GainSparsity and SymmetryForced Rigidity in the Plane
Discrete Comput Geom
GainSparsity and SymmetryForced Rigidity in the Plane
Tibor Jordán 0 1 2 3 4 5
Viktória E. Kaszanitzky 0 1 2 3 4 5
Shinichi Tanigawa 0 1 2 3 4 5
Viktória E. Kaszanitzky 0 1 2 3 4 5
Shinichi Tanigawa 0 1 2 3 4 5
0 Department of Operations Research, Eötvös University , Pázmány Péter sétány 1/C, 1117 Budapest , Hungary
1 Department of Operations Research, Eötvös University, and the MTAELTE Egerváry Research Group on Combinatorial Optimization, Pázmány Péter sétány 1/C , 1117 Budapest , Hungary
2 Editor in Charge: Günter M. Ziegler
3 Centrum Wiskunde & Informatica (CWI) , Postbus 94079, 1090 GB Amsterdam , The Netherlands
4 Research Institute for Mathematical Sciences, Kyoto University , Sakyoku, Kyoto 6068502 , Japan
5 Department of Mathematics and Statistics, Lancaster University , Lancaster LA1 4YF , UK
We consider planar barandjoint frameworks with discrete point group symmetry in which the joint positions are as generic as possible subject to the symmetry constraint. We provide combinatorial characterizations for symmetryforced rigidity of such structures with rotation symmetry or dihedral symmetry of order 2k with odd k, unifying and extending previous work on this subject. We also explore the matroidal background of our results and show that the matroids induced by the row independence of the orbit matrices of the symmetric frameworks are isomorphic to gain sparsity matroids defined on the quotient graph of the framework, whose edges are labeled by elements of the corresponding symmetry group. The proofs are based on new Henneberg type inductive constructions of the gain graphs that correspond to

the bases of the matroids in question, which can also be seen as symmetry preserving
graph operations in the original graph.
Mathematics Subject Classification
Primary 52C25; Secondary 05B35 · 68R10
1 Introduction
A ddimensional barandjoint framework (or, simply, a framework) is a straightline
realization of a finite simple graph G in Euclidean dspace. We think of a barandjoint
framework as a collection of fixedlength bars (corresponding to the edges of G) which
are connected at their ends by universal joints (corresponding to the vertices of G).
Frameworks can be used to model various structures with pairwise distance constraints
and whose rigidity property is of particular interest in applications ranging from civil
engineering [9,25] and crystallography [27] to sensor network localization [7] and
biochemistry [29]. In several applications the model frameworks may have symmetry,
which makes it important to explore the impact of symmetry on the flexibility and
rigidity of the framework.
In the past 10 years this research area has received an ever increasing attention
which has led to rigorous definitions, a clear separation of different directions and a
number of new results (see, e.g., [3,16,19]). One of the general goals of the research
is to extend Laman’s classical theorem on generically rigid planar frameworks (with
no symmetry conditions). The works initiated by Ross [17] and Malestein and Theran
[11] gave natural extensions of Laman’s theorem to periodic frameworks in the plane,
where the ingenious idea is to look at count conditions for quotient graphs with group
labelings.
This paper deals with finite barandjoint frameworks with point group
symmetry in the symmetryforced setting and extends Laman’s classical theorem as well as
its matroidal background and algorithmic implications, to planar frameworks with
rotational or dihedral symmetry, assuming that the joint positions are as generic as
possible subject to the symmetry conditions. In our symmetryforced setting, a
framework is said to be symmetryforced flexible if it has a nontrivial symmetric infinitesimal
motion. For the generic frameworks that we consider, this is equivalent to the
existence of a nontrivial symmetry preserving flex [21], and our main result characterizes
symmetric frameworks that admit nontrivial symmetry preserving flexes in terms of
simple count conditions of the underlying quotient grouplabeled graphs, which can
be checked in polynomial time by combinatorial algorithms.
By using the orbit rigidity matrix introduced by Schulze and Whiteley [23], we can
reformulate our problem in terms of the generic rank of a matrix in which each row
corresponds to an edge orbit and each vertex orbit has two columns. This in turn is
equivalent to characterizing independence in a matroid defined on the edge set of the
grouplabeled quotient graph, in which vertices and edges correspond to vertex and
edge orbits, respectively, and which concisely represents the graph structure with the
corresponding symmetry. Our main results characterize these matroids in the case of
rotation symmetry or dihedral symmetry D2k of order 2k with odd k. If the underlying
symmetry is cyclic, the matroid turns out to be a (k, l)gaincount matroid, in which
independence is defined by imposing certain sparsity conditions on the edge set of a
graph, whose edges are labeled by group elements. In the dihedral case the matroid
arises by a related, but more general construction.
Matroids of the former type can be obtained by matroidal operations (e.g. matroid
union and Dilworth truncation) from matroids that have been extensively studied
in matroid theory and are called frame matroids (or bias matroids) [31,32]. These
matroids, and their relatives, which also play a role in the theory of infinite periodic
frameworks [11,12,18], have been generalized in a recent paper [24] which unified
most of the existing results on symmetric and periodic frameworks, including our
cyclic case. However, the matroid of the dihedral case does not fit this general class.
We prove our results by developing Henneberg type inductive constructions for the
bases of our matroids and show that these operations preserve the rowindependence
of the orbit rigidity matrix. This approach, which has been used in many combinatorial
characterizations of rigidity theory, leads to the desired result. In our problems, due to
the more complex sparsity conditions and the group labeling, we also need some new
operations and extended geometric arguments, to handle the symmetry constraints.
The complete answer in the case of dihedral symmetry remains open. However,
most of our inductive steps (extending or reducing a symmetric framework or a labeled
graph, respectively) are valid also for dihedral groups D2k with even k, and can be
used to show that in the even case the irreducible graphs (frameworks), where our
reduction operations are not applicable, are very special. Interestingly, the smallest
such framework, which is predicted to be rigid by the matroidal count but is flexible is
the Bottema mechanism, a wellknown mechanism in the kinematics literature (see,
e.g., [30]).
For the case when the underlying symmetry is cyclic, the same combinatorial
characterizations were also given by Malestein and Theran [12,13] by a completely
different proof approach. The main contributions of this paper are (i) to develop a
concise approach to analyze the rigidity of symmetric frameworks based on inductive
constructions and (ii) to give the first combinatorial characterization for frameworks
with noncyclic symmetry, which is far more complicated than the cyclic case. After
the publication of the technical report [8] of this paper, our formulation and results on
inductive constructions were used for analyzing the infinitesimal rigidity of symmetric
frameworks [15,22] and the symmetricforced rigidity of symmetric frameworks on
surfaces [14]. Also the matroid construction given in Sect. 7 was recently generalized
in [6].
The structure of the paper is as follows. In the rest of this section we introduce
some basic notation. In Sect. 2 we define and investigate gain graphs, which are
directed multigraphs with edges labeled by elements of a group. Gain count matroids,
defined on gain graphs by sparsity conditions, are introduced in Sect. 3 along with the
necessary matroidal background. In Sect. 4 we develop our inductive construction for
the bases of a specific gain count matroid by using three operations and a single base
graph. In Sect. 5 we recall the basic definitions and results needed to study symmetric
frameworks, including the orbit rigidity matrix and the necessary count conditions.
In Sect. 6 we prove the first geometric lemmas and use them, together with results of
Sect. 4, to complete the characterization of rigid frameworks with cyclic symmetry. In
Sect. 7 we prove the inductive construction for the bases of our second matroid by using
five operations and four types of base graphs. In this case we need to handle graphs
of minimum degree four and hence we need more operations and longer arguments.
To make the paper more readable, the lengthy case, when the graph is fourregular,
is moved to the end of the paper, to Sect. 9. In Sect. 8 we prove additional geometric
lemmas and use them, together with the inductive construction of Sect. 7, to prove the
second main result, the characterization of rigid frameworks with dihedral symmetry
with odd k. We also present frameworks that meet the sparsity requirements but are
dependent and flexible when k is even. In Sect. 10 we briefly discuss the algorithmic
implications and make some further remarks.
In the rest of the introduction, let us introduce notation used throughout the paper.
Let E be a finite set. A partition P of E is a family of nonempty subsets of E such
that each element of E belongs to exactly one member of P. If E = ∅, the partition
of E is defined as the empty set. A subpartition of E is a partition of a subset of E .
Let G = (V , E ) be an undirected graph. For v ∈ V , let dG (v) be the degree of v in
G and NG (v) be the set of neighbors of v in G. For F ⊆ E , VG (F ) denotes the set of
endvertices of edges in F , and let G[F ] = (V (F ), F ), that is, the graph edgeinduced
by F . If the graph is clear from the context, the subscript G may be dropped. For
F ⊆ E and v ∈ V (F ), let dF (v) = dG[F](v).
A vertex subset X ⊂ V (G) (resp., an edge subset X ⊂ E (G)) is called a
separator (resp., a cut) if the removal of X disconnects G. A separator X with X  = 1 is
called a cutvertex. G is called kconnected (resp., kedgeconnected) if the size of
any separator (resp., any cut) is at least k. A separator (resp., a cut) is called
nontrivial if its removal disconnects G into at least two nontrivial connected components,
where a connected component is called trivial if it consists of a single vertex. G is
called essentially kconnected (resp., essentially kedgeconnected) if the size of any
nontrivial separator (resp., any nontrivial cut) is at least k.
For simplicity, some properties of edgeinduced subgraphs will be associated with
the corresponding edge sets as follows. Let F ⊆ E . F is called connected if G[F ]
is connected. A connected component of F is the edge set of a connected component
of G[F ]. C (F ) denotes the partition of F into connected components of F , and let
c(F ) = C (F ). F is called a forest if it contains no cycle and called a tree if it is a
connected forest. F is called a spanning tree of a graph G = (V , E ) if F is a tree with
F ⊆ E and V (F ) = V .
Let G = (V , E ) be a directed graph. A walk in G is a sequence W =
v0, e1, v1, e2, v2, . . . , vk−1, ek , vk of vertices and edges such that vi−1 and vi are
the endvertices of ei for every 1 ≤ i ≤ k. We often denote a walk as a sequence
of edges implicitly assuming the incidence at each vertex. For two walks W and W
for which the end vertex of W and the starting vertex of W coincide, we denote the
concatenation of W and W (that is, the walk W followed by W ) by W ∗ W . A walk
is called closed if the starting vertex and the end vertex coincide.
It is sometimes convenient to regard the empty set as a subgroup of a group. Let
D be a dihedral group. For a cyclic subgroup C of D, C¯ denotes the maximal cyclic
subgroup containing C.
2 Gain Graphs
In this section we shall review some basic properties of gain graphs. We refer the
reader to [5,31,32] for more details.
Let G = (V , E ) be a directed graph which may contain multiple edges and loops,
and let S be a group. An Sgain graph (G, φ) is a pair, in which each edge is associated
with an element of S by a gain function φ : E → S. The orientation of G is, in some
sense, arbitrary, and is used only as a reference orientation: the orientation of each
edge may be changed, provided that we also modify φ such that if the edge has gain
g in one direction then it has gain g−1 in the other direction. Therefore we often
do not distinguish between G and the underlying undirected graph and use notations
introduced in Sect. 1, implicitly referring to the underlying graph.
Let W be a walk in (G, φ). The gain of W is defined as φ (W ) = φ (e1) ·
φ (e2) · · · φ (ek ) if each edge is oriented in the forward direction through W , and
for a backward edge ei we replace φ (ei ) with φ (ei )−1 in the product. Note that
φ (W −1) = φ (W )−1.
Let (G, φ) be a gain graph. For v ∈ V (G) we denote by π1(G, v) the set of closed
walks starting at v. Similarly, for X ⊆ E (G) and v ∈ V (G), π1(X, v) denotes the
set of closed walks starting at v and using only edges of X , where π1(X, v) = ∅ if
v ∈/ V (X ).
Let X ⊆ E (G). The subgroup induced by X relative to v is defined as X φ,v =
{φ (W )  W ∈ π1(X, v)}. The subscript φ of X φ,v is sometimes omitted if it is
clear from the context. Note that, for any connected X ⊆ E (G) and two vertices
u, v ∈ V (X ), X ψ,u is conjugate to X ψ,v (see, e.g., [5, p. 88] for the proof).
2.1 The Switching Operation
For v ∈ V (G) and g ∈ S, a switching operation at v with g changes the gain function
φ on E (G) as follows.
φ (e) =
⎧ g · φ (e) · g−1 if e is a loop incident with v,
⎪⎪⎪⎨ g · φ (e) if e is a nonloop edge and is directed from v,
φ (e) · g−1 if e is a nonloop edge and is directed to v,
⎪
⎪⎪⎩ φ (e) otherwise.
(1)
We say that a gain function φ on edge set E (G) is equivalent to another gain function
φ on E (G) if φ can be obtained from φ by a sequence of switching operations.
The following two facts are fundamental (see, e.g., [5, Sect. 2.5.2] or [31, Sect. 5]
for the proofs).
Proposition 2.1 Let (G, φ) be a gain graph. Let φ be the gain function obtained
from φ by a switching operation. Then, for any X ⊆ E (G) and u ∈ V (G), X φ ,u is
conjugate to X φ,u .
Proposition 2.2 Let (G, φ) be a gain graph. Then, for any forest F ⊆ E (G), there is
a gain function φ equivalent to φ such that φ (e) = id for every e ∈ F .
2.2 Balanced and Cyclic Sets of Edges
As we shall see, the subgroup X ψ,v itself will not be important, when we define our
matroids induced by gains. We only need to know whether X ψ,v is trivial or not, or
whether it is cyclic or not. We now introduce notions to describe these properties.
Let (G, φ) be a gain graph. An edge subset F ⊆ E (G) is called balanced if F ψ,v
is trivial for every v ∈ V (F ). Note that F is balanced if and only if every cycle in F is
balanced. The latter property is the definition of the balancedness given by Zaslavsky
[31].
In the same way, an edge subset F ⊆ E (G) is called cyclic if F ψ,v is cyclic
for every v ∈ V (F ) (note that the terms balanced and cyclic are not exclusive). A
gain graph (G, φ) is called balanced (resp. cyclic) if E (G) is balanced (resp. cyclic),
respectively.
Proposition 2.2 suggests a simple way to check the above introduced properties of
X , in analogy with the fact that the cycle space of a graph is spanned by fundamental
cycles. For a connected X ⊆ E (G), take a spanning tree T of the edge induced
graph G[X ]. By Proposition 2.2 we can convert the gain function to an equivalent
gain function such that φ (e) = id for all e ∈ T . Now consider any closed walk
W ∈ π1(X, v), and denote W by W = v1v2, v2v3, . . . , vk vk+1, and let Wi = Pi ∗
{vi vi+1}∗ Pi−+11 for 1 ≤ i < k, where Pi denotes the path from v to vi in T . Then observe
φ (W ) = φ (W1)·φ (W2) · · · φ (Wk ). By φ (e) = id for all e ∈ T , we deduce that φ (W ) is
a product of elements in {φ (e) : e ∈ X \T }, implying that X φ,v ⊆ φ (e) : e ∈ X \T ,
where φ (e) : e ∈ X \T is the group generated by {φ (e) : e ∈ X \T }. Conversely,
φ (e) is contained in X φ,v for all e ∈ X \T . Thus, X φ,v = φ (e) : e ∈ X \T . In
particular, we proved the following.
Lemma 2.3 For a connected X ⊆ E (G) and a spanning tree T of G[X ], suppose
that φ (e) = id for all e ∈ T . Then, X φ,v = φ (e) : e ∈ X \T . In particular, the
following hold.
(i) X is unbalanced if and only if there is an edge in X \T whose gain is nonidentity.
(ii) X is cyclic if and only if all gains of X \T are contained in a cyclic subgroup of S.
The following technical lemmas will be used in the proof of our main theorem.
Lemma 2.4 Let (G, φ) be an Sgain graph, and X and Y be connected edge subsets
such that the graph (V (X ) ∩ V (Y ), X ∩ Y ) is connected.
(1) If X and Y are balanced, then X ∪ Y is balanced.
(2) If X is balanced and Y is cyclic, then X ∪ Y is cyclic.
(3) If X, Y are cyclic and X ∩ Y is unbalanced, then X ∪ Y is cyclic, provided that
for every nontrivial cyclic subgroup C of S there is a unique maximal cyclic
subgroup C¯ of S containing C.
Proof Since the graph (V (X ) ∩ V (Y ), X ∩ Y ) is connected, there is a spanning tree
T in G[X ∪ Y ] such that T ∩ X is a spanning tree of G[X ], T ∩ Y is a spanning tree
of G[Y ], and T ∩ X ∩ Y is a spanning tree of G[X ∩ Y ]. By Proposition 2.2, there is
a gain function φ equivalent to φ such that φ (e) = id for each e ∈ T .
If X and Y are balanced, Lemma 2.3 implies that φ (e) = id for all e ∈ X ∪ Y .
Thus (1) holds.
If X is balanced, then every label in X ∪ Y is contained in Y φ ,v by Lemma 2.3,
and hence X ∪ Y is cyclic if Y is cyclic. This implies (2).
If X, Y are cyclic and X ∩ Y is unbalanced, then there is an edge e ∈ X ∩ Y for
which φ (e) is nonidentity. Let C be a cyclic subgroup of S generated by φ (e) and C¯
be the maximal cyclic subgroup containing C. Since X and Y are cyclic, Lemma 2.3
implies that φ (e) ∈ C¯ holds for every e ∈ X and for every e ∈ Y . Therefore X ∪ Y is
cyclic.
Lemma 2.5 Let (G, φ) be a gain graph, and X and Y be connected balanced edge
subsets. If the number of connected components of the graph (V (X ) ∩ V (Y ), X ∩ Y )
is two, then X ∪ Y is cyclic.
Proof We take a spanning tree T of G[X ∪ Y ] such that T ∩ X is a spanning tree
of G[X ]. Since the number of connected components of (V (X ) ∩ V (Y ), X ∩ Y ) is
two, T ∩ Y consists of two connected components, denoted T1 and T2. {V (T1), V (T2)}
partitions Y into three subsets {Y1, Y2, Y3} such that Yi = {e ∈ Y : V ({e}) ⊆ V (Ti )}
for i = 1, 2 and Y3 = Y \(Y1 ∪ Y2).
By Proposition 2.2, we can take a gain function φ equivalent to φ such that φ (e) =
id for e ∈ T . Since X and Y are balanced, we have φ (e) = id for e ∈ X ∪ Y1 ∪ Y2.
Moreover, assuming that every edge in Y3 is oriented toward V (Y1), we have φ (e) =
φ ( f ) for all e, f ∈ Y3, since otherwise T1 ∪ T2 ∪ {e, f } contains an unbalanced cycle,
contradicting the fact that Y is balanced. Therefore X ∪ Y is cyclic.
3 Gain Count Matroids
3.1 Matroids Induced by Submodular Functions
Let E be a finite set. A function μ : 2E → R is called submodular if μ(X ) + μ(Y ) ≥
μ(X ∪ Y ) + μ(X ∩ Y ) for every X, Y ⊆ E . μ is monotone if μ(X ) ≤ μ(Y ) for any
X ⊆ Y . A monotone submodular function μ : 2E → Z induces a matroid on E ,
where F ⊆ E is independent if and only if I  ≤ μ(I ) for every nonempty I ⊆ F .
See e.g. [4, Sect. 13.4]. This matroid is denoted by M(μ).
For a monotone submodular function μ, let ν = μ − 1. Then, ν is monotone
submodular and induces the matroid M(ν). This matroid is referred to as the Dilworth
truncation of M(μ). Although the details are omitted here, the name of Dilworth
truncation is justified from a connection with Dilworth truncation for general matroids,
see [4] for more details.
Now we consider the union of two matroids induced by monotone submodular
functions μ1 and μ2. Since monotonicity and submodularity are both preserved under
the sum operation, μ1 + μ2 is monotone and submodular. In general, the union of
M(μ1) and M(μ2) is not equal to M(μ1 + μ2). We do have equality in some special
cases, for example, when μ1 = μ2 or when both μ1 and μ2 are nonnegative.
As an example, consider the union of two copies of the graphic matroid of a graph
G = (V , E ). It is the matroid induced by f2,2 defined by f2,2(F ) = 2V (F ) − 2 on
3.2 GainCount Matroids
In this paper we shall consider frame matroids on gain graphs. Let S be a group and
(G, φ) be an Sgain graph. The frame matroid of (G, φ) is defined such that F ⊆ E
is independent if and only if each connected component of F contains no cycle or just
one cycle, which is unbalanced if it exists [32]. If we define gS : 2E → Z by
where
gS (F ) =
(V (Fi ) − 1 + αS (Fi )),
2E , as f2,2/2 induces the graphic matroid on G. The 2dimensional generic rigidity
matroid is the one induced by f2,2 − 1, and hence it is the Dilworth truncation of the
union of two copies of the graphic matroid.
In general, for a graph G = (V , E ) and two integers k and l with k ≥ 1 and
l ≤ 2k − 1, let
fk,l (F ) = kV (F ) − l (F ⊆ E ).
G is called (k, l)sparse if F  ≤ fk,l (F ) for any nonempty F ⊆ E . The matroid
induced by fk,l is called the (k, l)count matroid on G. If l ≥ 0, M( fk,l ) is indeed the
one induced by fk,0, truncated l times. See e.g. [4] for more details. Below we shall
apply the same construction to the union of some copies of a frame matroid to define
gaincount matroids.
(2)
(3)
(4)
then the frame matroid is the matroid induced by gS . We omit the subscript S from
αS if it is clear from the context.
For an Sgain graph and two positive integers k and l with k ≤ l, we define
gk,l : 2E → Z by
gk,l (F ) = kgS (F ) − (l − k) (F ⊆ E ).
We call the matroid M(gk,l ) induced by gk,l a (k, l)gaincount matroid or gcount
matroid for short. This matroid is the union of k copies of the frame matroid, followed
by l − k Dilworth truncations. In this paper, we shall investigate the (2, 3)gcount
matroid and its variants.
The independence of M(gk,l ) can be described in a compact form (see [8] for the
proof, which is a rather straightforward calculation).
Lemma 3.1 Let (G, φ) be an Sgain graph with G = (V , E ). Then E is independent
in M(gk,l ) if and only if F  ≤ kV (F ) − l + kα(F ) for any nonempty F ⊆ E .
In this sense, we may define (k, l)gainsparsity as in the case of (k, l)sparsity of
undirected graphs as follows.
Definition 3.1 Let k and l be positive integers with k ≤ l and (G, φ) be an Sgain
graph with a graph G = (V , E ) and a group S. An edge set X ⊆ E is called (k,
l)gainsparse (or (k, l)gsparse for short) if F  ≤ gk,l (F ) for any nonempty F ⊆ X ,
i.e.,
• F  ≤ kV (F ) − l for every nonempty balanced F ⊆ X ;
• F  ≤ kV (F ) − l + k for every nonempty unbalanced F ⊆ X ,
and it is called (k, l)gaintight (or (k, l)gtight for short) if it is (k, l)gsparse with
X  = gk,l (X ).
If E is (k, l)gsparse then graph (G, φ) is said to be (k, l)gsparse, and (G, φ) is
called maximum (k, l)gtight if it is (k, l)gsparse with E  = kV  − l + k.
Remark 3.1 Note that the value of gk,l is invariant under switching operations, and
thus the induced matroid is uniquely determined up to equivalence of gain functions.
Remark 3.2 We can further consider the union of frame matroids of gain graphs
(G, φ1) and (G, φ2) with the same underlying graph but distinct gain functions. We
should remark that both graphic matroids and bicircular matroids are special cases
of frame matroids. The union of copies of graphic, frame and bicircular matroids on
an Sgain graph, followed by Dilworth truncations, can be described as the matroid
induced by a counting condition. For example, in the union of the graphic matroid and
the frame matroid of a gain graph (G, φ), followed by a single Dilworth truncation,
E (G) is independent if and only if F  ≤ 2V (F ) − 3 for any balanced set F ⊆ E (G)
and F  ≤ 2V (F ) − 2 for any nonempty F ⊆ E (G). This matroid was used by Ross
[17] for characterizing the generic rigidity of barandjoint frameworks on a torus.
Tanigawa [24] proposed a more general class of matroids extending matroid union
operations.
4 Constructive Characterization of Maximum (2, 3)gTight Graphs
4.1 Operations Preserving (2, 3)gSparsity
In this section we define three operations, called extensions, that preserve (2,
3)gsparsity. The first two operations generalize the wellknown Henneberg operations
[26,28] to gain graphs.
Let (G, φ) be an Sgain graph. The 0extension adds a new vertex v and two new
nonloop edges e1 and e2 to G such that the new edges are incident to v and the other
endvertices are two not necessarily distinct vertices of V (G). If e1 and e2 are not
parallel then their labels can be arbitrary. Otherwise the labels are assigned such that
φ (e1) = φ (e2), assuming that e1 and e2 are directed to v.
The 1extension first chooses an edge e and a vertex z, where e may be a loop and
z may be an endvertex of e. It subdivides e, with a new vertex v and new edges e1, e2
such that the tail of e1 is the tail of e and the tail of e2 is the head of e. The labels of the
new edges are assigned such that φ (e1) · φ (e2)−1 = φ (e). The 1extension also adds a
third edge e3 oriented to v. The label of e3 is assigned so that it is locally unbalanced,
i.e., every twocycle ei e j , if it exists, is unbalanced.
The loop 1extension adds a new vertex v to G and connects it to a vertex z ∈ V (G)
by a new edge with any label. It also adds a new loop l incident to v with φ (l) = id
(Fig. 1).
The 0extension and the 1extension were already considered by Ross [17] for
Z2gain graphs. In the covering graph each operation can be seen as a graph operation that
preserves the underlying symmetry. Some of them can be recognized as performing
socalled Henneberg operations [26,28] simultaneously. In the case of threefold rotation
symmetry, these operations are considered by Schulze [21].
Lemma 4.1 Let (G, φ) be a (2,3)gsparse graph. Applying the 0extension,
1extension or loop 1extension to (G, φ) results in a (2,3)gsparse graph (G , φ )
with V (G ) = V (G) + 1 and E (G ) = E (G) + 2.
Proof For a contradiction, suppose that G contains an edge set F ⊆ E (G ) for which
F  > 2V (F )−3+2α(F ). Let v be the new vertex added by the extension, and let Ev
be the set of edges incident to v. Since E (G )\Ev ⊆ E (G), Ev ∩ F = ∅. In particular,
v ∈ V (F ). Also, since the new labeling is assigned to be locally unbalanced, F is not
contained in Ev.
If G is constructed by a 1extension then let e be the subdivided edge of G and let
e1 and e2 be the resulting two new edges.
Let F = F \Ev. If G is constructed by a 1extension and {e1, e2} ⊆ F , then we
further insert e to F . We then have F  ≥ F  − 2, V (F ) = V (F ) − 1, and
α(F ) ≤ α(F ) in each case. These imply F  ≥ F  − 2 > 2V (F ) − 5 + 2α(F ) ≥
2V (F ) − 3 + 2α(F ), contradicting the (2, 3)gsparsity of G as ∅ = F ⊆ E (G).
We shall define the inverse moves of the operations above, which are called
reductions. For a vertex v and two incoming nonloop edges e1 = (u, v) and e2 = (w, v),
we denote by e1 · e2−1 a new edge from u to w with label φ (e1) · φ (e2)−1 (by extending
φ). If u = w then e1 · e2−1 is a loop. Each reduction corresponds to one of the following
operations on a gain graph (G, φ).
A 0reduction chooses a degree two vertex and deletes it from G.
A 1reduction chooses a vertex v with d(v) = 3 that is not incident to a loop. Let
e1, e2, e3 be the edges incident to v. Without loss of generality we may assume that
each ei is oriented to v. The 1reduction deletes v with the incident edges and adds
one of e1 · e2−1, e2 · e−1 and e3 · e1−1 as a new edge.
3
A loop 1reduction chooses a vertex incident to exactly one loop and one nonloop
edge and deletes the chosen vertex with the incident edges.
A 1reduction may destroy the (2, 3)gsparsity of a graph. We say that a reduction
(at a vertex v) is admissible if the resulting graph is (2, 3)gsparse.
4.2 Constructive Characterization
Lemma 4.2 Let (G, φ) be a (2,3)gsparse graph and v ∈ V (G) a vertex not incident
to a loop with d(v) = 3. Then there is an admissible 1reduction at v.
Proof Let E = E (G), G = G − v and E = E (G ). Let e1, e2, e3 be the edges
incident to v in G. Without loss of generality we may assume that each ei is oriented
to v. For simplicity we put ei, j = ei · e −j1.
Suppose for a contradiction that there is no admissible splitting at v, that is, none
of E + e1,2, E + e2,3 and E + e3,1 is independent in M(g2,3). Equivalently,
e1,2, e2,3, e3,1 ∈ clg(E ), where clg denotes the closure operator of M(g2,3). Let
X = {e1, e2, e3, e1,2, e2,3, e3,1}.
Claim 4.3 e1 ∈ clg(X − e1).
Proof We split the proof into three cases depending on the cardinality of N (v).
If N (v) = 3 then, by Proposition 2.2, we may assume φ (e1) = φ (e2) = φ (e3) =
id. We then have φ (e1,2) = φ (e2,3) = φ (e3,1) = id. Therefore X forms a balanced
K4, which is a circuit of M(g2,3). Thus, e1 ∈ clg(X − e1) holds.
If N (v) = 2 then we may assume that e1 and e2 are parallel. By Proposition 2.2, we
may assume that φ (e2) = φ (e3) = id. This implies φ (e1,3) = φ (e1) and φ (e2,3) = id.
Since G is (2, 3)gsparse, we have φ (e1) = id by φ (e2) = φ (e3) = id, which implies
that e1,2 is an unbalanced loop with φ (e1,2) = φ (e1). It can be easily checked, by
counting, that X is indeed a circuit in M(g2,3). Thus, e1 ∈ clg(X − e1) holds.
If N (v) = 1 then let X = {e1, e2, e3, e1,2}. We have X  = 2V (X ) and X is
a circuit of M(g2,3). Therefore e1 ∈ clg(X − e1) ⊂ clg(X − e1).
Since e1,2, e2,3, e3,1 ∈ clg(E ), by Claim 4.3, we have e1 ∈ clg(X −e1) ⊆ clg(E +
X − e1) = clg(E + e2 + e3) = clg(E − e1), which contradicts the (2, 3)gsparsity
of G.
The following constructive characterization of maximum (2, 3)gtight graphs is a
direct consequence of Lemmas 4.1 and 4.2 (see [8] for the concrete proof).
Theorem 4.4 An Sgain graph (G, φ) is maximum (2,3)gtight if and only if it can
be built up from an Sgain graph with one vertex and an unbalanced loop incident to
it with a sequence of 0extensions, 1extensions, and loop1extensions.
Remark 4.1 Theorem 4.4 for the case of threefold rotation symmetry is implicit in
[20]. For Z2gain graphs, the corresponding result with a slightly different count
condition (see Remark 3.2) was shown by Ross [17].
Further applications of Theorem 4.4 and other operations are recently discussed in
[14,15,22].
5 SymmetryForced Rigidity
In this section we define the notion of symmetryforced infinitesimal rigidity,
introduced by Schulze and Whiteley [23]. In Sect. 5.1, we first introduce Ssymmetric
graphs, whose automorphism group has a subgroup isomorphic to S. In Sect. 5.2 we
shall review the conventional notion of infinitesimal rigidity. In Sect. 5.3 we introduce
symmetryforced infinitesimal rigidity, which is only concerned with infinitesimal
motions invariant under the underlying symmetry. In Sect. 5.4 we introduce the orbit
rigidity matrix, which is the main tool for investigating symmetryforced infinitesimal
rigidity in the subsequent sections. In Sect. 5.5 we prove a necessary condition for
symmetric frameworks to be symmetryforced infinitesimally rigid.
5.1 SSymmetric Graphs
Let H be a simple graph. An automorphism of H is a permutation π : V (H ) →
V (H ) such that {u, v} ∈ E (H ) if and only if {π(u), π(v)} ∈ E (H ). The set of all
automorphisms of H forms a subgroup of the symmetric group of V (H ), known as
the automorphism group Aut(H ) of H .
Let S be a group. An action of S on H is a group homomorphism ρ : S → Aut(H ).
An action ρ is called free if ρ(g)(v) = v for any v ∈ V and any nonidentity g ∈ S.
We say that a graph H is (S, ρ)symmetric if S acts on H by ρ. If ρ is clear from
the context, we will simply denote ρ(g)(v) by g · v or gv. Note that, for g ∈ S and
u, v ∈ V , {u, v} ∈ E (H ) if and only if {gu, gv} ∈ E (H ), and hence there is an
induced action of S on E (H ) defined by g · {u, v} = {gu, gv}.
Let H be an (S, ρ)symmetric graph. The quotient graph H/S of H is a multigraph
on the set V (H )/S of vertex orbits, together with the set E (H )/S of edge orbits as the
edge set. An edge orbit may be represented by a loop in H/S. Figure 2 is an example
when S is a dihedral group.
Different graphs may have the same quotient graph. However, if we assume that ρ
is free, then a gain labeling makes the relation onetoone. To see this, we arbitrarily
choose a vertex v as a representative vertex from each vertex orbit. Then, each orbit
is written by Sv = {gv : g ∈ S}. If ρ is a free action, an edge orbit connecting Su
and Sv in H/S can be written by {{gu, ghv} : g ∈ S} for a unique h ∈ S. We then
orient the edge orbit from Su to Sv in H/S and assign to it the gain h. In this way,
we obtain the quotient Sgain graph, denoted (H/S, φ).
Conversely, any Sgain graph (G, φ) can be “lifted” as an (S, ρ)symmetric graph
with a free action ρ. To see this, we simply denote the pair (g, v) of g ∈ S and
v ∈ V (G) by gv. The covering graph (also known as the derived graph) of (G, φ) is
the simple graph with vertex set S × V (G) = {gv : g ∈ S, v ∈ V (G)} and the edge
r
r
id
Cπ
id
set {{gu, gφ (e)v} : e = (u, v) ∈ E (G), g ∈ S}. Clearly, S freely acts on the covering
graph, under which the quotient gain graph comes back to (G, φ). For more properties
of covering graphs, see e.g. [5].
5.2 Infinitesimal Rigidity
Before we investigate the rigidity theory of symmetric graphs we review the basic
notions of the conventional rigidity of graphs.
A ddimensional barandjoint framework (or simply a framework) is a pair (H, p)
of a simple graph H and a mapping p : V (H ) → Rd , called a jointconfiguration.
We denote the set { p(v) : v ∈ V (H )} of points by p(H ).
Infinitesimal rigidity is concerned with the dimension of the space of infinitesimal
motions. An infinitesimal motion of a framework (H, p) is defined as an assignment
m : V (H ) → Rd such that
m(u) − m(v), p(u) − p(v) = 0 for all {u, v} ∈ E (H ),
(5)
where ·, · denotes the standard inner product in the ddimensional Euclidean space.
The set of infinitesimal motions forms a linear space, denoted L(H, p).
In general, for a set P ⊆ Rd of points, an infinitesimal isometry of P is defined by
m : P → Rd such that
m(x ) − m(y), x − y = 0 for all x , y ∈ P.
The set of infinitesimal isometries forms a linear space, denoted by iso( P). Notice
that, for a skewsymmetric matrix S and t ∈ Rd , a mapping m : P → Rd defined by
m(x ) = Sx + t (x ∈ P)
is an infinitesimal isometry of P. Indeed, it is wellknown that any infinitesimal
isometry can be described in this form, and
dim iso( P) = d(k + 1) −
,
where k denotes the affine dimension of P.
Example 5.1 Let us consider the infinitesimal isometries of a point set P in the plane.
According to (6), we have
dim iso( P) =
3 if  P ≥ 2,
2 if  P = 1.
m(v) = Sp(v) + t (v ∈ V (H ))
For t ∈ R2, let mt (x ) = t (x ∈ P). Then, mt is an infinitesimal isometry, called a
translation. On the other hand, let mr (x ) = Cπ/2x (x ∈ P), where Cπ/2 denotes the
2 × 2 orthogonal matrix representing the fourfold rotation around the origin. Then,
mr is also an infinitesimal isometry, which we call an infinitesimal rotation. It is well
known that iso( P) is spanned by {mt , mt , mr } for two linearly independent vectors
t, t ∈ R2. See Fig. 3 for examples.
An infinitesimal motion m : V (H ) →
trivial if m can be expressed by
Rd of a framework (H, p) is said to be
(6)
(7)
for some skewsymmetric matrix S and t ∈ Rd . The set of all trivial motions forms
a linear subspace of L(H, p), denoted by tri(H, p). By definition, tri(H, p) is
isomorphic to iso( p(H )), and hence (6) gives the exact dimension of tri(H, p). (H, p)
is called infinitesimally rigid if L(H, p) = tri(H, p).
5.3 Symmetric Frameworks and SymmetryForced Infinitesimal Rigidity
A discrete point group (or simply a point group) is a finite discrete subgroup of O(Rd ),
the orthogonal group of dimension d, i.e., the set of d × d orthogonal matrices over R.
For d = 2, point groups are classified into two classes, groups Ck of kfold rotations
and dihedral groups D2k of order 2k. For a special case, D2 consists of a
mirrorreflection and the identity. In the subsequent discussion of this section, S denotes a
point group.
Suppose that H is (S, ρ)symmetric for a point group S. A jointconfiguration p
is said to be (S, ρ)symmetric (or, simply, Ssymmetric) if
gp(v) = p(gv) for all g ∈ S and for all v ∈ V (H ).
(8)
Such a pair ( H, p) is called an (S, ρ)symmetric framework (or simply an
Ssymmetric framework or a symmetric framework).
We shall consider “symmetrypreserving” infinitesimal motions of symmetric
frameworks. We say that an infinitesimal motion m : V ( H ) → Rd is symmetric
if
gm(v) = m(gv) for all g ∈ S and for all v ∈ V ( H ).
(9)
The set of Ssymmetric infinitesimal motions and the set of trivial ones form linear
subspaces of L ( H, p) and tri( H, p), denoted L S ( H, p) and triS ( H, p), respectively.
We say that ( H, p) is symmetryforced infinitesimally rigid if L S ( H, p) = triS ( H, p).
A set P of points is called Ssymmetric if g P = {gp : p ∈ P } = P for all g ∈ S.
An infinitesimal isometry m : P → Rd of an Ssymmetric point set P is called
Ssymmetric if gm(x ) = m(gx ) for all x ∈ P and g ∈ S. The set of Ssymmetric
infinitesimal isometries forms a linear subspace of iso( P ), denoted isoS ( P ). Clearly,
triS ( H, p) is isomorphic to isoS ( p( H )).
Example 5.2 Let us consider point groups in O(R2), which will be mainly discussed
in Sects. 6 and 8. Let P be an Ssymmetric point set in R2. See Fig. 3 for examples
of Ck symmetric infinitesimal isometries. In general, if  P  > 1,
dim isoCk ( P ) =
3 if k = 1,
1 if k ≥ 2,
and if P = { }
x ,
dim isoCk ( P ) =
2 if k = 1,
0 if k ≥ 2 (where x should be the origin).
Similarly, for the dihedral group D2k of order 2k,
dim isoD2k ( P) =
1 if k = 1,
0 if k ≥ 2.
A result of Schulze [21] motivates us to look at Ssymmetric infinitesimal rigidity,
which states that if (H, p) is not symmetryforced infinitesimally rigid on an Sgeneric
p, then (H, p) has a nontrivial continuous motion that preserves the (S, ρ)symmetry.
5.4 The Orbit Rigidity Matrix
Let (H, p) be an (S, ρ)symmetric framework in Rd . Due to (9), the system (5) of
linear equations (with respect to m) is redundant. Schulze and Whiteley [23] pointed
out that the system can be reduced to E (H )/S linear equations.
To see this, we first define a jointconfiguration p˜ of vertex orbits by p˜ : V (H )/S →
Rd . By taking a representative vertex v from each vertex orbit Sv, we set p˜(Sv) = p(v)
[then, the locations of the other nonrepresentative vertices are uniquely determined
by (8)].
In a similar way, we define an infinitesimal motion of (H/S, p˜) by m˜ : V (H )/S →
Rd . By using the representative vertices determined above, we fix a onetoone
correspondence between Ssymmetric infinitesimal motions of V (H ) and infinitesimal
motions of V (H )/S by m˜ (Sv) = m(v) for each vertex orbit Sv.
Let (H/S, φ) be the quotient Sgain graph of H . Recall that each (oriented) edge
orbit Se connecting Su and Sv with gain he can be written by Se = {{gu, ghev} :
g ∈ S}. The system (5) is hence written by
m(gu) − m(ghev), p(gu) − p(ghev) = 0 for all {gu, ghev} ∈ Se
(10)
over all edge orbits Se ∈ E (H )/S. Recall that the transpose of g is g−1 for any
g ∈ O(Rd ). By (8) and (9),
m(gu) − m(ghev), p(gu) − p(ghev)
= m(u) − hem(v), p(u) − he p(v)
= m(u), p(u) − he p(v) + m(v), p(v) − he−1 p(u)
= m˜ (Su), p˜(Su) − he p˜(Sv) + m˜ (Sv), p˜(Sv) − he−1 p˜(Su) .
Therefore, for p˜ : V (H )/S → Rd , a mapping m˜ : H/S →
motion of (H/S, p˜) if and only if
Rd is an infinitesimal
m˜ (Su), p˜(Su) − he p˜(Sv) + m˜ (Sv), p˜(Sv) − he−1 p˜(Su) = 0
(11)
for every oriented edge orbit Se with φ (Se) = he. By regarding (11) as a system of
linear equations of m˜ , the corresponding E (H )/S × dV (H )/Smatrix is called the
orbit rigidity matrix.
In general, for an Sgain graph (G, φ) and p˜ : V → Rd , we shall define the orbit
rigidity matrix as an E (G) × dV (G)matrix, in which each row corresponds to an
edge, each vertex is associated with a dtuple of columns, and the row corresponding
to e = (u, v) ∈ E (G) is written by
u
v
0 · · · 0 p˜(u) − φ (e) p˜(v) 0 · · · 0 p˜(v) − φ (e)−1 p˜(u) 0 · · · 0
if e is not a loop, and by
v
0 · · · 0 (2Id − φ (e) − φ (e)−1) p˜(v) 0 · · · 0
if e is a loop. The orbit rigidity matrix of (G, φ, p˜) is denoted by O(G, φ, p˜). From
the above discussion, it follows that the dimension of the Ssymmetric infinitesimal
motions can be computed from the rank of the orbit rigidity matrix of the corresponding
quotient gain graph, which is formally stated as follows:
Theorem 5.1 (Schulze and Whiteley [23]). Let (H, p) be an (S, ρ)symmetric
framework with a free action ρ. Then,
dim LS (H, p) = dV (H )/S − rank O(H/S, φ, p˜),
where (H/S, φ) is the quotient Sgain graph and p˜ is a jointconfiguration of vertex
orbits corresponding to p.
5.5 Necessary Condition for SymmetryForced Rigidity
Combining some observations given in Sect. 2, we can show a necessary condition
for the row independence of orbit rigidity matrices.
Lemma 5.2 Let (G, φ) be an Sgain graph with underlying graph G = (V , E ), and
let p : V → Rd . If O(G, φ, p) is row independent, then
F  ≤
{dV (Fi ) − dim iso Fi φ,w ( p(Fi ))}
Fi ∈C(F)
for all F ⊆ E and w ∈ V (Fi ), where p(Fi ) = {gp(v) : v ∈ V (Fi ), g ∈ S}.
Proof Let RF be the linear space spanned by the row vectors associated with F in
O(G, φ, p). Observe that each nonzero entry of the row vector associated with e ∈ F
is in the columns associated with V (F ). This means that RF is the direct sum of RF
for F ∈ C (F ), and hence it suffices to check the statement for a connected F with
V (F ) = V .
Clearly, dim RF ≤ dV . Since F  ≤ dim RF , we now show that dim R⊥F ≥
dim iso F φ,w ( p(F )), where R⊥F denotes the orthogonal complement of RF .
To see this we first check that a switching operation does not change the rank of
the orbit rigidity matrix. Let φ be the gain function obtained from φ by a switching
operation at v0 with g0 ∈ S. We define p : V → Rd by
p (u) =
p(u) if u = v0,
g0 p(u) if u = v0.
Note that p (F ) = {gp (v) : v ∈ V , g ∈ S} = p(F ). We now show
rank O(G, φ, p) = rank O(G, φ , p ).
Let us consider a nonloop edge e = (u, v0) oriented to v0 in G. The row
corresponding to e in O(G, φ , p ) is written by
u v0
0 · · · 0 p (u) − φ (e) p (v0) 0 · · · 0 p (v0) − φ (e)−1 p (u) 0 · · · 0
By (1), we have φ (e) = φ (e)g0−1. Thus, by using (12), the row of e becomes
u v0
0 · · · 0 p(u) − φ (e) p(v0) 0 · · · 0 g0( p(v0) − φ (e)−1 p(u)) 0 · · · 0
Similarly, for a nonloop edge e = (v0, u) oriented from v0 in G, the row of e becomes
exactly the same form as above. By using the same calculation, for a loop e incident
to v0 in G, the row of e in O(G , φ , p ) can be written as
(12)
(13)
v0
0 · · · 0 g0(2Id − φ (e) − φ (e)−1) p(v0) 0 · · · 0
By performing column operations within the d columns associated with v0, these
are converted to
u v0
0 · · · 0 p(u) − φ (e) p(v0) 0 · · · 0 p(v0) − φ (e)−1 p(u) 0 · · · 0
and
v0
0 · · · 0 (2Id − φ (e) − φ (e)−1) p(v0) 0 · · · 0
respectively, which implies that rank O(G, φ, p) = rank O(G, φ , p ). Therefore, the
row independence of the orbit rigidity matrix is invariant under switching operations.
Moreover, since p(F ) = p (F ), dim iso F φ,w ( p(F )) = dim iso F φ ,w ( p (F )). So it
suffices to prove the statement for O(G, φ , p ).
Let T be a spanning tree of G. Since we can freely apply switching operations, we
may assume that φ (e) = id for all e ∈ T . Then, by Lemma 2.3, F φ,w = φ (e) : e ∈
F \T for a vertex w ∈ V (F ).
Let us take any m ∈ iso F φ,w ( p(F )) and let m˜ : V → Rd be defined by m˜ (v) =
m( p(v)) for v ∈ V . We show that m˜ is in the orthogonal complement of RF . To check
it, let us consider any edge e = (u, v) ∈ F with gain h = φ (e). Since m ∈ iso( p(F )),
we have
p(u) − hp(v), m( p(u)) − m(hp(v)) = 0.
Since m is F φ,wsymmetric, we also have m(hp(v)) = hm( p(v)). Therefore, we
obtain
0 =
p(u) − hp(v), m( p(u)) − m(hp(v)) =
p(u) − hp(v), m˜ (u) − hm˜ (v) ,
implying that m˜ is in the orthogonal complement of RF . Consequently, dim R⊥F ≥
dim iso F φ,w ( p(F )), and hence F  ≤ dim RF ≤ dV  − dim iso F φ,w ( p(F )).
This, together with Theorem 5.1, directly implies a necessary condition for
symmetric frameworks to be symmetryforced infinitesimally rigid.
Recall that S is a finite family of orthogonal matrices. Let QS be the field generated
by Q and the entries of all the matrices in S. Since S is finite, almost all numbers
in R are transcendental over QS . For a given gain graph (G, φ), a mapping p˜ :
V (G) → Rd is called Sgeneric if the set of coordinates of p˜(v) for all v ∈ V (G) is
algebraically independent over QS . Similarly, for a given (S, ρ)symmetric graph H ,
an (S, ρ)symmetric jointconfiguration p : V (H ) → Rd is called Sgeneric if the
corresponding jointconfiguration p˜ of the vertex orbits is Sgeneric. An Ssymmetric
framework is called Sgeneric if the joint configuration is Sgeneric.
In Sects. 6 and 8 we will check that the condition of Lemma 5.2 is indeed sufficient
for generic symmetric frameworks in the plane with cyclic groups and dihedral groups
D2k with odd k, respectively.
6 Combinatorial Characterization of Generic Rigidity with Cyclic
Symmetry
In this section we shall prove a combinatorial characterization of the symmetryforced
rigidity of Sgeneric symmetric frameworks with cyclic point groups in the plane. The
following lemma is a key observation, which is an extension of the one given in [26,28]
for proving Laman’s theorem. The lemma is not limited to cyclic groups.
Lemma 6.1 Let (G, φ) be an Sgain graph for a point group S ⊂ O(R2). Let (G , φ )
be an Sgain graph obtained from (G, φ) by a 0extension, 1extension, or
loop1extension. If there is a mapping p : V (G) → R2 such that O(G, φ, p) is row
independent, then there is a mapping p : V (G ) → R2 such that O(G , φ , p ) is
row independent.
Proof If there is a p such that O(G, φ, p) is row independent, then O(G, φ, q) is row
independent for all Sgeneric q. Hence, we may assume that p is Sgeneric. We only
show the difficult case where (G , φ ) is constructed from (G, φ) by a 1extension (see
[8] for the easier case where (G , φ ) is constructed from (G, φ) by a 0extension or
a loop1extension).
Suppose that (G , φ ) is obtained from (G, φ) by a 1extension, by removing an
existing edge e and adding a new vertex v with three new nonloop edges e1, e2, e3
incident to v. We may assume that ei is outgoing from v. Let ui be the other endvertex of
ei , and let gi = φ (ei ) and pi = p(ui ) for i = 1, 2, 3. By the definition of 1extension,
we have φ (e) = g1−1g2.
Claim 6.2 The three points gi pi (i = 1, 2, 3) do not lie on a line.
Proof Since p is Sgeneric, u1 = u2 = u3 should hold if they lie on a line. Then
p1 = p2 = p3. By the definition of 1extensions, gi = g j if ui = u j . This implies
that g1 p1, g2 p2, g3 p3 are three distinct points on a circle. Thus, they do not lie on a
line.
We take p : V (G ) → R2 such that p (w) = p(w) for all w ∈ V (G), and
p (v) is a point on the line through g1 p1 and g2 p2 but is not equal to g1 p1 or g2 p2.
O(G , φ , p ) is described as follows: if u1 = u2
e3
e1
e2
E (G) − e
v u1
p (v) − g3 p3 ∗
p (v) − g1 p1 p1 − g1−1 p (v)
p (v) − g2 p2 0
0 O(G − e, φ, p)
u2
where the rightbottom block O(G −e, φ, p) denotes the orbit rigidity matrix obtained
from O(G, φ, p) by removing the row of e, whereas, if u1 = u2,
We consider the case when u1 = u2 (the case when u1 = u2 is similar). Since
p (v) lies on the line through g1 p1 and g2 p2, p (v) − gi p(ui ) is a scalar multiple
of g1 p1 − g2 p2 for i = 1, 2. Hence, by multiplying the rows of e1 and e2 by an
appropriate scalar, O(G , φ , p ) becomes
e3
e1
e2
E (G) − e
v u1
p (v) − g3 p3 ∗
g1 p1 − g2 p2 −g1−1(g1 p1 − g2 p2)
g1 p1 − g2 p2 0
0
u2
∗
0
∗
0
−g2−1(g1 p1 − g2 p2) 0
O(G − e, φ, p)
C
g2p2
g2p2
Let C be the circle whose center is the origin and which passes through g1 p1 (and
hence through g3 p1). We split the proof into two cases depending on whether g3g1−1
is the twofold rotation Cπ or not.
(iv1) Suppose that g3g1−1 = Cπ . Let C be a circle whose center is the origin and
the diameter is much larger than that of C . We shall relocate g2 p2 on C such that
g2 p2 is on the line through g1 p1 and the origin as shown in Fig. 10a. Then, if L and
L are parallel, we have only two possible locations q and q for g4 p2 (as shown in
Fig. 10a). Since the diameter of C can be arbitrarily large, D2k has no element that
sends g2 p2 to q or q . In other words, if p is generic, L and L are not parallel.
(iv2) Suppose that g3g1−1 = Cπ . Then g3 p1 is the antipodal point of g1 p1 in C as
shown in Fig. 10b. Let C be a circle whose center is the origin and the diameter is
slightly larger than that of C . We shall relocate g2 p2 on C such that L is the tangent
of C at g1 p1 (see Fig. 10b). Then, we have only two possible locations q and q for
g4 p2 as L and L are parallel and g4 p2 is on C , where q is the antipodal point of
g2 p2 with respect to the origin and q is the reflection of q2 p2 along the line parallel
to L and through the origin. When p is generic, L is not parallel to any reflection lines
in D2k , implying g4 p2 = q . Hence, g4 p2 = q. This means that g4g2−1 is also the
twofold rotation Cπ .
Recall that Cπ is in the center of O(R2), i.e., gCπ = Cπ g for any g ∈ O(R2).
Hence, by g3g1−1 = Cπ , we have g1−1g3 = g1−1Cπ g1 = Cπ . Symmetrically, by
g4g2−1 = Cπ , we have g2−1g4 = Cπ . This however implies that g1−1g3 = g2−1g4,
which contradicts (19).
Following the statement of Claim 8.8, we shall split the proof into two cases.
(Case 1) Suppose that L and L are not parallel. Let q be the intersection of L and L .
By Claim 8.8(i), we have q = gi pi . We define p : V (G ) → R2 by p (w) = p(w)
for w ∈ V (G) and p (v) = q for the added vertex v. Then, O(G , φ , p ) can be
written as follows:
e1
e2
e3
e4
E(G) − e − f
where O(G − e − f, φ, p) is the matrix obtained from O(G, φ, p) by removing the
rows of e and f . Consider the rows associated with e1 and e2. Since q is on L, q − gi pi
is a scalar multiple of g1 p1 − g2 p2, and hence these two rows can be transformed to
the following form by row operations: if u1 = u2
Notice that, in each case, the row of e2 is converted to that of e in O(G, φ, p). In a
symmetric manner, the rows of e3 and e4 can be converted to the above form, simply
by replacing 1 and 2 with 3 and 4, respectively. Thus, O(G , φ , p ) is converted to
v
e1 g1 p1 − g2 p2
e3 g3 p3 − g4 p4
E(G) 0
∗
∗
O(G, φ, p)
The rightbottom block O(G, φ, p) is row independent while the lefttop block is also
row independent since L and L are not parallel. In other words, O(G , φ , p ) is row
independent.
(Case 2) Suppose that L and L are parallel. By Claim 8.8, L = L , p1 = p2,
p3 = p4, and g−1g2 and g3−1g4 are reflections. Let q be any point on L with q = g1 p1
1
and q = g2 p1. We define p : V (G ) → R2 by p (w) = p(w) for w ∈ V (G) and
p (v) = q for the new vertex v. Then, the orbit rigidity matrix is described as follows:
e1
e2
e3
e4
E(G) − e − f
and then to
Since g1−1g2 is a reflection, we have g1−1g2 = g2−1g1. Hence, by adding the half of
the second row to the first row, we obtain
Since q is on the line L, q − gi pi is a scalar multiple of (g1 − g2) p1 for i = 1, 2.
Hence, the rows of e1 and e2 can be converted to
Next we consider the rows of e3 and e4. By subtracting the row of e3 from that of e4,
we obtain
Since L and L are parallel, {(g1 − g2) p1, (g3 − g4) p3} is linearly dependent. Thus,
by subtracting the row of e1 from that of e4, we have
Moreover, since g4−1g3 is a reflection, Lemma 8.3 implies that (I2 − g4−1g3)g3−1q is
a scalar multiple of (I2 − g4−1g3) p3, and hence (g3−1 − g4−1)q is a scalar multiple of
(I2 − g−1g3) p3. Therefore, by using g3−1g4 = g4−1g3, the row of e4 can be converted
4
by a scalar multiplication to
v u1 u2
e4 0 0 (2I2 − g3−1g4 − g4−1g3) p3 0
In total, O(G , φ , p ) is changed to the following form by rowoperations:
e1
e3
e2
e4
E(G) − e − f
v
(g1 − g2)p1
q − g3 p3
0
0
0
u1
0
p3 − g3−1q
(2I2 − g1−1g2 − g2−1g1)p1
0
u3 V (G)
0 0
0 0
0 0
(2I2 − g−1g4 − g4−1g3)p3 0
3
O(G − e − f, φ, p)
The rightbottom block together with the rows of e2 and e4 forms O(G, φ, p), which
is row independent. Also, since q is on L, but not on L , {(g1 − g2) p1, q − g3 p3} is
linearly independent. Therefore, O(G , φ , p) is row independent.
Combining Theorem 7.12, Lemmas 6.1, 8.1, 8.4, 8.5, and 8.7, we can now complete
the proof of Theorem 8.2.
8.2 SymmetryForced Infinitesimal Motions with D2kSymmetry for Even k
Notice that all the lemmas given in the last subsection are independent of the parity
of k. Therefore, we obtain the following statement even for a dihedral group D2k with
even k: for a generic (D2k , ρ)symmetric framework (H, p) with even k and a free
action ρ, (H, p) is symmetryforced infinitesimally rigid if the quotient gain graph
can be constructed from a disjoint union of base graphs by 0extensions, 1extensions,
loop1extensions, 2extensions and loop2extensions. However, as we have seen in
Fig. 9, there are infinitely many gain graphs that cannot be constructed from base
graphs. By Theorem 7.9, minimal examples are D2k sparse double cycles Cn2. Below,
we show that some of them indeed have symmetric infinitesimal motions.
For Cn2, the vertex set is denoted by {1, . . . , n} and the edges of the 2cycle between
i and i + 1 (mod n) are denoted by ei,1 and ei,2 for i = 1, . . . , n.
Theorem 8.9 Let D4 be the dihedral group of order 4, which consists of the identity
I2, the twofold rotation Cπ , and two reflections r and r . Let (G, φ) be a D4sparse
Cn2 such that
• φ (ei,1) = id and φ (ei,2) = r for i = 1, . . . , n − 1;
• φ (en,1) = Cπ and φ (en,2) = r .
Then, for any D4generic p : V (G) → R2, rank O(G, φ, p) = 2n if and only if n is
odd.
Proof Let p : i ∈ V (G) → (xi , yi ) ∈ R2 be a D4generic mapping. Then Cπ p(i ) =
(−xi , −yi ), r p(i ) = (−xi , yi ), r p(i ) = (xi , −yi ). The rows of O(G, φ, p) are as
follows:
i i + 1 i i + 1
ei,1 0 xi − xi+1 xi+1 − xi 0 0 yi − yi+1 yi+1 − yi 0
ei,2 0 xi − xi+1 xi+1 − xi 0 0 yi + yi+1 yi+1 + yi 0
n 1 n 1
en,1 0 xn + x1 x1 + xn 0 0 yn + y1 y1 + yn 0
en,2 0 xn + x1 x1 + xn 0 0 yn − y1 y1 − yn 0
and
where the left and the right half sides correspond to x  and ycoordinates, respectively.
For each i , we subtract the first row from the second row and then multiply the first
row by an appropriate scalar. We then have, for each i = 1, . . . , n − 1,
Cπ
r r r
id
id
id
r
r
id
Cπ
r
r
id
(a)
(b)
and
In other words, O (G, φ , p) is converted to the following form:
y2 y1
y3 y2
. . . . . .
yn
yn yn−1
y1
The determinant of this matrix is 2(1 − (−1)n−1)
and only if n is even.
n
i=1 yi , which is equal to zero if
See Fig. 11 for examples of frameworks given in Theorem 8.9. For n = 2, the
covering graph is K4,4 and the corresponding framework is known as Bottema’s mechanism
(see [23, Sect. 7.2.1]).
9 Proof of Theorem 7.8
In this section we prove Theorem 7.8. For simplicity of the description, a Dgain graph
is called essential if it is Dsparse, 4regular, not a base graph, and not a double cycle.
Lemma 7.6 shows that 2extensions and loop2extensions preserve Dsparsity, and
hence what we have to prove is the following theorem.
Theorem 9.1 Any essential Dgain graph (G, φ) has a vertex at which a 2reduction
or a loop2reduction is admissible.
For simplicity, in the subsequent discussion we omit gain functions φ when referring
to gain graphs if it is clear from the context. Also an edge (u, v) from u to v is simply
denoted by uv, and a Dtight set is called a tight set.
The proof of Theorem 9.1 consists of four parts. In Sect. 9.1, we shall prove useful
lemmas for subsequent discussion. In Sect. 9.2, we prove Theorem 9.1 for the following
graphs:
• graphs consisting of only special vertices (Lemma 9.5), where a vertex is called
special if it is incident with a loop or two parallel classes of edges;
• graphs that are not 2connected (Lemma 9.6),
• “almost” nearcyclic graphs (Lemma 9.8), defined below,
• graphs that are not essentially 4edgeconnected (Lemma 9.9),
• graphs having a vertex v with N (v) = 2.
In Sect. 9.3 we discuss graphs not belonging to the above classes. In Sect. 9.4 we put
everything together to complete the proof of Theorem 9.1.
9.1 Preliminary Facts
The following fundamental properties of 4regular graphs will be frequently used.
• A 4regular graph is Eulerian. Hence, a 4regular connected graph is
2edgeconnected.
• Let G = (V , E ) be a graph with maximum degree at most 4. Then, for any X ⊆ V ,
iG (X ) ≤ 2X  − dG (X )/2 , where iG (X ) denotes the number of edges induced
by X . In particular, if G is 4regular, iG (X ) = 2X  − dG (X )/2.
The next lemma asserts that if the maximum degree is at most 4, then Dsparsity
is equivalent to the following simpler properties:
(C1) F  ≤ 2V (F ) − 3 for every nonempty balanced set F ⊆ E ;
(C2) G is not cyclic.
Lemma 9.2 Let G = (V , E ) be a Dgain graph with maximum vertex degree at most
4. If G is connected, then G is Dsparse if and only if
(i) G is not 4regular and condition (C1) is satisfied, or
(ii) G is 4regular and conditions (C1) and (C2) are satisfied.
If G is not connected, then G is Dsparse if and only if each connected component is
Dsparse.
Proof If the maximum degree is at most 4, F  ≤ 2V (F ) for any F ⊆ E . In
particular, if G is connected, we have F  ≤ iG (V (F )) ≤ 2V (F )− dG (V (F ))/2 ≤
2V (F ) − 1 for any F ⊆ E with V (F ) = V . Therefore, F  ≥ 2V (F ) holds if and
only if G is 4regular and F = E .
Thus, to prove Theorem 9.1, we shall investigate whether (C1) and (C2) are satisfied
after the reductions. The next lemma will be used when (C2) is not satisfied. We say
that (G, φ) is almost nearcyclic if there are two incident edges e and f such that
G − e − f is cyclic.
Lemma 9.3 Let (G, φ) be a connected 4regular Dsparse graph with G = (V , E )
and v be a vertex in G that is not incident to a loop. Let e1, e2, e3, e4 be the edges
incoming to v, and suppose that G − v + e1 · e2−1 + e3 · e4−1 is connected and cyclic.
Then, there is an equivalent gain function φ to φ and a cyclic subgroup C of D such
that
• φ (e) ∈ C for every e ∈ E \{e3, e4}, and
• φ (e3) ∈/ C¯ and φ (e4) ∈/ C¯.
In particular, G is almost nearcyclic.
Proof Let G = G − v + e1 · e2−1 + e3 · e4−1. Since G is connected and cyclic,
by Lemma 2.3, there is an equivalent gain function φ to φ and a cyclic subgroup
C of D such that φ (e) ∈ C for all e ∈ E (G ). Let a = φ (e1 · e2−1) ∈ C and
a = φ (e3 · e4−1) ∈ C. Then, by using some elements b1, b2 ∈ D, we can express
φ (ei ) by
φ (e1) = ab1, φ (e2) = b1, φ (e3) = a b2, φ (e4) = b2.
We further perform a switching operation at v with b1. We consequently have an
equivalent gain function φ to φ such that
φ (e1) = a, φ (e2) = id, φ (e3) = a b, φ (e4) = b,
where b = b2b1−1. Notice that φ (e) ∈ C for all e ∈ E \{e3, e4}. Since G is not cyclic,
we must have b ∈/ C¯, implying that φ (e3) ∈/ C¯ and φ (e4) ∈/ C¯.
The following technical lemma is one of the key observations. A vertex in a
4regular graph is called special if it is incident with a loop or two parallel classes of
edges with N (v) = 2.
Lemma 9.4 Let (G, φ) be a connected 4regular Dsparse graph with G = (V , E )
and V  ≥ 3, v be a vertex in G that is not special, and e1, e2, e3, e4 be the edges
incoming to v. If G − e3 − e4 or G − v + e1 · e2−1 + e3 · e4−1 is connected and cyclic,
then at least one of the following holds:
(a) G is nearcyclic.
(b) G − v + e1 · e3−1 + e2 · e4−1 is Dsparse.
(c) v is a cutvertex in G and G − v + e1 · e3−1 + e2 · e4−1 is connected.
Proof For simplicity, we denote ei, j = ei · e −j1 for i, j ∈ {1, 2, 3, 4}. We assume that
neither (a) nor (b) occur and show that (c) holds.
We claim that there are an equivalent gain function φ to φ and a cyclic subgroup
C of D such that φ (e) ∈ C holds for e ∈ E \{e3, e4} and φ (e3) ∈/ C¯ and φ (e4) ∈/ C¯.
To see this, first observe that if G − v + e1 · e2−1 + e3 · e4−1 is connected and cyclic,
then Lemma 9.3 implies the claim. On the other hand, if G − e3 − e4 is connected and
cyclic, then by Lemma 2.3, there is an equivalent φ to φ and a cyclic subgroup C of
D such that φ (e) ∈ C for e ∈ E \{e3, e4}. Since G is neither cyclic nor nearcyclic,
we have φ (e3) ∈/ C¯, and φ (e4) ∈/ C¯.
Note that φ (e1,3) ∈/ C¯ and φ (e2,4) ∈/ C¯.
Let us consider G − v. Since G − v is cyclic with E (G − v) = 2V (G − v) − 2,
G − v is (2, 3)gsparse. Applying Lemma 7.7 with φ (e1,3) ∈/ C¯, we deduce that
G − v + e1,3 is Dsparse. Let G = G − v + e1,3 + e2,4. Since (b) does not hold, G
is not Dsparse. By Lemma 9.2, G (or a connected component of G ) violates (C1)
or (C2).
Case 1: If (C1) is violated, then G − v + e1,3 contains a balanced tight set F such
that V (F ) contains the endvertices of e2,4 and F + e2,4 is balanced. Let s and t be the
endvertices of e2,4, which are possibly the same vertex. By Lemma 7.1, if F  > 1,
F contains a path from s to t that does not pass through e1,3. Recall that the gain
of each edge in this path is included in C, and the concatenation of the path and e2,4
forms an unbalanced closed walk in F + e2,4, contradicting that F + e2,4 is balanced.
Therefore, F  = 1 holds; in particular, since s, t ∈ V (F ) and F + e2,4 is balanced, it
follows that F = {e1,3} and {e1,3, e2,4} forms a balanced 2cycle in G . This implies
that v is special in G, contradicting the assumption of the lemma.
Case 2: We next consider the case when (C2) is violated in G . Suppose that v is
not a cutvertex. Note that, since E (G − v) = 2V (G − v) − 2, G − v contains
an unbalanced cycle C , whose gain is included in C. Let s and t be the endvertices of
e2,4, which are possibly the same vertex. Since G − v is connected, there is a path P
from s to a vertex in V (C ). We consider a closed walk W1 that first passes through
P starting at s, then goes around C , and comes back to s through P−1. We then have
φ (W1) ∈ C. Also, since G − v is connected, G − v has a path P connecting s
and t . The concatenation of P with e2,4 forms a closed walk W2 starting at s with
φ (W2) ∈/ C¯. Thus, {φ (W1), φ (W2)} generates a noncyclic group. Hence, G satisfies
(C2), a contradiction. Thus, v is a cutvertex in G.
Suppose that G is not connected. Then, by the 4regularity of G, G consists of two
connected components, denoted G1 and G2 with e1,3 ∈ E (G1) and e2,4 ∈ E (G2).
We have already seen that G − v + e1,3 is Dsparse, and hence its subgraph G1
is Dsparse. However, since G1 is 4regular, G1 is indeed maximum Dtight. By the
symmetry between e1,3 and e2,4, G2 is also maximum Dtight, and thus G is maximum
Dtight, a contradiction. Thus G should be connected, which implies (c).
9.2 Special Cases
Recall that a vertex is said to be special if it is incident with a loop or two parallel
classes of edges. A graph which consists of only special vertices is called a special
graph. Special graphs are classified into the following three classes C 2, Cn◦ and P2
n n
for n ≥ 2 (Fig. 12): As defined in Sect. 7.2, Cn2 is the graph obtained from the cycle of
n vertices by replacing each edge by two parallel copies; Cn◦ is the cycle of n vertices,
each of which is incident to a loop; Pn2 is the graph obtained from a path of n vertices
by replacing each edge by two parallel copies and adding one loop to each endvertex
of the path.
Lemma 9.5 Let (G, φ) be an essential Dgain graph whose underlying graph G =
(V , E ) is special. Then there is a vertex at which a 2reduction or a loop2reduction
is admissible.
Proof Since (G, φ) is essential, the underlying graph is either Pn2 or C ◦.
n
Suppose that the underlying graph is P2. We perform the loop2reduction at a
n
vertex incident to a loop l. The resulting graph is P2
n−1 and clearly it satisfies (C1). If it
does not satisfy (C2), then the resulting graph is cyclic and there is a cyclic subgroup
C of D such that the gain of every cycle in G except for the loop l is in C. This in turn
implies that G − l is cyclic, contradicting the assumption that G is essential.
Suppose that the underlying graph is C ◦. We may assume n ≥ 3 since C2◦ = P2.
n 2
We perform the 2reduction at a vertex incident to a loop l. The resulting Dgain graph,
denoted G , has the underlying graph Cn◦−1.
If G does not satisfy (C2), then the gain of each cycle in G except for the loop
l is included in a cyclic subgroup C of D, which again contradicts the fact that G is
essential.
It can be easily observed that G satisfies (C1) if n > 3. For n = 3, (C1) is violated
if the 2cycle of G is balanced, but in such a case the triangle in the original graph
G is balanced, and G turns out to be a fancy triangle, contradicting the fact that G is
essential.
The next lemma solves the case when the graph is not 2connected.
Lemma 9.6 Let G = (V , E ) be a connected essential Dgain graph with V  ≥ 2.
Suppose that G is not 2connected. Then a 2reduction is admissible at some vertex.
Proof By Lemma 9.5, we may assume that G is not equal to P2V . Then G has a
cutvertex v which is not special. We show that a 2reduction at v is admissible. Note that
G −v consists of two connected components by the 4regularity of G. Let e1, e7, e3, e4
be the edges incident to v, all of them are directed to v. From the 2edgeconnectivity
of G, we can assume, without loss of generality, that the endvertices of e1 and e3 are
included in a connected component of G − v while those of e2 and e4 are included in
the other component.
Consider the 2reduction at v through (e1, e2) and (e3, e4). Let G be the resulting
graph. Note that G is connected. Let us check that G satisfies (C1). To see this,
recall that any balanced tight set consisting of more than one edge is 2connected by
Lemma 7.1. Note also that e3 · e4−1 is not parallel to e1 · e2−1 as v is not special. Since
the endvertices of e3 · e4−1 belong to different connected components in G − v and
e1 · e2−1 is the bridge in G − v + e1 · e2−1, G − v + e1 · e2−1 has no balanced tight set
F such that V (F ) contains both endvertices of e3 · e4−1. This implies that G satisfies
(C1).
Therefore, if G satisfies (C2), then G is Dsparse by Lemma 9.2, and a 2reduction
is admissible at v. Suppose that G does not satisfy (C2). Then, G is connected and
cyclic. To apply Lemma 9.4, we next consider the 2reduction at v through (e1, e3)
and (e2, e4). The resulting graph, denoted by G , is disconnected. Lemma 9.4 thus
implies that G is Dsparse (recall that G is not nearcyclic since G is essential).
Thus, in the subsequent discussion, we may focus on 2connected graphs. The next
lemma solves the case when G has a special vertex not incident to a loop.
Lemma 9.7 Let G = (V , E ) be a 2connected essential Dgain graph. Suppose
that G has a special vertex not incident to a loop. Then, G has a vertex at which a
2reduction is admissible.
Proof Let w be a special vertex not incident to a loop. By definition of special vertices,
N (w) = 2 and w is incident to two parallel classes of edges. Since G = Cn2, G
contains two adjacent vertices u and v such that v is not special and u is special and
not incident to a loop (where u is possibly equal to w). Depending on the size of
N ({u, v}), we have two possible cases as shown in Fig. 13.
Let us denote the edges incident to u by e1, e2, e3, e4, where e1 and e2 are linking
from v to u and e3 and e4 are linking from a vertex in V \{u, v} to u. We perform the
2reduction at u through (e1, e2) and (e3, e4). Since both new edges are unbalanced loops
and adding unbalanced loops does not violate (C1), the resulting graph G satisfies
(C1). Therefore, if the 2reduction is not admissible at u, then G does not satisfy (C2),
and hence G − e1 − e2 is cyclic by Lemma 9.3.
Let a, b, c ∈ V such that N (v) = {u, a, b} and N (u) = {v, c}. Since N (u) = 2
with v ∈ N (u), without loss of generality we may assume a ∈/ N (u) (where b = c
possibly holds). Recall that G − e1 − e2 is connected and cyclic, and hence we can
apply Lemma 9.4 to deduce that the 2reduction at v through (bv, e1) and (av, e2)
is admissible. Indeed, since G is not nearcyclic and v is neither a cutvertex nor a
special vertex, Lemma 9.4 implies that this 2reduction at v is admissible.
The next lemma solves the case when G is almost nearcyclic.
Lemma 9.8 Let G = (V , E ) be a 2connected essential Dgain graph with at least
two vertices. Suppose that G is almost nearcyclic. Then a 2reduction or a
loop2reduction is admissible at some vertex in G.
Proof Since G is almost nearcyclic, there are two edges e1 and e2 for which e1 and
e2 are incident to a vertex v and G − e1 − e2 is cyclic.
Suppose that v is not special. Then, since v is not a cutvertex, a 2reduction is
admissible at v by Lemma 9.4. Therefore, let us consider the case when v is special.
If v is not incident to a loop, then Lemma 9.7 directly implies the claim. We can thus
assume that v is incident to a loop.
Suppose that both e1 and e2 are nonloop edges. By Lemma 2.3, we may assume
that the label of each edge in G − e1 − e2 is contained in a cyclic subgroup C of D.
By further performing a switching operation at v with φ (e1), φ is converted such that
φ (e1) = id and φ (e) ∈ C for all edges e not incident to v. This implies that if we
remove e2 and the loop incident to v from G, the resulting graph is cyclic. In other
words, it suffices to consider the case when e1 or e2 is a loop.
We hence assume that e1 is the loop incident to v. Let e3 be the remaining nonloop
edge incident to v, where φ (e3) ∈ C. Observe that the gain of the nonloop edge e2 is
not included in C¯, since otherwise G −e1 becomes cyclic, contradicting the assumption
that G is essential. Therefore, φ (e2 · e3−1) ∈/ C¯, and the loop2reduction at v adds the
edge e2 · e3−1 to the cyclic (2, 3)gsparse graph G − v. By Lemma 7.7, the resulting
gain graph is Dsparse.
By using Lemma 9.8, we can now prove an important consequence for graphs that
are not essentially 4edgeconnected.
Lemma 9.9 Let G = (V , E ) be a 2connected essential Dgain graph with V  =
n ≥ 4. Suppose that G is not essentially 4edgeconnected. Then, G has a vertex at
which a 2reduction or a loop2reduction is admissible.
Proof Since G is 2edgeconnected and is not essentially 4edgeconnected, there
exists a subset X of V for which X  > 1, V \X  > 1 and dG (X ) = 2. Since G is not
Cn◦, we can suppose that B(X ) contains a vertex v not incident to a loop, where B(X )
denotes a set of vertices of X adjacent to some vertices of V \X . By the 2connectivity,
v is not a cutvertex. Hence, denoting the four edges incident to v by e1, . . . , e4, we
may assume that e1, e2, e3 are included in the subgraph induced by X while e4 is not.
Note that v is a vertex of degree 3 in G − e4, and hence, by Lemma 7.5, a
1reduction at v is admissible in G − e4. Without loss of generality, we may assume
that G − v + e1 · e2−1 (obtained by a 1reduction at v in G − e4) is Dsparse.
We now consider adding e3 · e4−1 to G − v + e1 · e2−1 to complete the 2reduction
at v. Let G = G − v + e1 · e2−1 + e3 · e4−1, and suppose that G does not satisfy (C1).
Since any balanced tight set F is 2edgeconnected if F  > 1, there is no balanced
tight set F for which V (F ) contains both endvertices of e3 · e4−1 unless F  = 1. If
G − v + e1 · e2−1 has a balanced set F such that F  = 1 and V (F ) contains both
endvertices of e3 · e4−1, then the edge in F , denoted by f , is incident to e3 and e4 and
connects between X and V \X . However, since dG (X ) = 2, X  > 1 and V \X  > 1,
the vertex incident to e4 and f turns out to be a cutvertex of G, contradicting the
2connectivity of G. Thus, G satisfies (C1).
If G does not satisfy (C2), it is cyclic. By Lemma 9.3, G is almost nearcyclic,
and we can apply Lemma 9.8 to conclude that a 2reduction or a loop2reduction is
admissible at some vertex v.
The final special case is when G has a vertex v with N (v) = 2.
Lemma 9.10 Let G = (V , E ) be a 2connected essential Dgain graph. Suppose that
G has a vertex v with N (v) = 2 that is not incident to a loop. Then, there is a vertex
at which a 2reduction is admissible.
Proof If v is special, Lemma 9.7 implies the claim.
If v is not special, then there are three parallel edges between v and a neighbor
of v. By the 4regularity, if V  ≥ 4, G is not essentially4edgeconnected, and thus
Lemma 9.9 implies the statement.
If V  = 3, G is equal to the graph (shown in Fig. 15) of three vertices V =
{u, v, w}, three parallel edges e1, e2, e3 between u and v, a loop l attached to w, and
two remaining edges uw and vw, denoted by f1 and f2, respectively. We may assume
φ ( f1) = φ ( f2) = id. Let C be the subgroup generated by φ (l). Since G is not cyclic,
there is an unbalanced cycle whose gain is not included in C¯.
If a triangle, say e1 f1 f2 has a gain not included in C¯, then the 2reduction at u through
(e1, f1) and (e2, e3) results in a Dsparse P2. Otherwise, removing e2 and e3 results in
2
a cyclic graph. Then G is almost nearcyclic, and Lemma 9.8 implies the statement.
9.3 The Remaining Cases
In a graph G, the star of a vertex v means the subgraph of G whose vertex set is
N (v) ∪ {v} and the edge set is the set of edges incident to v. A hat subgraph is
a balanced subgraph whose underlying graph is a hat. See Fig. 14 for an example.
The following claim, together with the previous lemmas, will complete the proof of
Theorem 9.1.
Theorem 9.11 Let G = (V , E ) be a 2connected, essentially 2edgeconnected, and
essential graph with V  ≥ 3. Suppose also that G is not almost nearcyclic. Then, for
every vertex v ∈ V that is not incident to a loop with N (v) ≥ 3, either a 2reduction
at v is admissible or the star of v is contained in a hat subgraph.
In Sect. 9.3.1, we focus on the case of N (v) = 4. Lemma 9.12 says that if the
2reduction is not admissible then G has an obstacle around v. We will investigate
Fig. 14 A hat subgraph
id id
id
Fig. 15 The special graph given
in the proof of Lemma 9.10
intersection properties of obstacles. The corresponding results for the case of N (v) =
3 will be given in Sect. 9.3.2. In Sect. 9.3.3, we prove Theorem 9.11 based on the
intersection properties of obstacles.
In the rest of this section, clD denotes the closure operator of the underlying matroid
MD(G, φ).
9.3.1 Obstacles Around a Vertex v with N (v) = 4
Throughout Sect. 9.3.1, (G, φ) denotes a Dgain graph satisfying the assumptions
of Theorem 9.11, v denotes a vertex with N (v) = 4, N (v) = {a, b, c, d}, and Ev
denotes the set of edges incident to v.
An edge subset F is called subtight if F  = 2V (F ) − 4 and F is balanced. We
first make a simple observation which describes the situation where 2reductions are
not admissible.
Lemma 9.12 Suppose that the 2reduction through (av, vb) and (cv, vd) is not
admissible. Then there is an edge subset F ⊆ E \Ev satisfying one of the
following properties:
(i) F is balanced tight with a, b ∈ V (F ) and av · vb ∈ clD(F );
(ii) F is balanced tight with c, d ∈ V (F ) and cv · vd ∈ clD(F );
(iii) F is subtight with a, b, c, d ∈ V (F ), F + av · vb is balanced tight, and cv · vd ∈
clD(F + av · vb).
Proof Let us first consider the graph G = G−v+av·vb. If G is not Dsparse, then, by
Lemma 9.2, E \Ev has a balanced tight set F with a, b ∈ V (F ) and av · vb ∈ clD(F ),
which satisfies property (i).
Hence, let us assume that G is Dsparse. If G + cv · vd is cyclic, Lemma 9.3
implies that G is almost nearcyclic, contradicting the assumption that G is not almost
nearcyclic. Therefore, G + cv · vd satisfies (C2). By Lemma 9.2, there exists a
balanced tight set F ⊆ E \Ev ∪ {av · vb} with c, d ∈ V (F ) and cv · vd ∈ clD(F ).
Depending on whether av · vb ∈ F or not, we find a desired subset of the statement; if
av · vb ∈/ F then F is the one satisfying property (ii); otherwise F − av · vb satisfies
property (iii) (we remark that, in the latter case, V (F − av · vb) contains a, b, c, d
since F is 2edgeconnected).
Since the first and the second cases of the statement of Lemma 9.12 are symmetric,
we basically have two types of obstacles: for a vertex v and N (v) = {a, b, c, d},
F ⊆ E \Ev is called an obstacle of type 1 (for the 2reduction through (av, vb) and
(cv, vd)) if F satisfies (i) or (ii) of Lemma 9.12; F is called an obstacle of type 2 if F
satisfies (iii).
Lemma 9.13 Suppose that X is an obstacle of type 2 for the 2reduction through
(av, vb) and (cv, vd). Then, the following holds for X :
• X ∪ Ev = 2V (X ∪ Ev) − 2;
• There is an equivalent gain function φ to φ such that φ (e) = id for e ∈ X ∪
{va, vb}, and φ (vc) = φ (vd) = id;
• X ∪ Ev is cyclic.
Proof By definition, X  = 2V (X ) − 4, and hence X ∪ Ev = 2V (X ∪ Ev) − 2
by N (v) ⊆ V (X ).
Since cv · vd ∈ clD(X + av · vb) and X + av · vb is balanced, X + av · vb + cv · vd
is also balanced. Hence, by Lemma 2.3, there is an equivalent gain function φ to φ
such that φ (e) = id for e ∈ X and φ (av · vb) = φ (cv · vd) = id. We thus have
φ (av) = φ (bv) = g and φ (cv) = φ (dv) = g for some g, g ∈ D. By performing a
switching operation at v with g if necessary, we may assume that φ (av) = φ (bv) =
id and φ (cv) = φ (dv) = g g−1. If g g−1 = id, X ∪ Ev becomes a balanced
set with X ∪ Ev > 2V (X ∪ Ev) − 3, contradicting the Dsparsity of G. Thus,
φ (cv) = φ (dv) = id, and X ∪ Ev is cyclic.
In the same manner we also have the following technical lemma.
Lemma 9.14 Let X and Y be obstacles for the 2reduction through (av, vb) and
(cv, vd) and through (av, vc) and (bv, vd), respectively. Suppose that X is type 2
and X ∪ Y is cyclic. Then, X ∪ Y ∪ Ev is cyclic.
Proof Since X is balanced and X ∪ Y is cyclic, for some cyclic subgroup C of D,
there is an equivalent gain function φ to φ such that φ (e) = id for every e ∈ X
and φ (e) ∈ C for every e ∈ Y by Lemma 2.3. Moreover, since X + av · vb and
X + av · vb + cv · vd are balanced, we have φ (av · vb) = φ (cv · vd) = id. As
in the previous proof, by applying a switching operation at v, we may assume that
φ (va) = φ (vb) = id and φ (vc) = φ (vd).
By the definition of the obstacles (whether type 1 or type 2), Y + Y + av · vc or
Y + bv · vd is connected and balanced. Hence φ (av · vc) ∈ C¯ or φ (bv · vd) ∈ C¯,
which implies φ (vc) = φ (vd) ∈ C¯. Thus, every label of X ∪ Y ∪ Ev is included in
¯.
C
The following lemmas describe different relations among obstacles.
Lemma 9.15 Let X and Y be obstacles for the 2reduction through (av, vb) and
(cv, vd) and through (av, vc) and (bv, vd), respectively. If X ∩ Y = ∅, then X ∪ Y
is not a balanced set.
Proof Suppose for a contradiction that X ∪ Y is a balanced set with X ∩ Y = ∅.
(Case 1) If both X and Y are of type 1, X ∪ Y is tight by Lemma 7.2 and hence
X ∪ Y  = 2V (X ∪ Y ) − 3. Without loss of generality, we may assume that a, b, c ∈
V (X ∪ Y ), av · vb ∈ clD(X ) and av · vc ∈ clD(Y ). Since X ∪ Y is balanced,
there is an equivalent gain function φ to φ such that φ (e) = id for e ∈ X ∪ Y .
Moreover, since av · vb ∈ clD(X ) and av · vc ∈ clD(Y ), we have φ (av) = φ (bv) =
φ (cv). This implies that X ∪ Y ∪ {av, bv, cv} is a balanced set. However, since
X ∪ Y ∪ {av, bv, cv} > 2V (X ∪ Y ∪ {av, bv, cv}) − 3, the existence of such a
balanced set contradicts the Dsparsity of G.
(Case 2) Let us consider the case when X is type 2. By definition of obstacles
(whether type 1 or type 2), Y + av · vc or Y + bv · vd is balanced and
2edgeconnected. Without loss of generality, we assume that Y + av · vc is balanced and
2edgeconnected. By Lemma 9.13, there exists an equivalent gain function φ to φ
such that φ (e) = id for e ∈ X ∪ {va, vb} and φ (vc) = φ (vd) = id. Moreover, since
X ∪ Y is balanced, we may assume that φ (e) = id for e ∈ Y . Since φ (av · vc) = id
but φ (e) = id for e ∈ Y , Y + av · vc is unbalanced, a contradiction.
Lemma 9.16 Let X and Y be obstacles for the 2reductions through (av, vb) and
(cv, vd) and through (av, vc) and (bv, vd), respectively. If X  > 1 and Y  > 1,
then X ∩ Y = ∅.
Proof Without loss of generality, we assume a ∈ V (X ) ∩ V (Y ). Recall that each
balanced tight set is 2connected if the size is more than one. By the 4regularity of
G, each vertex of N (v) has degree three in G − v. Hence, if X and Y are type 1 with
X  > 1 and Y  > 1, then X ∩ Y contains an edge incident to a.
If X is type 2, then X + av · vb is balanced tight with a, b, c, d ∈ V (X + av · vb)
by definition. Hence, if Y is type 1, then X ∩ Y contains an edge incident to c or d.
If both X and Y are type 2, then X ∩ Y contains an edge incident to d.
Lemma 9.17 Let X , Y , Z be obstacles for the 2reductions through (av, vb) and
(cv, vd), through (av, vc) and (bv, vd), and through (av, vd) and (bv, vc),
respectively. If there is no hat subgraph containing the star of v, then X ∩ Y = ∅, Y ∩ Z = ∅
or Z ∩ X = ∅ holds.
Proof Note that a type 2 obstacle consists of more than one edge. If two of X, Y and
Z are not singleton sets, then the lemma follows from Lemma 9.16. Hence we may
assume that Y  = Z  = 1, and denote Y = {ey } and Z = {ez }. Clearly, ey = ez .
(Case 1) Let us first consider the case when X is also a singleton set. Let X = {ex }.
Depending on the relative position of ex , ey and ez , we have two situations: (I) ex , ey
and ez share a vertex or (II) ex , ey and ez form a triangle.
In case (I), the star of v is included in a hat subgraph. Indeed, if denoting without
loss of generality ex = ab, ey = ac, and ez = ad, {ex , ey , ez , va, vb, vc, vd} forms
a hat if it is balanced. Since X, Y and Z are obstacles, we have φ (ex ) = φ (av · vb),
φ (ey ) = φ (av · vc) and φ (ez ) = φ (av · vd), and hence this subgraph is indeed
balanced.
In case (II), without loss of generality, we assume ex = ab, ey = bc and ez = ca.
Then {ex , ey , ez , va, vb, vc} forms K4. Since φ (ex ) = φ (av · vb), φ (ey ) = φ (bv · vc)
and φ (ez ) = φ (cv · va), this K4 does not have any unbalanced cycle. Therefore, Case
(II) cannot happen because of the Dsparsity of G, as a balanced K4 is not Dsparse.
(Case 2) Next, we consider the case when X  > 1. We further split the proof into
two subcases depending on whether X is type 1 or type 2.
If X is type 2, then X ∪ Ev = 2V (X ∪ Ev) − 2 by Lemma 9.13. Also, by
Lemma 9.13, there exists an equivalent gain function φ to φ such that φ (e) = id for
e ∈ X ∪ {va, vb} and φ (vc) = φ (vd) = id. Denote φ (vc) by g. Since Y and Z are
obstacles, we have φ (ey ) = φ (ez ) = g, which in particular implies ey , ez ∈/ X . By
N (v) ⊆ V (X ) and ey = ez , X ∪ Y ∪ Z ∪ Ev = 2V (X ∪ Y ∪ Z ∪ Ev), which in turn
implies E = X ∪ Y ∪ Z ∪ Ev. Notice that the label of each edge in X ∪ Y ∪ Z ∪ Ev
is either the identity or g. In other words, X ∪ Y ∪ Z ∪ Ev is cyclic, contradicting the
Dsparsity of G.
The remaining case is when X is type 1. Without loss of generality we assume
a, b ∈ V (X ). By X  > 1 and Lemma 7.1, dX (a) ≥ 2 and dX (b) ≥ 2. Since ey is
either ac or bd and ez is either ad or bc, it suffices to consider the following two cases
by symmetry: (i)(ey , ez ) = (ac, ad), and (ii)(ey , ez ) = (ac, bc).
In subcase (i), X ∩ Y or X ∩ Z contains an edge incident to a as dX (a) ≥ 2 and
dG−v(a) = 3.
In subcase (ii), notice that, {av, bv, cv, ey , ez , av · vb} is a circuit of the underlying
Dsparsity matroid since it forms a balanced K4. By av · vb ∈ clD(X ), we have
cv ∈ clD(X + av + bv + ey + ez ) ⊆ clD(E − cv), contradicting the independence
of E . Therefore, this case does not occur and the proof is complete.
9.3.2 Obstacles Around a Vertex v with N (v) = 3
In this subsection we shall investigate obstacles for a 2reduction at a vertex v with
N (v) = 3. Most of the arguments are similar to the previous subsection. Throughout
Sect. 9.3.2, (G, φ) denotes a Dgain graph satisfying the assumptions of Theorem 9.11,
v denotes a vertex with N (v) = 3, N (v) = {a, b, c}, and Ev denotes the set of edges
incident to v. Without loss of generality, we assume that there are parallel edges e1
and e2 between v and a, and we denote Ev = {e1, e2, vb, vc}.
We again have three possible ways for a 2reduction at v. In each case, there exists
an obstacle if the operation is not admissible. The proof of the following claim is
identical to that of Lemma 9.12 and hence is omitted.
Lemma 9.18 Suppose that the 2reduction through (e1, vb) and (e2, vc) is not
admissible. Then there is an edge subset F ⊆ E \Ev satisfying one of the following
properties:
(i) F is balanced tight with a, b ∈ V (F ) and e1 · vb ∈ clD(F );
(ii) F is balanced tight with a, c ∈ V (F ) and e2 · vc ∈ clD(F );
(iii) F is subtight with a, b, c ∈ V (F ), F + e1 · vb is balanced tight, and e2 · vc ∈
clD(F + e1 · vb).
For the 2reduction through (e1, e2) and (bv, vc), we encounter an even simpler
situation.
Lemma 9.19 Suppose that the 2reduction through (e1, e2) and (bv, vc) is not
admissible. Then there is a balanced tight set F ⊆ E \Ev with b, c ∈ V (F ) and
bv · vc ∈ clD(F ).
Proof Note that e1·e2−1 is a loop. G−v+e1·e2−1 is Dsparse by Lemma 9.2 since adding
an unbalanced loop does not affect (C1). Note that G −v+e1 ·e2−1 +bv·vc is connected.
If G−v+e1·e2−1+bv·vc does not satisfy (C2), then Lemma 9.3 implies that G is almost
nearcyclic, which contradicts our assumption on G. If G − v + e1 · e2−1 + bv · vc
does not satisfy (C1), then G − v + e1 · e2−1 contains a balanced tight set F with
b, c ∈ V (F ) and bv · vc ∈ clD(F ). Since a balanced tight set does not contain a loop
by Lemma 7.1, we have F ⊆ E \Ev.
According to Lemmas 9.18 and 9.19, we can define the type of an obstacle as in the
previous subsection. Lemma 9.19 also says that we only encounter type 1 obstacles for
the 2reduction through (e1, e2) and (bv, vc). The next two lemmas are counterparts
of Lemmas 9.14 and 9.15, respectively, with identical proofs, which are omitted.
Lemma 9.20 Let X and Y be obstacles for distinct 2reductions at v. If X is type 2
and X ∪ Y is cyclic, then X ∪ Y ∪ Ev is cyclic.
Lemma 9.21 Let X and Y be obstacles for distinct 2reductions at v. Then,
if X ∩ Y = ∅, then X ∪ Y is balanced.
To prove the counterpart of Lemma 9.17, we need the following two additional
lemmas.
Lemma 9.22 Suppose that Z is an obstacle of type 1 for the 2reduction through
(e1, e2) and (bv, vc). Then, there is an equivalent gain function φ to φ such that
φ (e) = id for e ∈ Z ∪ {vb, vc}.
Proof Z + bv · vc is balanced. Hence, by Lemma 2.3, there is an equivalent gain
function φ to φ such that φ (e) = id for e ∈ Z + bv · vc. By performing a switching
operation at v with φ (bv) if necessary, we may assume that φ (bv) = φ (vc) = id.
Lemma 9.23 Let X be an obstacle of type 2 for the 2reduction through (e1, vb) and
(e2, vc). Suppose further that there is no obstacle of type 1 for the 2reduction through
(e1, vb) and (e2, vc). Then dX (a) + dX (b) + dX (c) ≥ 5 holds.
Proof Let X = X +e1·vb. By definition, X is balanced tight with a, b, c ∈ V (X ) and
X  > 1. Such a balanced tight set is 2connected and essentially 3edgeconnected
by Lemma 7.1. We thus have dX (u) ≥ 2 for u ∈ {a, b, c}.
Suppose that dX (a) = dX (b) = 2. Since X is essentially 3edgeconnected and
e1 ·vb is incident to a and b, X must be a triangle on a, b, c. This means that X contains
an edge linking from a to c, denoted by e . Recall that X + e2 · vc is balanced by
definition of type 2 obstacles. However, since e and e2 · vc are parallel, for X + e2 · vc
to be balanced, {e , e2 · vc} has to be a balanced 2cycle, that is, {e } is a type 1 obstacle
for the 2reduction through (e1, vb) and (e2, vc), contradicting the assumption of the
lemma.
Therefore, dX (a) ≥ 3 or dX (b) ≥ 3, implying dX (a) + dX (b) + dX (c) ≥ 7.
Since X = X + e1 · vb, we obtain dX (a) + dX (b) + dX (c) ≥ 5.
Lemma 9.24 Let X , Y , Z be obstacles for the 2reductions through (e1, vb) and
(e2, vc), through (e1, vc) and (e2, vb), and through (e1, e2) and (bv, vc), respectively.
Then, X ∩ Y = ∅, Y ∩ Z = ∅, or Z ∩ X = ∅ holds.
ey e1 e2 ex
e1 e2 ex ey
id v id
id
c
b
c
b
c
Proof We split the proof into two cases depending on whether a type 1 obstacle exists
for the 2reduction through (e1, vb) and (e2, vc).
(Case 1) Suppose that there is no type 1 obstacle for the 2reduction through (e1, vb)
and (e2, vc). Then, X is type 2. By Lemma 9.23, dX (a) + dX (b) + dX (c) ≥ 5 holds.
If dX (a) ≥ 2, then X ∩ Y contains an edge incident to a since dG−v(a) = 2 and
dY (a) ≥ 1. If dX (a) = 1, then we have dX (b) ≥ 2 and dX (c) ≥ 2. Since dG−v(b) =
dG−v(c) = 3, Z  = 1 holds if X ∩ Z = ∅. However, in this case, we have dX∪Z (b) =
dX∪Z (c) = 3, and thus X ∩ Y or Y ∩ Z contains an edge incident to b or c.
In a symmetric manner, we are done in the case when a type 1 obstacle does not
exist for the 2reduction through (e1, vc) and (e2, vb).
(Case 2) We now consider the case when both X and Y are type 1. If X  > 1 or
Y  > 1, then X or Y is 2connected, and hence X ∩ Y contains an edge incident
to a as dG−v(a) = 2. We thus assume X  = Y  = 1 and X = Y . Let us denote
X = {ex } and Y = {ey }. Without loss of generality, we assume that ex connects from
a to b. Also, by Lemma 9.22, we may assume φ (e) = id for e ∈ Z ∪ {vb, vc}. Since
e1 · vb ∈ clD(X ), we have φ (ex ) = φ (e1 · vb) = φ (e1). The proof is completed by a
further case analysis: (i) ey connects from a to c or (ii) ey connects from a to b (see
Fig. 16).
In case (i), we have e1 · vc ∈ clD(Y ) by definition. Therefore, φ (ey ) = φ (e1 ·
vc) = φ (e1). Notice that {e1, vb, vc, ex , ey , bv · vc} forms a K4 without unbalanced
cycles by φ (ey ) = φ (e1) = φ (ex ). Moreover, since bv · vc ∈ clD(Z ), we obtain
e1 ∈ clD({vb, vc, ex , ey , bv · vc}) ⊆ clD(E − e1). This contradicts the independence
of E in the underlying Dsparsity matroid.
Let us consider case (ii). If Z  > 1, then X ∩ Z or Y ∩ Z contains an edge incident
to b as Z is type 1 and dZ (b) ≥ 2. Suppose that Z  = 1, X ∩ Y = ∅, X ∩ Z = ∅ and
Y ∩ Z = ∅. Then X ∪ Y ∪ Z ∪ Ev induces a subgraph in which v, a and b have degree
four. So, if V  > 4, then c becomes a cutvertex, contradicting the 2connectivity of
G. On the other hand, if V  = 4, then G becomes the graph shown in Fig. 16(ii’).
In this case removing e2 and ey results in a cyclic graph (where any cycle except
the loop is balanced by φ (e1) = φ (ex )). This means that G is almost nearcyclic, a
contradiction.
9.3.3 Proof of Theorem 9.11
Proof of Theorem 9.11 Suppose that no 2reduction is admissible at v. Then we have
three obstacles X , Y and Z for the three possible 2reductions at v. Suppose further
that the star of v is not contained in a hat subgraph. Then, by Lemmas 9.17 and 9.24,
we may assume without loss of generality that X ∩ Y = ∅ holds.
If X ∪ Y  ≥ 2V (X ∪ Y ) − 1, then V (X ∪ Y ) ∪ {v} = V must hold since G is
essentially 4edgeconnected. We then have X ∪ Y ∪ Ev ≥ 2V  + 1, contradicting
the Dsparsity of G.
Therefore we have
X ∪ Y  ≤ 2V (X ∪ Y ) − 2.
To derive a contradiction, we next show that the number of connected components in
(V (X ) ∩ V (Y ), X ∩ Y ) is equal to two. To see this, let c0 be the number of trivial
connected components (i.e., singleton vertex components) in (V (X ) ∩ V (Y ), X ∩ Y )
and let c1 be the number of nontrivial connected components in it. Then,
X  + Y  ≥ 2V (X ) − 4 + 2V (Y ) − 4 = 2V (X ∪ Y ) + 2V (X ∩ Y ) + 2c0 − 8,
(21)
X ∩ Y  ≤ 2V (X ∩ Y ) − 3c1,
where the last inequality comes from F  ≤ 2V (F )−3 for any nonempty F ⊆ X ∩Y .
From (20–22), we obtain 2c0 + 3c1 ≤ 6. On the other hand by X ∩ Y = ∅ we also
have c1 ≥ 1. Hence we get c1 + c2 ≤ 2, and the number of connected components in
the graph (V (X ) ∩ V (Y ), X ∩ Y ) is at most two.
If the number of connected components in (V (X )∩ V (Y ), X ∩Y ) is one, then, since
X and Y are connected and balanced, Lemma 2.4(1) implies that X ∪ Y is balanced,
which contradicts Lemmas 9.15 and 9.21.
Thus the number of connected components in (V (X ) ∩ V (Y ), X ∩ Y ) is two. Then
2c0 + 3c1 ≥ 5. Hence by (21) and (22) we have
X ∪ Y  ≥ 2V (X ∪ Y ) − 3.
Also by Lemma 2.5 X ∪ Y is cyclic. This implies that X ∪ Y is not tight, as X ∪ Y
cannot be cyclic tight by (20).
If both X and Y are type 1, then X ∪ Y is tight by Lemma 7.2, which does not
happen. Hence X or Y is type 2, and Lemmas 9.14 and 9.20 imply that X ∪ Y ∪ Ev
is also cyclic. Also by (23) and N (v) ⊆ X ∪ Y (as X or Y is type 2) we obtain
X ∪ Y ∪ Ev ≥ 2V (X ∪ Y ∪ Ev) − 1. Thus, due to the essential 4edgeconnecitivity
of G, V (X ∪ Y ∪ Ev) ≥ V  − 1 must hold.
If V (X ∪ Y ∪ Ev) = V , then X ∪ Y ∪ Ev = E  − 1, and hence G is near cyclic,
as X ∪ Y ∪ Ev is cyclic. On the other hand, if V (X ∪ Y ∪ Ev) = V − u for some
u ∈ V , then u is incident to a loop and two nonloop edges by 4regularity. Observe
that removing this loop and one of the two nonloop edges results in a cyclic graph.
This means that G is almost nearcyclic.
In both cases G turns out to be almost nearcyclic, which contradicts the assumption
on G. This completes the proof.
(22)
(23)
9.4 Proof of the Main Theorem
We are now ready to prove Theorem 9.1, which also completes the proof of
Theorem 7.8.
Proof of Theorem 9.1 By Lemmas 9.5, 9.6, 9.8 and 9.9, we may assume that G is
2connected, essentially 4edgeconnected, not special, and not almost nearcyclic.
Also, by Lemma 9.10, we may assume that every vertex v with N (v) = 2 is incident
to a loop.
Since G is not special, G has a vertex v that is not incident to a loop. Then
N (v) ≥ 3. By Theorem 9.11, either the 2reduction at v is admissible or the star of v
is contained in a hat subgraph H . Suppose the latter holds. We denote the vertices of
H by a1, a2, b1, b2, b3, and assume that a1 and a2 have degree four in H (and hence
a1 or a2 is v). Since H is balanced, we may assume that all labels in H are identity.
Moreover, since G is not a fancy hat, we may assume that b1 is not incident to a loop.
We prove that some 2reduction at b1 is admissible. Suppose that no 2reduction is
admissible at b1. Then, by Theorem 9.11, the star of b1 is contained in a hat subgraph
H . Note that H is different from H .
We claim that H contains a triangle on b1, ai , b j for some i ∈ {1, 2} and j ∈ {2, 3}.
To see this first suppose that a1a2 ∈/ E (H ). Then, since each vertex has degree at
least 2 in H , we have a1b2 ∈ E (H ) or a1b3 ∈ E (H ) by NG (a1) = {a2, b1, b2, b3}
and a1a2 ∈/ E (H ). This also implies b1b2 ∈ E (H ) or b1b3 ∈ E (H ), respectively,
as b1 is incident to all the vertices of H . Thus H has a triangle on b1, a1, b j for some
j ∈ {2, 3}.
If a1a2 ∈ E (H ), then H contains a triangle on b1, a1, a2. In a hat subgraph, two
vertices of each triangle have degree four, which implies N (ai ) ⊆ V (H ) for some
i ∈ {1, 2}. Therefore, ai b2 ∈ E (H ) and b1b2 ∈ E (H ), and hence b1b2ai forms a
triangle.
Consequently, without loss of generality, we may assume that H contains a triangle
on b1, b2, a1. Recall that a hat subgraph is balanced. Since φ (a1b1) = φ (a1b2) = id,
we obtain φ (b1b2) = id as H contains a triangle on a1, b1, b2. Observe then that
{a1, a2, b1, b2} induces a K4 in which the label of each edge is the identity. This
contradicts the Dsparsity of G. Consequently, the 2reduction at b1 is admissible.
10 Concluding Remarks
The main results of this paper (Theorems 6.3 and 8.2) give rise to efficient algorithms
for testing generic symmetric rigidity with rotation symmetry or dihedral symmetry
D2k with odd k. This can be done by computing the rank of the quotient graphs in the
corresponding matroids M(g2,3) or MD(G, φ). Here we briefly describe the main
algorithmic ideas and show that testing independece in these matroids can be done in
polynomial time.
Let (G, φ) be a gain graph with G = (V , E ). First consider M(g2,3), in which E
is independent if and only if (i) G is (2, 1)sparse and (ii) every nonempty balanced
subset F ⊆ E is (2, 3)sparse, cf. Lemma 3.1. There exist efficient algorithms for
testing (k, l)sparsity for any pair of integers k, l, see e.g. [2,10], so checking (i) is
easy. Observe that G satisfies (ii) if and only if every minimally non(2, 3)sparse
graph (also called a (2, 3)circuit or an M circuit) is unbalanced. Suppose that G
satisfies (i) and consider one of its M components, i.e. a subgraph H of G induced by
a connected component of the (2, 3)sparsity matroid of G (see [2, 7] for more details
on M components). Each (2, 3)circuit is a subgraph of some M component, so we
may deal with them separately. The key observation is that within H the complements
of the (2, 3)circuits are pairwise edgedisjoint. Since the M components are pairwise
edgedisjoint, this shows that the number of (2, 3)circuits in G is O (n) and they
can easily be enumerated. Then it remains to test whether each of these circuits is
unbalanced, which can be done by switching and using Lemma 2.3 (similar arguments
are given in [1]).
Next consider MD(G, φ), in which E is independent if and only if (i) G is (2,
0)sparse and (ii) every cyclic subset F ⊆ E is (2, 1)sparse, and (iii) every balanced
subset F ⊆ E is (2, 3)sparse. As above, testing (2, 0)sparsity is easy. We can again
observe that G satisfies (ii) if and only if every minimally non(2, 1)sparse graph (a
(2, 1)circuit) is noncyclic. Suppose that G satisfies (i). Then it is easy to see that
these circuits are edgedisjoint, which shows that we have O (n) circuits to check. As
above, they can easily be enumerated, and we can use switching and Lemma 2.3 to see
whether they are all noncyclic. So suppose G satisfies (ii) as well. As above, it remains
to check whether every (2, 3)circuit is unbalanced. Let H be an M component of G.
It is not hard to see that H − e is (2, 1)sparse for all e ∈ E ( H ). Thus, by using the
arguments above, it follows that we have O (n2) circuits to enumerate and test, which
can also be done efficiently by the same techniques.
Acknowledgments This work was supported JSPS GrantinAid for Scientific Research (A) 25240004
and JSPS GrantinAid for Young Scientist (B) 24740058, the Hungarian Scientific Research Fund grant
no. K81472, and the National Development Agency of Hungary, grant no. CK 80124, based on a source
from the Research and Technology Innovation Fund.
1. Berardi, M., Heeringa, B., Malestein, J., Theran, L.: Rigid components in fixedlattice and cone
frameworks. In: Proceedings of CCCG 2011, Toronto, p. 6 (2011)
2. Berg, A.R., Jordán, T.: Algorithms for graph rigidity and scene analysis. In: Proceedings of 11th Annual
European Symposium on Algorithms (ESA). LNCS, vol. 2832, pp. 78–89 (2003)
3. Borcea, C., Streinu, I.: Periodic frameworks and flexibility. Proc. R. Soc. Lond. A 466(2121), 2633–
2649 (2010)
4. Frank, A.: Connections in Combinatorial Optimization. Oxford Lecture Series in Mathematics and Its
Applications. Oxford University Press, Oxford (2011)
5. Gross, J.L., Tucker, T.W.: Topological Graph Theory. Dover, New York (1987)
6. Ikeshita, R., Tanigawa, S.: Count matroids of grouplabeled graphs. arXiv:1507.01259 (2015)
7. Jackson, B., Jordán, T.: Connected rigidity matroids and unique realizations of graphs. J. Comb. Theory
Ser. B 94, 1–29 (2005)
8. Jordán, T., Kaszanitzky, V., Tanigawa, S.: Gainsparsity and symmetryforced rigidity in the plane.
EGRES Technical Report, TR201217 (2012)
9. Kanno, Y., Ohsaki, M., Murota, K., Katoh, N.: Group symmetry in interiorpoint methods for
semidefinite program. Opt. Eng. 2, 293–320 (2001)
10. Lee, A., Streinu, I.: Pebble game algorithms and sparse graphs. Discrete Math. 308(8), 1425–1437
(2008)
11. Malestein, J., Theran, L.: Generic combinatorial rigidity of periodic frameworks. Adv. Math. 233(1),
291–331 (2013)
12. Malestein, J., Theran, L.: Frameworks with forced symmetry II: orientationpreserving crystallographic
groups. Geom. Dedicata 170, 219–262 (2014)
13. Malestein, J., Theran, L.: Frameworks with forced symmetry I: reflections and rotations. Discrete
Comput. Geom. 54, 339–367 (2015)
14. Nixon, A., Schulze, B.: Symmetryforced rigidity of frameworks on surfaces. arXiv:1312.1480 (2013)
15. Nixon, A., Schulze, B., Sljoka, A., Whiteley, W.: Symmetry adapted Assur decompositions. Symmetry
6(3), 516–550 (2014)
16. Owen, J., Power, S.: Frameworks, symmetry and rigidity. Int. J. Comput. Geom. Appl. 20, 723–750
(2010)
17. Ross, E.: Geometric and combinatorial rigidity of periodic frameworks as graphs on the torus. PhD
Thesis, York University, Toronto (2011). http://www.math.yorku.ca/~ejross/RossThesis.pdf
18. Ross, E.: Inductive constructions for frameworks on a twodimensional fixed torus. Discrete Comput.
Geom. 54, 78–109 (2015)
19. Schulze, B.: Combinatorial and geometric rigidity with symmetry constraints. PhD Thesis, York
University (2009). http://www.math.yorku.ca/Who/Faculty/Whiteley/SchulzePhDthesis.pdf
20. Schulze, B.: Symmetric versions of Laman’s theorem. Discrete Comput. Geom. 44(4), 946–972 (2010)
21. Schulze, B.: Symmetry as a sufficient condition for a finite flex. SIAM J. Discrete Math. 24(4), 1291–
1312 (2010)
22. Schulze, B., Tanigawa, S.: Infinitesimal rigidity of symmetric frameworks. SIAM J. Discrete Math.
29(3), 1259–1286 (2015)
23. Schulze, B., Whiteley, W.: The orbit rigidity matrix of a symmetric framework. Discrete Comput.
Geom. 46(3), 561–598 (2011)
24. Tanigawa, S.: Matroids of gain graphs in discrete applied geometry. Trans. Am. Math. Soc. 367(2015),
8597–8641 (2015)
25. Tarnai, T.: Simultaneous static and kinematic indeterminacy of space trusses with cyclic symmetry.
Int. J. Solids Struct. 16, 347–359 (1980)
26. Tay, T.S., Whiteley, W.: Generating isostatic graphs. Struct. Topol. 11, 21–68 (1985)
27. Wegner, F.: Rigidunit modes in tetrahedral crystals. J. Phys. Condens. Matter 19(40), 406218 (2007)
28. Whiteley, W.: Some Matroids from Discrete Applied Geometry. Matroid Theory (Seattle, WA, 1995),
171–311. Contemporary Mathematics. American Mathematical Society, Providence, RI (1996)
29. Whiteley, W.: Counting out to the flexibility of molecules. Phys. Biol. 2, S116–S126 (2005)
30. Wunderlich, W.: Projective invariance of shaky structures. Acta Mech. 42, 171–181 (1982)
31. Zaslavsky, T.: Biased graphs. I. Bias, balance, and gains. J. Comb. Theory Ser. B 47(1), 32–52 (1989)
32. Zaslavsky, T.: Biased graphs. II. The three matroids. J. Comb. Theory Ser. B 51(1), 46–72 (1991)