iii). ¤
Whenever jΦ(C3) 6= 0, the analogous result to Corollary 5.5.6 does not
hold. In fact, given any independent framework (G, p) ∈ R(G,C3,Φ), where
jΦ(C3) 6= 0, there does not exist any isostatic framework (G′, p′) ∈ R(G′,C3,Φ′)
so that G ⊆ G′, V (G) = V (G′), and Φ′ : C3 → Aut(G′) is defined by
Φ′(x) = Φ(x) for all x ∈ C3, because an isostatic framework in R(G′,C3,Φ′)
must satisfy jΦ′(C3) = 0.
..
.
. ...
.
. .
Figure 5.51: Independent frameworks in R(G,C3,Φ) with jΦ(C3) = 1. These
frameworks cannot be contained in an isostatic framework that has the same
joints and also C3 symmetry.
Note also that if jΦ(C3) 6= 0, then a characterization of (C3, Φ)-generically
independent graphs in terms of symmetric 3Forest2 partitions, analogous to
the one given in Corollary 5.5.5, does not exist, because a vertex that is fixed
by Φ(C3) can only belong to either none or all three of the forests of a (C3, Φ)
3Forest2 partition. However, we conjecture that for jΦ(C3) 6= 0, the following
characterizations of (C3, Φ)-generically independent graphs hold.
Conjecture 5.5.7 Let G be a graph, C3 = Id, C3, C23 be a symmetry group
in dimension 2, and Φ : C3 → Aut(G) be a homomorphism, so that jΦ(C3) 6= 0
271
and the set R(G,C3,Φ) contains a framework (G, p) with the property that the
points p(v), v ∈ V (G), span all of R2. The following are equivalent:
(i) R(G,C3,Φ) 6= ∅ and G is (C3, Φ)-generically independent;
(ii) |E(G)| ≤ 2|V (G)| − 3− 2jΦ(C3), |E(H)| ≤ 2|V (H)| − 3 for all H ⊆ G
with |V (H)| ≥ 2, and for every subgraph H of G that is invariant
under Φ(C3), contains exactly m vertices that are fixed by Φ(C3), where
0 ≤ m ≤ jΦ(C3), and satisfies |V (H)| ≥ m + 3, we have |E(H)| ≤2|V (H)| − 3− 2m;
(iii) there exists a sequence
(G0, Φ0), (G1, Φ1), . . . , (Gk, Φk) = (G, Φ)
such that
(a) G0 is the graph with jΦ(C3) vertices and no edges, Gi+1 is a (C3, Φi)
partial vertex addition of order 0 or 1, a (C3, Φi) vertex addition,
a (C3, Φi) edge split, a (C3, Φi) ∆ addition, or a (C3, Φi) ∆ ex-
tension of Gi with V (Gi+1) = V (Gi) ∪ vi+1, wi+1, zi+1 for all
i = 0, 1, . . . , k − 1;
(b) Φ0 : C3 → Aut(G0) is the homomorphism that sends each x ∈ C3
to the identity automorphism and for all i = 0, 1, . . . , k− 1, Φi+1 :
C3 → Aut(Gi+1) is the homomorphism defined by Φi+1(x)|V (Gi) =
Φi(x) for all x ∈ C3 and Φi+1(C3)|vi+1,wi+1,zi+1 = (vi+1 wi+1 zi+1).
272
Infinitesimal rigidity
Suppose first that for a graph G, we have jΦ(C3) = 0. If G has a (C3, Φ′)-
generically isostatic subgraph G′ with V (G) = V (G′), where Φ′ : C3 →Aut(G′) is defined by Φ′(x) = Φ(x) for all x ∈ C3, then G is clearly (C3, Φ)-
generically infinitesimally rigid. We conjecture that the converse of this result
also holds and that it can be proven using symmetric 3Forest2 partitions of
graphs.
Whenever jΦ(C3) 6= 0, however, the analogous result does not hold. In fact,
given any infinitesimally rigid framework (G, p) ∈ R(G,C3,Φ), where jΦ(C3) 6= 0,
there does not exist any isostatic framework (G′, p′) ∈ R(G′,C3,Φ′) so that
G′ ⊆ G, V (G) = V (G′), and Φ′ : C3 → Aut(G′) is defined by Φ′(x) = Φ(x)
for all x ∈ C3, because an isostatic framework in R(G′,C3,Φ′) must satisfy
jΦ′(C3) = 0.
..
.
. ...
.
. .
..
.
Figure 5.52: Infinitesimally rigid frameworks in R(G,C3,Φ) with jΦ(C3) = 1.
These frameworks cannot contain an isostatic framework that has the same
joints and also C3 symmetry.
In order to obtain characterizations of (C3, Φ)-generically infinitesimally
rigid graphs in cases where jΦ(C3) 6= 0, one needs to apply methods beyond
those developed in this thesis. One might ask, for example, whether the
Lovasz-Yemini matroid partition algorithm (see [48]) can be appropriately
symmetrized. These questions, however, have not yet been explored.
273
Chapter 6
Symmetry as a sufficient
condition for a flex
Recall from Section 2.2.2 that if a framework (G, p) is infinitesimally rigid,
then (G, p) is also rigid. The converse of this result does not hold. However,
in the end of Section 2.2.5 we summarized some results which assert that
under certain conditions, the existence of an infinitesimal flex of (G, p) also
implies the existence of a flex of (G, p). In fact, it follows from Theorem
2.2.15 and Corollary 2.2.16 that for ‘almost all’ realizations of the graph G,
rigidity and infinitesimal rigidity are equivalent. If (G, p) is a symmetric
framework with a non-trivial point group, however, then the joints of (G, p)
typically lie in special, non-generic (and frequently also non-regular) posi-
tions, so that these results of Section 2.2.5 cannot be applied to (G, p).
In this chapter, we show that if (G, p) is a symmetric framework in the
set R(G,S,Φ), where S is a non-trivial symmetry group and Φ : S → Aut(G)
is a homomorphism, then the symmetry of (G, p) can be exploited to obtain
274
sufficient conditions for the existence of a flex of (G, p). We establish these
conditions by symmetrizing the methods in [3] and by using the fact that the
rigidity matrix of (G, p) can be block-diagonalized as described in Section
4.1.3. As a corollary of these results, one obtains the Proposition 1 stated
(but not proven) in [43].
In Section 6.3, we use these symmetry-adapted results to prove the exis-
tence of a flex for a number of interesting and famous examples of symmetric
frameworks in both 2D and 3D.
The structures analyzed in [35] and [64] can also be proven to be flexible
using the methods of this chapter.
The development of a symmetry-based rigidity analysis not only yields
additional sufficient conditions for the existence of a flex of a given frame-
work, but it also enables us to obtain valuable information about the sym-
metry properties of a detected flex. In particular, using the results of this
chapter, we can determine whether a framework (G, p) in R(G,S,Φ) possesses
a ‘symmetry-preserving’ flex, i.e., a flex which moves the joints of (G, p) on
differentiable displacement paths in such a way that all the resulting frame-
works remain in the set R(G,S,Φ).
Each of the frameworks in Section 6.3 which we prove to be flexible with
the results of this chapter possesses such a symmetry-preserving flex, and it
is precisely this kind of flex that the new methods detect in each case. While
detection of these flexes is not new, the verification of the flexes uses this
new approach, and is much simpler than previous methods. New flexes can
also be detected, and some will be presented in [58]. See also Chapter 7 for
further comments.
275
6.1 Alternate definitions of rigidity via the
edge function
We begin by introducing the edge function of a graph G and then refor-
mulating some of the basic definitions in rigidity theory stated in Section 2.2
in terms of this function. The introduction of symmetric versions of these
‘reformulated’ definitions will allow us to examine the rigidity properties of
symmetric frameworks in Sections 6.2 and 6.3.
Definition 6.1.1 Let G be a graph with V (G) = v1, v2, . . . , vn. For a fixed
ordering of the edges of G, we define the edge function fG : Rdn → R|E(G)| by
fG
(p(v1), . . . , p(vn)
)=
(. . . , ‖p(vi)− p(vj)‖2, . . .
),
where vi, vj ∈ E(G) and p(vi) ∈ Rd for all i = 1, . . . , n.
If (G, p) is a d-dimensional framework with n vertices, then f−1G
(fG(p)
)
is the set of all configurations q of n points in Rd with the property that
corresponding bars of the frameworks (G, p) and (G, q) have the same length.
In particular, we clearly have f−1Kn
(fKn(p)
) ⊆ f−1G
(fG(p)
), where Kn is the
complete graph on V (G).
The definitions of a motion, a flex, and a rigid motion of (G, p) (see
Definitions 2.2.3 and 2.2.4) can be rewritten in terms of the edge function of
G as follows.
Definition 6.1.2 Let G be a graph with n vertices and let (G, p) be a frame-
work in Rd. A motion of (G, p) is a differentiable path x : [0, 1] → Rdn such
276
that x(0) = p and x(t) ∈ f−1G
(fG(p)
)for all t ∈ [0, 1].
A motion x of (G, p) is a rigid motion if x(t) ∈ f−1Kn
(fKn(p)
)for all
t ∈ [0, 1] and a flex of (G, p) if x(t) /∈ f−1Kn
(fKn(p)
)for all t ∈ (0, 1].
The next result gives some alternate definitions of a flexible framework
all of which are equivalent to Definition 2.2.5, as shown in [3].
Theorem 6.1.1 [3] Let (G, p) be a framework in Rd with n vertices. The
following are equivalent:
(i) (G, p) is flexible;
(ii) there exists a flex x : [0, 1] → Rdn of (G, p);
(iii) there exists a motion x : [0, 1] → Rdn of (G, p) such that x(t) /∈f−1
Kn
(fKn(p)
)for some t ∈ (0, 1];
(iv) for every neighborhood Np of p ∈ Rdn, we have f−1Kn
(fKn(p)
) ∩ Np &
f−1G
(fG(p)
) ∩Np.
Remark 6.1.1 In Definition 6.1.2, we may replace the term ‘differentiable
path’ by the terms ‘continuous path’ or ‘analytic path’. The fact that all of
these definitions are equivalent is a consequence of some basic results from
algebraic geometry [3, 54, 83].
Let fG : Rdn → R|E(G)| be the edge function of a graph G. Then it is
an easy but important observation that the rigidity matrix of a framework
(G, p) is (up to a constant) the Jacobian matrix dfG(p) of fG, evaluated at
the point p ∈ Rdn.
277
Recall from Section 2.2.5 that a point p ∈ Rdn is said to be a regular
point of a graph G if there exists a neighborhood Np of p in Rdn so that
rank(R(G, p)
) ≥ rank(R(G, q)
)for all q ∈ Np. This definition may now
also be reformulated in terms of the edge function fG as follows.
Definition 6.1.3 A point p ∈ Rdn is a regular point of a graph G if
there exists a neighborhood Np of p in Rdn so that rank(dfG(p)
)=
max rank(dfG(q)
)| q ∈ Np.
6.2 Detection of symmetric flexes
Our main goal in this section is to find sufficient conditions for the exis-
tence of a ‘symmetry-preserving’ flex of a symmetric framework. As we will
see, the existence of a flex that preserves some, but not all of the symmetries
of a given framework can be predicted in an analogous way.
Let G be a graph with V (G) = v1, . . . , vn, S be a symmetry group in
dimension d with r pairwise non-equivalent irreducible linear representations
I1, . . . , Ir, Φ : S → Aut(G) be a homomorphism, and (G, p) be a framework
in R(G,S,Φ). In this chapter, I1 will always denote the trivial irreducible linear
representation of S, i.e., I1 denotes the linear representation of degree one
with the property that I1(x) is the identity transformation for all x ∈ S.
Recall from Section 4.1.3 that V(I1)e denotes the H ′
e-invariant subspace of
Rdn which corresponds to I1. So, p ∈ V(I1)e if and only if H ′
e(x)(p) = p for all
x ∈ S.
Further, recall from Section 3.2 that if (G, p) is a framework in R(G,S,Φ),
278
then p ∈ Rdn is an element of the subspace U =⋂
x∈S Lx,Φ of Rdn, where
Lx,Φ = ker(M(x) −PΦ(x)
)for all x ∈ S.
Note that it follows immediately from the definitions of U and V(I1)e that
U = V(I1)e , because p ∈ U if and only if M(x)p = PΦ(x)p for all x ∈ S if and
only if (PΦ(x))TM(x)p = p for all x ∈ S if and only if He(x)p = p for all
x ∈ S if and only if p ∈ V(I1)e .
So, since we are interested in flexes of (G, p) ∈ R(G,S,Φ) that preserve the
symmetry of (G, p), we need to restrict the edge functions fG of G and fKn of
Kn to the subspace V(I1)e of Rdn. In the following, we let fG : V
(I1)e → R|E(G)|
denote the restriction of fG to V(I1)e , and fKn : V
(I1)e → R(n
2) denote the
restriction of fKn to V(I1)e . The Jacobian matrices of fG and fKn , evaluated
at a point p ∈ V(I1)e , are denoted by dfG(p) and dfKn(p), respectively.
Definition 6.2.1 An element p ∈ V(I1)e is said to be a regular point of G in
V(I1)e if there exists a neighborhood Np of p in V
(I1)e so that rank
(dfG(p)
)=
max rank(dfG(q)
)| q ∈ Np. A regular point of Kn in V(I1)e is defined
analogously.
Definition 6.2.2 An (S, Φ)-symmetry-preserving flex of a framework
(G, p) ∈ R(G,S,Φ) is a differentiable path x : [0, 1] → V(I1)e such that x(0) = p
and x(t) ∈ f−1G
(fG(p)
) \ f−1Kn
(fKn(p)
)for all t ∈ (0, 1].
Lemma 6.2.1 Let G be a graph, S be a symmetry group, Φ : S →Aut(G) be a homomorphism, and (G, p) be a framework in R(G,S,Φ). If p
is a regular point of G in V(I1)e , then there exists a neighborhood Np of
p in V(I1)e such that f−1
G
(fG(p)
) ∩ Np is a smooth manifold of dimension
dim(V
(I1)e
)− rank(dfG(p)
).
279
Proof. The result follows immediately from Proposition 2 (and subsequent
remark) in [3]. ¤
Theorem 6.2.2 Let G be a graph with n vertices, S be a symmetry group,
Φ : S → Aut(G) be a homomorphism, and (G, p) be a framework in R(G,S,Φ).
If p is a regular point of G in V(I1)e and also a regular point of Kn in V
(I1)e ,
then
(i) rank(dfG(p)
)= rank
(dfKn(p)
)if and only if (G, p) has no (S, Φ)-
symmetry-preserving flex;
(ii) rank(dfG(p)
)< rank
(dfKn(p)
)if and only if (G, p) has an (S, Φ)-
symmetry-preserving flex.
Proof. Since p is a regular point of both G and Kn in V(I1)e , it follows from
Lemma 6.2.1 that there exist neighborhoods Np and N ′p of p in V
(I1)e so that
f−1G
(fG(p)
)∩Np is a manifold of dimension dim(V
(I1)e
)− rank(dfG(p)
)and
f−1Kn
(fKn(p)
) ∩N ′p is a manifold of dimension dim
(V
(I1)e
)− rank(dfKn(p)
).
Since f−1Kn
(fKn(p)
) ∩ N ′′p is a submanifold of f−1
G
(fG(p)
) ∩ N ′′p , where N ′′
p =
Np ∩N ′p, it follows that
rank(dfG(p)
) ≤ rank(dfKn(p)
).
Clearly, rank(dfG(p)
)= rank
(dfKn(p)
)if and only if there exists a neigh-
borhood N∗p of p in V
(I1)e such that f−1
Kn
(fKn(p)
) ∩ N∗p = f−1
G
(fG(p)
) ∩ N∗p .
Therefore, if rank(dfG(p)
)= rank
(dfKn(p)
), then there does not exist an
(S, Φ)-symmetry-preserving flex of (G, p).
If rank(dfG(p)
)< rank
(dfKn(p)
), then every neighborhood of p in V
(I1)e
280
contains elements of f−1G
(fG(p)
) \ f−1Kn
(fKn(p)
), and hence, by the proof of
Proposition 1 in [3] (and references therein), there exists an (S, Φ)-symmetry-
preserving flex of (G, p). This completes the proof. ¤
In order to make further use of Theorem 6.2.2, we need the following
fundamental observations.
Recall from Section 4.1.3 that with respect to the bases Be and Bi, the
rigidity matrix of a framework (G, p) in R(G,S,Φ) has the block form
R(G, p) =
R1(G, p) 0
. . .
0 Rr(G, p)
, (6.1)
where for t = 1, . . . , r, the block Rt(G, p) corresponds to the irreducible
linear representation It of S, and the size of the block Rt(G, p) depends
on the dimensions of the subspaces V(It)e of Rdn and V
(It)i of R|E(G)|. (In
particular, the block Rt(G, p) is an empty matrix if and only if both of the
coefficients λt and µt in equations (4.2) and (4.6) are equal to zero.)
Since with respect to the bases Be and Bi, the Jacobian matrix of fG,
evaluated at p, is (up to a constant) the matrix R(G, p), it follows that with
respect to the bases B(I1)e and Bi, the Jacobian matrix of fG, evaluated at
the point p ∈ V(I1)e , is (up to a constant) the matrix
R1(G, p)
0
...
0
.
Thus, we have
rank(R1(G, p)
)= rank
(dfG(p)
). (6.2)
281
Furthermore, note that if Kn is the complete graph on the vertex set
V (G), then with respect to the bases Be and Bi, where Bi is an appropriate
extension of the basis Bi, the rigidity matrix of (Kn, p) has a block form
analogous to the one of R(G, p) in (6.1), namely
R(Kn, p) =
R1(Kn, p) 0
. . .
0 Rr(Kn, p)
.
Clearly, Rt(G, p) is a submatrix of Rt(Kn, p) for all t = 1, . . . , r. Moreover,
analogously to (6.2), we have
rank(R1(Kn, p)
)= rank
(dfKn(p)
). (6.3)
Recall from Definition 4.1.8 that if u ∈ V(I1)e , then u is said to be sym-
metric with respect to I1. So, if we think of the vector u ∈ Rdn as a set
of displacement vectors with one vector at each joint of (G, p) ∈ R(G,S,Φ),
then u is symmetric with respect to I1 if and only if all of the displacement
vectors remain unchanged under all symmetry operations in S. A vector
u ∈ Rdn that is symmetric with respect to I1 can therefore also be termed
fully (S, Φ)-symmetric [35, 43] (see also Figure 6.1).
Theorem 6.2.3 Let G be a graph, S be a symmetry group in dimension d,
Φ : S → Aut(G) be a homomorphism, and (G, p) be a framework in R(G,S,Φ)
with the property that the points p(v), v ∈ V (G), span all of Rd. If p is a
regular point of G in V(I1)e and also a regular point of Kn in V
(I1)e and there
exists a fully (S, Φ)-symmetric infinitesimal flex of (G, p), then there also
exists an (S, Φ)-symmetry-preserving flex of (G, p).
282
.. .
.
.(a)
.
...
.
.
.
.(b)
.
. .
.
. .
.
.(c)
Figure 6.1: Fully (S, Φ)-symmetric infinitesimal motions of frameworks: (a)
a fully (Cs, Φ)-symmetric infinitesimal rigid motion of (K3, p) ∈ R(K3,Cs,Φ);
(b) a fully (Cs, Φ)-symmetric infinitesimal flex of (K3,3, p) ∈ R(K3,3,Cs,Φ); (c) a
fully (C3, Φ)-symmetric infinitesimal flex of (Gtp, p) ∈ R(Gtp,C3,Φ). Since each
of the above frameworks is an injective realization, the type Φ is uniquely
determined in each case.
Proof. Let Kn be the complete graph on V (G). Since (G, p) has a fully
(S, Φ)-symmetric infinitesimal flex and the points p(v), v ∈ V (G), span all
of Rd, we have nullity(R1(G, p)
)> nullity
(R1(Kn, p)
), and hence
rank(R1(G, p)
)< rank
(R1(Kn, p)
). (6.4)
Since, by (6.2), we have rank(R1(G, p)
)= rank
(dfG(p)
)and, by (6.3), we
have rank(R1(Kn, p)
)= rank
(dfKn(p)
), it follows from (6.4) that
rank(dfG(p)
)< rank
(dfKn(p)
).
The result now follows from Theorem 6.2.2. ¤
The above results concerning the subspace V(I1)e of Rdn may be transferred
analogously to the affine subspaces of Rdn of the form p + V(It)e , where t 6= 1.
283
More precisely, if we define a point q ∈ p + V(It)e to be a regular point of
a graph G in p + V(It)e if there exists a neighborhood Nq of q in p + V
(It)e so
that rank(dfG(q)
)= max rank
(dfG(q′)
)| q′ ∈ Nq, where fG denotes the
restriction of the edge function fG to p+V(It)e , then the following results can
be proved completely analogously to the Theorems 6.2.2 and 6.2.3.
Theorem 6.2.4 Let G be a graph with n vertices, S be a symmetry group,
Φ : S → Aut(G) be a homomorphism, and (G, p) be a framework in R(G,S,Φ).
If p is a regular point of G in p + V(It)e and also a regular point of Kn in
p + V(It)e , then
(i) rank(dfG(p)
)= rank
(dfKn(p)
)if and only if (G, p) does not have a
flex x with x(t) ∈ p + V(It)e for all t ∈ [0, 1];
(ii) rank(dfG(p)
)< rank
(dfKn(p)
)if and only if (G, p) has a flex x with
x(t) ∈ p + V(It)e for all t ∈ [0, 1].
Theorem 6.2.5 Let G be a graph, S be a symmetry group in dimension d,
Φ : S → Aut(G) be a homomorphism, and (G, p) be a framework in R(G,S,Φ)
with the property that the points p(v), v ∈ V (G), span all of Rd. If p is a
regular point of G in p+V(It)e and also a regular point of Kn in p+V
(It)e and
there exists an infinitesimal flex u of (G, p) with u ∈ V(It)e , then there also
exists a flex x of (G, p) with x(t) ∈ p + V(It)e for all t ∈ [0, 1].
Note that if we define ker (It) = x ∈ S| It(x) = id, where id is the
identity transformation, then ker (It) is a normal subgroup of S (see [42],
for example). Therefore, Theorems 6.2.4 and 6.2.5 provide us with sufficient
284
conditions for the existence of a flex of (G, p) that preserves the sub-symmetry
of (G, p) given by ker (It) and Φ|ker (It).
An important property of the subspace V(I1)e which does not hold for
the affine subspaces p + V(It)e , where t 6= 1, is that, by Corollary 4.1.3, for
every q ∈ V(I1)e , the rigidity matrix R(G, q) has the same block structure
as the rigidity matrix R(G, p). Thus, p ∈ V(I1)e is a regular point of G in
V(I1)e if and only if there exists a neighborhood Np of p in V
(I1)e so that
rank(R1(G, p)
) ≥ rank(R1(G, q)
)for all q ∈ Np.
Similarly, p ∈ V(I1)e is a regular point of Kn in V
(I1)e if and only if
there exists a neighborhood Np of p in V(I1)e so that rank
(R1(Kn, p)
) ≥rank
(R1(Kn, q)
)for all q ∈ Np.
The fact that regular points of G and Kn in V(I1)e can be characterized in
this way is essential to proving all the remaining results of this section. These
results will turn out to be very useful for practical applications of Theorem
6.2.3, as we will see in the final section of this chapter.
Theorem 6.2.6 Let G be a graph with n vertices, S be a symmetry group
in dimension d, Φ : S → Aut(G) be a homomorphism, and (G, p) be a
framework in R(G,S,Φ). If the points p(v), v ∈ V (G), span all of Rd, then p
is a regular point of Kn in V(I1)e .
Proof. Since the points p(v), v ∈ V (G), span all of Rd, there exists a
neighborhood Np of p in V(I1)e so that for all q ∈ Np, the points q(v), v ∈
V (G), also span all of Rd. It follows from the results of Section 4.2.1 that for
all q ∈ Np, the dimension of the subspace of Rdn consisting of all fully (S, Φ)-
symmetric infinitesimal rigid motions of (G, p) is equal to the dimension of
285
the subspace of Rdn consisting of all fully (S, Φ)-symmetric infinitesimal rigid
motions of (G, q). Therefore, we have rank(R1(Kn, p)
)= rank
(R1(Kn, q)
)
or equivalently, by (6.3), rank(dfKn(p)
)= rank
(dfKn(q)
)for all q ∈ Np.
Thus, p is a regular point of Kn in V(I1)e . ¤
By Theorem 6.2.6, the condition that p is a regular point of Kn in V(I1)e
may be omitted in Theorem 6.2.3.
Theorem 6.2.7 Let G be a graph, S be a symmetry group, Φ : S → Aut(G)
be a homomorphism, and (G, p) be a framework in R(G,S,Φ). If p is (S, Φ)-
generic, then p is a regular point of G in V(I1)e .
Proof. Suppose G is a graph with n vertices and S is a symmetry group
in dimension d with r pairwise non-equivalent irreducible representations
I1, . . . , Ir. Fix a basis BU = u1, . . . , uk of U = V(I1)e =
⋂x∈S Lx,Φ and let
p = t1u1 + . . . + tkuk. Then the symmetry-adapted indeterminate rigidity
matrix RBU(n, d) for R(G,S,Φ) (corresponding to BU) is a matrix in the vari-
ables t′1, . . . , t′k. More precisely, the entries of RBU
(n, d) are elements of the
quotient field of the integral domain R[t′1, . . . , t′k]. Over this field we can again
do linear algebra. We let R(G)BU
(n, d) denote the submatrix of RBU(n, d) that
corresponds to the submatrix R(G, p) of R(Kn, p), i.e., R(G)BU
(n, d) is obtained
from RBU(n, d) by deleting those rows that do not correspond to edges of G.
If we replace each variable t′i in R(G)BU
(n, d) with ti, then, by Remark 3.2.1, we
obtain the rigidity matrix R(G, p). Therefore,
rank(R(G, p)
) ≤ rank(R
(G)BU
(n, d)).
Since (G, p) is (S, Φ)-generic, we also have
rank(R(G, p)
) ≥ rank(R
(G)BU
(n, d)),
286
and hence
rank(R(G, p)
)= rank
(R
(G)BU
(n, d)). (6.5)
Now, let Te be the matrix of the basis transformation from the canonical basis
of the R-vector space Rdn to the basis Be, and let Ti be the matrix of the basis
transformation from the canonical basis of the R-vector space R|E(G)| to the
basis Bi, so that the matrix R(G, p) = T−1i R(G, p)Te is block-diagonalized as
in (6.1). Then, by Corollary 4.1.3, the matrix R(G)BU
(n, d) = T−1i R
(G)BU
(n, d)Te
has the same block form as R(G, p). For t = 1, . . . , r, let R(G)t (n, d) denote
the block of R(G)BU
(n, d) that corresponds to the block Rt(G, p) of R(G, p).
Since the rank of a matrix is invariant under a basis transformation, and
since the rank of a block-diagonalized matrix is equal to the sum of the
ranks of its blocks, it follows from equation (6.5) thatr∑
t=1
rank(Rt(G, p)
)= rank
(R(G, p)
)
= rank(R(G, p)
)
= rank(R
(G)BU
(n, d))
= rank(R
(G)BU
(n, d))
=r∑
t=1
rank(R
(G)t (n, d)
).
Since we clearly have rank(Rt(G, p)
) ≤ rank(R
(G)t (n, d)
)for each t, it fol-
lows that rank(Rt(G, p)
)= rank
(R
(G)t (n, d)
)for each t. This gives the
result. ¤
Corollary 6.2.8 Let G be a graph, S be a symmetry group in dimension d,
Φ : S → Aut(G) be a homomorphism, and (G, p) be a framework in R(G,S,Φ)
with the property that the points p(v), v ∈ V (G), span all of Rd. If (G, p)
is (S, Φ)-generic and (G, p) has a fully (S, Φ)-symmetric infinitesimal flex,
then there also exists an (S, Φ)-symmetry-preserving flex of (G, p).
287
Proof. The result follows immediately from Theorems 6.2.3, 6.2.6, and 6.2.7.
¤
In Section 6.3, we will use Corollary 6.2.8 to prove the existence of an
(S, Φ)-symmetry-preserving flex for a variety of symmetric frameworks.
Note that it follows from our discussion in Chapter 3 that the frameworks
in Figures 6.1 (b) and (c) are not (S, Φ)-generic, so that Corollary 6.2.8
cannot be applied to these frameworks. In fact, it can be verified that none of
the frameworks in Figure 6.1 possesses any flex, let alone an (S, Φ)-symmetry-
preserving flex.
Note that Corollary 6.2.8 is a symmetrized version of Corollary 2.2.16 in
Section 2.2.5. Next, we show that a symmetrized version of Corollary 2.2.17
can also be obtained from the previous results.
Corollary 6.2.9 Let G be a graph, S be a symmetry group in dimension d,
Φ : S → Aut(G) be a homomorphism, and (G, p) be a framework in R(G,S,Φ)
with the property that the points p(v), v ∈ V (G), span all of Rd. If the
block R1(G, p) of the block-diagonalized rigidity matrix R(G, p) has linearly
independent rows and (G, p) has a fully (S, Φ)-symmetric infinitesimal flex,
then there also exists an (S, Φ)-symmetry-preserving flex of (G, p).
Proof. Since the block R1(G, p) has linearly independent rows, p is a regular
point of G in V(I1)e . The result now follows from Theorems 6.2.3 and 6.2.6.
¤
Corollary 6.2.9 confirms the observation made by R. Kangwai and S.
Guest in [43]. Note that the condition that the block matrix R1(G, p) has
linearly independent rows is equivalent to the condition that the framework
288
(G, p) has no fully (S, Φ)-symmetric non-zero self-stress, i.e., a non-zero self-
stress in the subspace V(I1)i of R|E(G)|. In particular, it follows that if (G, p) is
independent (i.e., (G, p) does not possess any non-zero self-stress) and there
exists a fully (S, Φ)-symmetric infinitesimal flex of (G, p), then there also
exists an (S, Φ)-symmetry-preserving flex of (G, p).
In order to apply Corollary 6.2.9 to a given framework (G, p), we need
to compute the rank of the submatrix block R1(G, p). This can be done by
finding the block-diagonalized rigidity matrix R(G, p) with the methods and
algorithms described in [24, 50], for example.
The rank of the submatrix block R1(G, p) can also be determined directly
by finding the rank of an appropriate ‘orbit rigidity matrix’ whose columns
and rows correspond to a set of representatives for the orbits of the group
action from S × V (G) to V (G) that sends (x, v) to Φ(x)(v) (see Remark
3.2.2) and a set of representatives for the orbits of the group action from
S × E(G) to E(G) that sends (x, e) to Φ(x)(e), respectively. The kernel of
this matrix is the space of fully (S, Φ)-symmetric infinitesimal motions and
the cokernel of this matrix is the space of fully (S, Φ)-symmetric self-stresses.
Further details will be presented in [58].
6.3 Examples of flexible frameworks
We now consider a number of examples of flexible symmetric frameworks
in dimensions 2 and 3. For each of these frameworks, we use the results of
the previous section to prove the existence of an (S, Φ)-symmetry-preserving
flex.
289
6.3.1 Examples in 2D
We begin with a very simple example of an independent framework in R2
with point group Cs that possesses a symmetry-preserving flex.
Example 6.3.1 Let K2,2 be the complete bipartite graph with partite sets
v1, v2 and v3, v4, Cs be a symmetry group in dimension 2, and Φ : Cs →Aut(K2,2) be the homomorphism defined by
Φ(Id) = id
Φ(s) = (v1)(v2)(v3 v4).
The framework (K2,2, p) ∈ R(K2,2,Cs,Φ) shown in Figure 6.2 is clearly inde-
...p1
..p2
..p3 . .p4
Figure 6.2: A fully (Cs, Φ)-symmetric infinitesimal flex of the independent
framework (K2,2, p).
pendent and the symmetry-extended version of Maxwell’s rule applied to