GainSparsity and SymmetryForced Rigidity in the Plane
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Abstract
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.
Keywords
Infinitesimal rigidity Frameworks Symmetry Rigidity of graphs Rigidity matroids Grouplabeled graphs Frame matroidsMathematics Subject Classification
Primary 52C25 Secondary 05B35 68R101 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 \({{{\mathcal {D}}}}_{2k}\) 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 \({{{\mathcal {D}}}}_{2k}\) 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 \({{{\mathcal {P}}}}\) of E is a family of nonempty subsets of E such that each element of E belongs to exactly one member of \({{{\mathcal {P}}}}\). If \(E=\emptyset \), 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 \(v\in V\), let \(d_G(v)\) be the degree of v in G and \(N_G(v)\) be the set of neighbors of v in G. For \(F\subseteq E\), \(V_G(F)\) 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 \(F\subseteq E\) and \(v\in V(F)\), let \(d_F(v)=d_{G[F]}(v)\).
A vertex subset \(X\subset V(G)\) (resp., an edge subset \(X\subset E(G)\)) 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 \(F\subseteq E\). 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 \(F\subseteq E\) and \(V(F)=V\).
Let \(G=(V,E)\) be a directed graph. A walk in G is a sequence \(W=v_0,e_1,v_1,e_2,v_2,\ldots ,\) \(v_{k1}, e_k,v_k\) of vertices and edges such that \(v_{i1}\) and \(v_{i}\) are the endvertices of \(e_i\) for every \(1\le i\le k\). 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 \({{{\mathcal {D}}}}\) be a dihedral group. For a cyclic subgroup \({{{\mathcal {C}}}}\) of \({{{\mathcal {D}}}}\), \(\bar{{{\mathcal {C}}}}\) denotes the maximal cyclic subgroup containing \({{{\mathcal {C}}}}\).
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 \({{{\mathcal {S}}}}\) be a group. An \({{{\mathcal {S}}}}\)gain graph \((G,\phi )\) is a pair, in which each edge is associated with an element of \({{{\mathcal {S}}}}\) by a gain function \(\phi :E\rightarrow {{{\mathcal {S}}}}\). 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 \(\phi \) 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 \((G,\phi )\). The gain of W is defined as \(\phi (W)=\phi (e_1)\cdot \phi (e_2)\cdots \phi (e_k)\) if each edge is oriented in the forward direction through W, and for a backward edge \(e_i\) we replace \(\phi (e_i)\) with \(\phi (e_i)^{1}\) in the product. Note that \(\phi (W^{1})=\phi (W)^{1}\).
Let \((G,\phi )\) be a gain graph. For \(v\in V(G)\) we denote by \(\pi _1(G,v)\) the set of closed walks starting at v. Similarly, for \(X\subseteq E(G)\) and \(v\in V(G)\), \(\pi _1(X,v)\) denotes the set of closed walks starting at v and using only edges of X, where \(\pi _1(X,v)=\emptyset \) if \(v\notin V(X)\).
Let \(X\subseteq E(G)\). The subgroup induced by X relative to v is defined as \(\langle X\rangle _{\phi ,v}=\{\phi (W)\mid W\in \pi _1(X,v)\}\). The subscript \(\phi \) of \(\langle X\rangle _{\phi ,v}\) is sometimes omitted if it is clear from the context. Note that, for any connected \(X\subseteq E(G)\) and two vertices \(u,v\in V(X)\), \(\langle X\rangle _{\psi ,u}\) is conjugate to \(\langle X\rangle _{\psi ,v}\) (see, e.g., [5, p. 88] for the proof).
2.1 The Switching Operation
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 \((G,\phi )\) be a gain graph. Let \(\phi '\) be the gain function obtained from \(\phi \) by a switching operation. Then, for any \(X\subseteq E(G)\) and \(u\in V(G)\), \(\langle X\rangle _{\phi ',u}\) is conjugate to \(\langle X\rangle _{\phi ,u}\).
Proposition 2.2
Let \((G,\phi )\) be a gain graph. Then, for any forest \(F\subseteq E(G)\), there is a gain function \(\phi '\) equivalent to \(\phi \) such that \(\phi '(e)=\mathrm{id}\) for every \(e\in F\).
2.2 Balanced and Cyclic Sets of Edges
As we shall see, the subgroup \(\langle X\rangle _{\psi ,v}\) itself will not be important, when we define our matroids induced by gains. We only need to know whether \(\langle X\rangle _{\psi ,v}\) is trivial or not, or whether it is cyclic or not. We now introduce notions to describe these properties.
Let \((G,\phi )\) be a gain graph. An edge subset \(F\subseteq E(G)\) is called balanced if \(\langle F\rangle _{\psi ,v}\) is trivial for every \(v\in V(F)\). 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 \(F\subseteq E(G)\) is called cyclic if \(\langle F\rangle _{\psi ,v}\) is cyclic for every \(v\in V(F)\) (note that the terms balanced and cyclic are not exclusive). A gain graph \((G,\phi )\) 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 \(X\subseteq E(G)\), 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 \(\phi (e)=\mathrm{id}\) for all \(e\in T\). Now consider any closed walk \(W\in \pi _1(X,v)\), and denote W by \(W=v_1v_2, v_2v_3, \ldots , v_kv_{k+1}\), and let \(W_i=P_i*\{v_iv_{i+1}\}*P_{i+1}^{1}\) for \(1\le i< k\), where \(P_i\) denotes the path from v to \(v_i\) in T. Then observe \(\phi (W)=\phi (W_1)\cdot \phi (W_2)\cdots \phi (W_k)\). By \(\phi (e)=\mathrm{id}\) for all \(e\in T\), we deduce that \(\phi (W)\) is a product of elements in \(\{\phi (e):e\in X{\setminus } T\}\), implying that \(\langle X\rangle _{\phi ,v}\subseteq \langle \phi (e):e\in X{\setminus } T\rangle \), where \(\langle \phi (e):e\in X{\setminus } T\rangle \) is the group generated by \(\{ \phi (e):e\in X{\setminus } T\}\). Conversely, \(\phi (e)\) is contained in \(\langle X\rangle _{\phi ,v}\) for all \(e\in X{\setminus } T\). Thus, \(\langle X\rangle _{\phi ,v}=\langle \phi (e):e\in X{\setminus } T\rangle \). In particular, we proved the following.
Lemma 2.3
 (i)
X is unbalanced if and only if there is an edge in \(X{\setminus } T\) whose gain is nonidentity.
 (ii)
X is cyclic if and only if all gains of \(X{\setminus } T\) are contained in a cyclic subgroup of \({{{\mathcal {S}}}}\).
The following technical lemmas will be used in the proof of our main theorem.
Lemma 2.4
 (1)
If X and Y are balanced, then \(X\cup Y\) is balanced.
 (2)
If X is balanced and Y is cyclic, then \(X\cup Y\) is cyclic.
 (3)
If X, Y are cyclic and \(X\cap Y\) is unbalanced, then \(X\cup Y\) is cyclic, provided that for every nontrivial cyclic subgroup \({{{\mathcal {C}}}}\) of \({{{\mathcal {S}}}}\) there is a unique maximal cyclic subgroup \(\bar{{{\mathcal {C}}}}\) of \({{{\mathcal {S}}}}\) containing \({{{\mathcal {C}}}}\).
Proof
Since the graph \((V(X)\cap V(Y), X\cap Y)\) is connected, there is a spanning tree T in \(G[X\cup Y]\) such that \(T\cap X\) is a spanning tree of G[X], \(T\cap Y\) is a spanning tree of G[Y], and \(T\cap X\cap Y\) is a spanning tree of \(G[X\cap Y]\). By Proposition 2.2, there is a gain function \(\phi '\) equivalent to \(\phi \) such that \(\phi '(e)=\mathrm{id}\) for each \(e\in T\).
If X and Y are balanced, Lemma 2.3 implies that \(\phi '(e)=\mathrm{id}\) for all \(e\in X\cup Y\). Thus (1) holds.
If X is balanced, then every label in \(X\cup Y\) is contained in \(\langle Y\rangle _{\phi ',v}\) by Lemma 2.3, and hence \(X\cup Y\) is cyclic if Y is cyclic. This implies (2).
If X, Y are cyclic and \(X\cap Y\) is unbalanced, then there is an edge \(e\in X\cap Y\) for which \(\phi '(e)\) is nonidentity. Let \({{{\mathcal {C}}}}\) be a cyclic subgroup of \({{{\mathcal {S}}}}\) generated by \(\phi '(e)\) and \(\bar{{{\mathcal {C}}}}\) be the maximal cyclic subgroup containing \({{{\mathcal {C}}}}\). Since X and Y are cyclic, Lemma 2.3 implies that \(\phi '(e)\in \bar{{{\mathcal {C}}}}\) holds for every \(e\in X\) and for every \(e\in Y\). Therefore \(X\cup Y\) is cyclic. \(\square \)
Lemma 2.5
Let \((G,\phi )\) be a gain graph, and X and Y be connected balanced edge subsets. If the number of connected components of the graph \((V(X)\cap V(Y),X\cap Y)\) is two, then \(X\cup Y\) is cyclic.
Proof
We take a spanning tree T of \(G[X\cup Y]\) such that \(T\cap X\) is a spanning tree of G[X]. Since the number of connected components of \((V(X)\cap V(Y),X\cap Y)\) is two, \(T\cap Y\) consists of two connected components, denoted \(T_1\) and \(T_2\). \(\{V(T_1), V(T_2)\}\) partitions Y into three subsets \(\{Y_1,Y_2,Y_3\}\) such that \(Y_i=\{e\in Y:V(\{e\})\subseteq V(T_i)\}\) for \(i=1,2\) and \(Y_3=Y{\setminus } (Y_1\cup Y_2)\).
By Proposition 2.2, we can take a gain function \(\phi '\) equivalent to \(\phi \) such that \(\phi '(e)=\mathrm{id}\) for \(e\in T\). Since X and Y are balanced, we have \(\phi '(e)=\mathrm{id}\) for \(e\in X\cup Y_1\cup Y_2\). Moreover, assuming that every edge in \(Y_3\) is oriented toward \(V(Y_1)\), we have \(\phi '(e)=\phi '(f)\) for all \(e,f\in Y_3\), since otherwise \(T_1\cup T_2\cup \{e,f\}\) contains an unbalanced cycle, contradicting the fact that Y is balanced. Therefore \(X\cup Y\) is cyclic. \(\square \)
3 Gain Count Matroids
3.1 Matroids Induced by Submodular Functions
Let E be a finite set. A function \(\mu :2^E\rightarrow {\mathbb {R}}\) is called submodular if \(\mu (X)+\mu (Y)\ge \mu (X\cup Y)+\mu (X\cap Y)\) for every \(X,Y\subseteq E\). \(\mu \) is monotone if \(\mu (X)\le \mu (Y)\) for any \(X\subseteq Y\). A monotone submodular function \(\mu :2^E\rightarrow {\mathbb {Z}}\) induces a matroid on E, where \(F\subseteq E\) is independent if and only if \(I\le \mu (I)\) for every nonempty \(I\subseteq F\). See e.g. [4, Sect. 13.4]. This matroid is denoted by \({{{\mathcal {M}}}}(\mu )\).
For a monotone submodular function \(\mu \), let \(\nu =\mu 1\). Then, \(\nu \) is monotone submodular and induces the matroid \({{{\mathcal {M}}}}(\nu )\). This matroid is referred to as the Dilworth truncation of \({{{\mathcal {M}}}}(\mu )\). 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 \(\mu _1\) and \(\mu _2\). Since monotonicity and submodularity are both preserved under the sum operation, \(\mu _1+\mu _2\) is monotone and submodular. In general, the union of \({{{\mathcal {M}}}}(\mu _1)\) and \({{{\mathcal {M}}}}(\mu _2)\) is not equal to \({{{\mathcal {M}}}}(\mu _1+\mu _2)\). We do have equality in some special cases, for example, when \(\mu _1=\mu _2\) or when both \(\mu _1\) and \(\mu _2\) 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 \(f_{2,2}\) defined by \(f_{2,2}(F)=2V(F)2\) on \(2^E\), as \(f_{2,2}/2\) induces the graphic matroid on G. The 2dimensional generic rigidity matroid is the one induced by \(f_{2,2}1\), and hence it is the Dilworth truncation of the union of two copies of the graphic matroid.
3.2 GainCount Matroids
The independence of \({{{\mathcal {M}}}}(g_{k,l})\) can be described in a compact form (see [8] for the proof, which is a rather straightforward calculation).
Lemma 3.1
Let \((G,\phi )\) be an \({{{\mathcal {S}}}}\)gain graph with \(G=(V,E)\). Then E is independent in \({{{\mathcal {M}}}}(g_{k,l})\) if and only if \(F\le kV(F)l+k\alpha (F)\) for any nonempty \(F\subseteq E\).
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

\(F\le kV(F)l\) for every nonempty balanced \(F\subseteq X\);

\(F\le kV(F)l+k\) for every nonempty unbalanced \(F\subseteq X\),
If E is (k, l)gsparse then graph \((G,\phi )\) is said to be (k, l)gsparse, and \((G,\phi )\) is called maximum (k, l)gtight if it is (k, l)gsparse with \(E=kVl+k\).
Remark 3.1
Note that the value of \(g_{k,l}\) 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 \((G,\phi _1)\) and \((G,\phi _2)\) 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 \({{{\mathcal {S}}}}\)gain 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 \((G,\phi )\), followed by a single Dilworth truncation, E(G) is independent if and only if \(F\le 2V(F)3\) for any balanced set \(F\subseteq E(G)\) and \(F\le 2V(F)2\) for any nonempty \(F\subseteq E(G)\). 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 \((G,\phi )\) be an \({{{\mathcal {S}}}}\)gain graph. The 0extension adds a new vertex v and two new nonloop edges \(e_{1}\) and \(e_{2}\) 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 \(e_{1}\) and \(e_{2}\) are not parallel then their labels can be arbitrary. Otherwise the labels are assigned such that \(\phi (e_{1})\ne \phi (e_{2})\), assuming that \(e_1\) and \(e_2\) 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 \(e_{1},e_{2}\) such that the tail of \(e_1\) is the tail of e and the tail of \(e_2\) is the head of e. The labels of the new edges are assigned such that \(\phi (e_{1})\cdot \phi (e_{2})^{1}=\phi (e)\). The 1extension also adds a third edge \(e_{3}\) oriented to v. The label of \(e_3\) is assigned so that it is locally unbalanced, i.e., every twocycle \(e_ie_j\), if it exists, is unbalanced.
The 0extension and the 1extension were already considered by Ross [17] for \({\mathbb {Z}}^2\)gain 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 \((G,\phi )\) be a (2,3)gsparse graph. Applying the 0extension, 1extension or loop 1extension to \((G,\phi )\) results in a (2,3)gsparse graph \((G',\phi ')\) with \(V(G')=V(G)+1\) and \(E(G')=E(G)+2\).
Proof
For a contradiction, suppose that \(G'\) contains an edge set \(F\subseteq E(G')\) for which \(F>2V(F)3+2\alpha (F)\). Let v be the new vertex added by the extension, and let \(E_v\) be the set of edges incident to v. Since \(E(G'){\setminus } E_v\subseteq E(G)\), \(E_v\cap F\ne \emptyset \). In particular, \(v\in V(F)\). Also, since the new labeling is assigned to be locally unbalanced, F is not contained in \(E_v\).
If \(G'\) is constructed by a 1extension then let e be the subdivided edge of G and let \(e_1\) and \(e_2\) be the resulting two new edges.
Let \(F'=F{\setminus } E_v\). If \(G'\) is constructed by a 1extension and \(\{e_1,e_2\}\subseteq F\), then we further insert e to \(F'\). We then have \(F'\ge F2\), \(V(F')=V(F)1\), and \(\alpha (F')\le \alpha (F)\) in each case. These imply \(F'\ge F2>2V(F)5+2\alpha (F)\ge 2V(F')3+2\alpha (F')\), contradicting the (2, 3)gsparsity of G as \(\emptyset \ne F'\subseteq E(G)\). \(\square \)
We shall define the inverse moves of the operations above, which are called reductions. For a vertex v and two incoming nonloop edges \(e_1=(u,v)\) and \(e_2=(w,v)\), we denote by \(e_1\cdot e_2^{1}\) a new edge from u to w with label \(\phi (e_1)\cdot \phi (e_2)^{1}\) (by extending \(\phi \)). If \(u=w\) then \(e_1\cdot e_2^{1}\) is a loop. Each reduction corresponds to one of the following operations on a gain graph \((G,\phi )\).
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 \(e_1,e_2,e_3\) be the edges incident to v. Without loss of generality we may assume that each \(e_i\) is oriented to v. The 1reduction deletes v with the incident edges and adds one of \(e_1\cdot e_2^{1}\), \(e_2\cdot e_3^{1}\) and \(e_3\cdot e_1^{1}\) as a new edge.
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 \((G,\phi )\) be a (2,3)gsparse graph and \(v\in V(G)\) a vertex not incident to a loop with \(d(v)=3\). Then there is an admissible 1reduction at v.
Proof
Let \(E=E(G)\), \(G'=Gv\) and \(E'=E(G')\). Let \(e_1,e_2,e_3\) be the edges incident to v in G. Without loss of generality we may assume that each \(e_i\) is oriented to v. For simplicity we put \(e_{i,j}=e_i\cdot e_j^{1}\).
Suppose for a contradiction that there is no admissible splitting at v, that is, none of \(E'+e_{1,2}\), \(E'+e_{2,3}\) and \(E'+e_{3,1}\) is independent in \({\mathcal {M}}(g_{2,3})\). Equivalently, \(e_{1,2},e_{2,3},e_{3,1}\in \mathrm {cl}_{g}(E')\), where \(\mathrm {cl}_{g}\) denotes the closure operator of \({{{\mathcal {M}}}}(g_{2,3})\). Let \(X=\{e_1,e_2,e_3,e_{1,2},e_{2,3},e_{3,1}\}\).
Claim 4.3
\(e_1\in \mathrm{cl}_g(Xe_1)\).
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 \(\phi (e_1)=\phi (e_2)=\phi (e_3)=\mathrm {id}\). We then have \(\phi (e_{1,2})=\phi (e_{2,3})=\phi (e_{3,1})=\mathrm {id}\). Therefore X forms a balanced \(K_4\), which is a circuit of \({{{\mathcal {M}}}}(g_{2,3})\). Thus, \(e_1\in \mathrm{cl}_g(Xe_1)\) holds.
If \(N(v)=2\) then we may assume that \(e_1\) and \(e_2\) are parallel. By Proposition 2.2, we may assume that \(\phi (e_2)=\phi (e_3)=\mathrm{id}\). This implies \(\phi (e_{1,3})=\phi (e_1)\) and \(\phi (e_{2,3})=\mathrm{id}\). Since G is (2, 3)gsparse, we have \(\phi (e_1)\ne \mathrm{id}\) by \(\phi (e_2)=\phi (e_3)=\mathrm{id}\), which implies that \(e_{1,2}\) is an unbalanced loop with \(\phi (e_{1,2})=\phi (e_1)\). It can be easily checked, by counting, that X is indeed a circuit in \({{{\mathcal {M}}}}(g_{2,3})\). Thus, \(e_1\in \mathrm{cl}_g(Xe_1)\) holds.
If \(N(v)=1\) then let \(X'=\{e_1,e_2,e_3,e_{1,2}\}\). We have \(X'=2V(X')\) and \(X'\) is a circuit of \({{{\mathcal {M}}}}(g_{2,3})\). Therefore \(e_1\in \mathrm{cl}_g(X'e_1)\subset \mathrm{cl}_g(Xe_1)\). \(\square \)
Since \(e_{1,2},e_{2,3},e_{3,1}\in \mathrm {cl}_g(E')\), by Claim 4.3, we have \(e_1\in \mathrm {cl}_g(Xe_1)\subseteq \mathrm {cl}_g(E'+Xe_1)=\mathrm {cl}_g(E'+e_2+e_3)=\mathrm {cl}_g(Ee_1)\), which contradicts the (2, 3)gsparsity of G. \(\square \)
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 \({{{\mathcal {S}}}}\)gain graph \((G,\phi )\) is maximum (2,3)gtight if and only if it can be built up from an \({{{\mathcal {S}}}}\)gain 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 \({\mathbb {Z}}^2\)gain 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 \({{{\mathcal {S}}}}\)symmetric graphs, whose automorphism group has a subgroup isomorphic to \({{{\mathcal {S}}}}\). 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 \({{{\mathcal {S}}}}\)Symmetric Graphs
Let H be a simple graph. An automorphism of H is a permutation \(\pi :V(H)\rightarrow V(H)\) such that \(\{u,v\}\in E(H)\) if and only if \(\{\pi (u),\pi (v)\}\in E(H)\). The set of all automorphisms of H forms a subgroup of the symmetric group of V(H), known as the automorphism group \(\mathrm{Aut}(H)\) of H.
Let \({{{\mathcal {S}}}}\) be a group. An action of \({{{\mathcal {S}}}}\) on H is a group homomorphism \(\rho :{{{\mathcal {S}}}}\rightarrow \mathrm{Aut}(H)\). An action \(\rho \) is called free if \(\rho (g)(v)\ne v\) for any \(v\in V\) and any nonidentity \(g\in {{{\mathcal {S}}}}\). We say that a graph H is \(({{{\mathcal {S}}}},\rho )\)symmetric if \({{{\mathcal {S}}}}\) acts on H by \(\rho \). If \(\rho \) is clear from the context, we will simply denote \(\rho (g)(v)\) by \(g\cdot v\) or gv. Note that, for \(g\in {{{\mathcal {S}}}}\) and \(u,v\in V\), \(\{u,v\}\in E(H)\) if and only if \(\{gu,gv\}\in E(H)\), and hence there is an induced action of \({{{\mathcal {S}}}}\) on E(H) defined by \(g\cdot \{u,v\}=\{gu,gv\}\).
Different graphs may have the same quotient graph. However, if we assume that \(\rho \) 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 \({{{\mathcal {S}}}}v=\{gv:g\in {{{\mathcal {S}}}}\}\). If \(\rho \) is a free action, an edge orbit connecting \({{{\mathcal {S}}}}u\) and \({{{\mathcal {S}}}}v\) in \(H/{{{\mathcal {S}}}}\) can be written by \(\{\{gu,ghv\}:g\in {{{\mathcal {S}}}}\}\) for a unique \(h\in {{{\mathcal {S}}}}\). We then orient the edge orbit from \({{{\mathcal {S}}}}u\) to \({{{\mathcal {S}}}}v\) in \(H/{{{\mathcal {S}}}}\) and assign to it the gain h. In this way, we obtain the quotient \({{{\mathcal {S}}}}\)gain graph, denoted \((H/{{{\mathcal {S}}}},\phi )\).
Conversely, any \({{{\mathcal {S}}}}\)gain graph \((G,\phi )\) can be “lifted” as an \(({{{\mathcal {S}}}},\rho )\)symmetric graph with a free action \(\rho \). To see this, we simply denote the pair (g, v) of \(g\in {{{\mathcal {S}}}}\) and \(v\in V(G)\) by gv. The covering graph (also known as the derived graph) of \((G,\phi )\) is the simple graph with vertex set \({{{\mathcal {S}}}}\times V(G)=\{gv:g\in {{{\mathcal {S}}}}, v\in V(G)\}\) and the edge set \(\{\{gu,g\phi (e)v\}:e=(u,v)\in E(G), g\in {{{\mathcal {S}}}}\}\). Clearly, \({{{\mathcal {S}}}}\) freely acts on the covering graph, under which the quotient gain graph comes back to \((G,\phi )\). 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 \(p:V(H)\rightarrow {\mathbb {R}}^d\), called a jointconfiguration. We denote the set \(\{p(v):v\in V(H)\}\) of points by p(H).
Example 5.1
5.3 Symmetric Frameworks and SymmetryForced Infinitesimal Rigidity
A discrete point group (or simply a point group) is a finite discrete subgroup of \({{{\mathcal {O}}}}({\mathbb {R}}^d)\), the orthogonal group of dimension d, i.e., the set of \(d\times d\) orthogonal matrices over \({\mathbb {R}}\). For \(d=2\), point groups are classified into two classes, groups \({{{\mathcal {C}}}}_k\) of kfold rotations and dihedral groups \({{{\mathcal {D}}}}_{2k}\) of order 2k. For a special case, \({{{\mathcal {D}}}}_2\) consists of a mirrorreflection and the identity. In the subsequent discussion of this section, \({{{\mathcal {S}}}}\) denotes a point group.
Example 5.2
A result of Schulze [21] motivates us to look at \({{{\mathcal {S}}}}\)symmetric infinitesimal rigidity, which states that if (H, p) is not symmetryforced infinitesimally rigid on an \({{{\mathcal {S}}}}\)generic p, then (H, p) has a nontrivial continuous motion that preserves the \(({{{\mathcal {S}}}},\rho )\)symmetry.
5.4 The Orbit Rigidity Matrix
Let (H, p) be an \(({{{\mathcal {S}}}},\rho )\)symmetric framework in \({\mathbb {R}}^d\). 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 \(E(H)/{{{\mathcal {S}}}}\) linear equations.
To see this, we first define a jointconfiguration \({\tilde{p}}\) of vertex orbits by \({\tilde{p}}:V(H)/{{{\mathcal {S}}}}\rightarrow {\mathbb {R}}^d\). By taking a representative vertex v from each vertex orbit \({{{\mathcal {S}}}}v\), we set \({\tilde{p}}({{{\mathcal {S}}}}v)=p(v)\) [then, the locations of the other nonrepresentative vertices are uniquely determined by (8)].
In a similar way, we define an infinitesimal motion of \((H/{{{\mathcal {S}}}},{\tilde{p}})\) by \({\tilde{m}}:V(H)/{{{\mathcal {S}}}}\rightarrow {\mathbb {R}}^d\). By using the representative vertices determined above, we fix a onetoone correspondence between \({{{\mathcal {S}}}}\)symmetric infinitesimal motions of V(H) and infinitesimal motions of \(V(H)/{{{\mathcal {S}}}}\) by \({\tilde{m}}({{{\mathcal {S}}}}v)=m(v)\) for each vertex orbit \({{{\mathcal {S}}}}v\).
Theorem 5.1
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
Proof
Let \(R_F\) be the linear space spanned by the row vectors associated with F in \(O(G,\phi ,p)\). Observe that each nonzero entry of the row vector associated with \(e\in F\) is in the columns associated with V(F). This means that \(R_F\) is the direct sum of \(R_{F'}\) for \(F'\in C(F)\), and hence it suffices to check the statement for a connected F with \(V(F)=V\).
Clearly, \(\dim R_F\le dV\). Since \(F\le \dim R_F\), we now show that \(\dim R_F^{\bot }\ge \dim \mathrm{iso}_{\langle F\rangle _{\phi ,w}}(p(F))\), where \(R_F^{\bot }\) denotes the orthogonal complement of \(R_F\).
Let T be a spanning tree of G. Since we can freely apply switching operations, we may assume that \(\phi (e)=\mathrm{id}\) for all \(e\in T\). Then, by Lemma 2.3, \(\langle F\rangle _{\phi ,w}=\langle \phi (e):e\in F{\setminus } T\rangle \) for a vertex \(w\in V(F)\).
This, together with Theorem 5.1, directly implies a necessary condition for symmetric frameworks to be symmetryforced infinitesimally rigid.
Recall that \({{{\mathcal {S}}}}\) is a finite family of orthogonal matrices. Let \({\mathbb {Q}}_{{{\mathcal {S}}}}\) be the field generated by \({\mathbb {Q}}\) and the entries of all the matrices in \({{{\mathcal {S}}}}\). Since \({{{\mathcal {S}}}}\) is finite, almost all numbers in \({\mathbb {R}}\) are transcendental over \({\mathbb {Q}}_{{{\mathcal {S}}}}\). For a given gain graph \((G,\phi )\), a mapping \({\tilde{p}}:V(G)\rightarrow {\mathbb {R}}^d\) is called \({{{\mathcal {S}}}}\)generic if the set of coordinates of \({\tilde{p}}(v)\) for all \(v\in V(G)\) is algebraically independent over \({\mathbb {Q}}_{{{\mathcal {S}}}}\). Similarly, for a given \(({{{\mathcal {S}}}},\rho )\)symmetric graph H, an \(({{{\mathcal {S}}}},\rho )\)symmetric jointconfiguration \(p:V(H)\rightarrow {\mathbb {R}}^d\) is called \({{{\mathcal {S}}}}\)generic if the corresponding jointconfiguration \({\tilde{p}}\) of the vertex orbits is \({{{\mathcal {S}}}}\)generic. An \({{{\mathcal {S}}}}\)symmetric framework is called \({{{\mathcal {S}}}}\)generic if the joint configuration is \({{{\mathcal {S}}}}\)generic.
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 \({{{\mathcal {D}}}}_{2k}\) 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 \({{{\mathcal {S}}}}\)generic 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 \((G,\phi )\) be an \({{{\mathcal {S}}}}\)gain graph for a point group \({{{\mathcal {S}}}}\subset {{{\mathcal {O}}}}({\mathbb {R}}^2)\). Let \((G',\phi ')\) be an \({{{\mathcal {S}}}}\)gain graph obtained from \((G,\phi )\) by a 0extension, 1extension, or loop1extension. If there is a mapping \(p:V(G)\rightarrow {\mathbb {R}}^2\) such that \(O(G,\phi ,p)\) is row independent, then there is a mapping \(p':V(G')\rightarrow {\mathbb {R}}^2\) such that \(O(G',\phi ',p')\) is row independent.
Proof
If there is a p such that \(O(G,\phi ,p)\) is row independent, then \(O(G,\phi ,q)\) is row independent for all \({{{\mathcal {S}}}}\)generic q. Hence, we may assume that p is \({{{\mathcal {S}}}}\)generic. We only show the difficult case where \((G',\phi ')\) is constructed from \((G,\phi )\) by a 1extension (see [8] for the easier case where \((G',\phi ')\) is constructed from \((G,\phi )\) by a 0extension or a loop1extension).
Suppose that \((G',\phi ')\) is obtained from \((G,\phi )\) by a 1extension, by removing an existing edge e and adding a new vertex v with three new nonloop edges \(e_1, e_2, e_3\) incident to v. We may assume that \(e_i\) is outgoing from v. Let \(u_i\) be the other endvertex of \(e_i\), and let \(g_i=\phi '(e_i)\) and \(p_i=p(u_i)\) for \(i=1,2,3\). By the definition of 1extension, we have \(\phi (e)=g_1^{1}g_2\).
Claim 6.2
The three points \(g_ip_i \ (i=1,2,3)\) do not lie on a line.
Proof
Since p is \({{{\mathcal {S}}}}\)generic, \(u_1=u_2=u_3\) should hold if they lie on a line. Then \(p_1=p_2=p_3\). By the definition of 1extensions, \(g_i\ne g_j\) if \(u_i=u_j\). This implies that \(g_1p_1, g_2p_2, g_3p_3\) are three distinct points on a circle. Thus, they do not lie on a line. \(\square \)
We are now ready to prove a combinatorial characterization. The same statement was also proved in [12] for rotation groups and in [13] for reflection group by completely different proofs.
Theorem 6.3
Let \({{{\mathcal {C}}}}\subset {{{\mathcal {O}}}}({\mathbb {R}}^2)\) be a cyclic point group, that is, either a group of kfold rotations or a group of a reflection, and let (H, p) be a generic \(({{{\mathcal {C}}}},\rho )\)symmetric framework in the plane with a free action \(\rho \). Then (H, p) is symmetryforced infinitesimally rigid if and only if the quotient \({{{\mathcal {C}}}}\)gain graph contains a spanning maximum (2, 3)gtight subgraph.
Proof
By Theorem 5.1 it suffices to show that for the quotient \({{{\mathcal {C}}}}\)gain graph \((H/{{{\mathcal {C}}}},\phi )\) and any \({{{\mathcal {C}}}}\)generic \({\tilde{p}}:V(H/{{{\mathcal {C}}}})\rightarrow {\mathbb {R}}^2\), \(O(H/{{{\mathcal {C}}}},\phi ,{\tilde{p}})\) is row independent if and only if \((H/{{{\mathcal {C}}}},\phi )\) is (2, 3)gsparse. Let us simply denote \(G=H/{{{\mathcal {C}}}}\).
(“If part”) It suffices to consider the case when G is maximum (2, 3)gtight. The proof is done by induction on V(G). For \(V(G)=1\), G consists of single vertex with an unbalanced loop. Then \(O(G,\phi ,{\tilde{p}})\) consists of a nonzero row, which implies that \(O(G,\phi ,{\tilde{p}})\) is rowindependent.
For \(V(G)>1\), by Theorem 4.4, G can be built up from a \({{{\mathcal {C}}}}\)gain graph with one vertex and an unbalanced loop with a sequence of 0extensions, 1extensions, and loop1extensions. Thus, there is a maximum (2, 3)gtight graph \((G',\phi ')\) from which \((G,\phi )\) is constructed by a 0extension, 1extension, or loop1extension. By induction, there is a \(p'\) such that \(O(G,\phi ',p')\) is row independent. Thus, Lemma 6.1 implies that there is a p such that \(O(G,\phi ,p)\) is row independent, which in turn implies that \(O(G,\phi ,q)\) is row independent for all \({{{\mathcal {C}}}}\)generic q.
7 Constructive Characterization of Maximum \({{{\mathcal {D}}}}\)Tight Graphs
In the previous sections we gave a constructive characterization of (2, 3)gsparse graphs and their realizations as symmetryforced rigid frameworks in the plane with cyclic point group symmetry. We next move to noncyclic point groups, that is, dihedral groups of order 2k that we denote by \({{{\mathcal {D}}}}_{2k}\) (or simply by \({{{\mathcal {D}}}}\)). The corresponding matroid, that we construct in the next subsection, is slightly different from the (2, 3)gcount matroid, as we need to take into account the fact that the underlying group is not cyclic.
7.1 \({{{\mathcal {D}}}}\)Sparsity
Definition 7.1
Let \((G,\phi )\) be a \({{{\mathcal {D}}}}\)gain graph. An edge set \(X\subseteq E(G)\) is called \({{{\mathcal {D}}}}\)sparse if \(F\le f_{{{\mathcal {D}}}}(F)\) for any nonempty \(F\subseteq X\), and it is called \({{{\mathcal {D}}}}\)tight if it is \({{{\mathcal {D}}}}\)sparse with \(X=f_{{{\mathcal {D}}}}(X)\).
If E(G) is \({{{\mathcal {D}}}}\)sparse then also \((G,\phi )\) is said to be \({{{\mathcal {D}}}}\)sparse, and \((G,\phi )\) is called maximum \({{{\mathcal {D}}}}\)tight if it is \({{{\mathcal {D}}}}\)sparse with \(E(G)=2V(G)\).
By a simple degree of freedom counting argument based on Example 5.2 and Lemma 5.2, it is not difficult to see that \({{{\mathcal {D}}}}\)sparsity is a necessary condition for orbit rigidity matrices to be row independent in the case of dihedral symmetry (a formal proof will be given in Lemma 8.1). To prove the sufficiency, the first question is whether \({{{\mathcal {D}}}}\)sparsity defines a collection of independent sets of a matroid. This will be proved in this subsection.
We will use the following technical lemmas on properties of \({{{\mathcal {D}}}}\)tight sets.
Lemma 7.1
 (i)
If F is cyclic, then F is connected.
 (ii)
If F is balanced with \(F> 1\), then F has neither parallel edges nor loops and is 2connected and essentially 3edgeconnected.
Proof
Since G is \({{{\mathcal {D}}}}\)sparse and \(\beta \) is monotone nondecreasing, we have \(F\le \sum _{F'\in C(F)}f_{{{\mathcal {D}}}}(F')\le 2V(F)(3\beta (F))c\), where c denotes the number of connected components in F. Hence, if F is not connected and \(\beta (F)<3\), then \(F<2V(F)3+\beta (F)\), implying that F is not \({{{\mathcal {D}}}}\)tight. Therefore if \(\beta (F)<3\) then F is connected.
Suppose further that F is balanced. Then we have \(\beta (X)=0\) for any \(X\subseteq F\). This means that \(X\le f_{2,3}(X)\) for any nonempty \(X\subseteq F\), and \(F=f_{{{\mathcal {D}}}}(F)=2V(F)3=f_{2,3}(F)\). In other words, F is independent in the generic 2rigidity matroid \({{{\mathcal {M}}}}(f_{2,3})\) of G[F]. It is known that, in the generic 2rigidity matroid, an independent set F with \(F=f_{2,3}(F)\) and \(F>1\) has neither parallel edges nor a loop and is 2connected and essentially 3edgeconnected (see e.g. [7]). \(\square \)
Lemma 7.2
Let \((G,\phi )\) be a \({{\mathcal {D}}}\)sparse graph with \(G=(V,E)\). Let \(X,Y\subseteq E\) be \({{{\mathcal {D}}}}\)tight edge sets with \(X\cap Y\ne \emptyset \). Then \(X\cup Y\) is \({{{\mathcal {D}}}}\)tight.
Proof
Without loss of generality, assume \(\beta (X)\ge \beta (Y)\).
Let \(d=2V(X\cup Y)X\cup Y\). Note that \(X\cup Y\) is \({{{\mathcal {D}}}}\)tight if one of the following holds: (i) \(d=0\), (ii) \(d\le 1\) and \(X\cup Y\) is cyclic, or (iii) \(d\le 3\) and \(X\cup Y\) is balanced.
If \(\beta (X)=3\), then (15) implies that \(d=0\) and hence \(X\cup Y\) is \({{{\mathcal {D}}}}\)tight.
Therefore we assume \(\beta (X)<3\). Then X and Y are connected by Lemma 7.1. We split the proof into two cases depending on the value of \(\beta (X)\).
(Case 2) If \(\beta (X)=0\), then \(\beta (Y)=0\) and we have \(d\le 62c_03c_1\) by (15). By \(c_1\ge 1\), we have three possible pairs \((c_0,c_1)=(0,1), (1,1), (0,2)\). If \((c_0,c_1)=(0,1)\), then \(d\le 3\) and Lemma 2.4 implies that \(X\cup Y\) is balanced. Thus, \(X\cup Y\) is a balanced \({{{\mathcal {D}}}}\)tight set. If \((c_0,c_1)=(1,1)\) or \((c_0,c_1)=(0,2)\), then \(d\le 1\) and Lemma 2.5 implies that \(X\cup Y\) is cyclic. Thus, \(X\cup Y\) is a cyclic \({{{\mathcal {D}}}}\)tight set.
This completes the proof. \(\square \)
Lemma 7.3
Let \((G,\phi )\) be a \({{{\mathcal {D}}}}\)gain graph with \(G=(V,E)\) and X and Y be \({{{\mathcal {D}}}}\)tight sets with \(X\subseteq Y\subseteq E\). For \(e\in E{\setminus } Y\), if \(f_{{{\mathcal {D}}}}(X)=f_{{{\mathcal {D}}}}(X+e)\), then \(f_{{{\mathcal {D}}}}(Y)=f_{{{\mathcal {D}}}}(Y+e)\).
Proof
Since \(f_{{{\mathcal {D}}}}(X)=f_{{{\mathcal {D}}}}(X+e)\), the endvertices of e are contained in V(X), implying \(V(Y+e)=V(Y)\). If X or Y is not cyclic, then we have \(\beta (Y)=\beta (Y+e)=3\), meaning that \(f_{{{\mathcal {D}}}}(Y)=f_{{{\mathcal {D}}}}(Y+e)\).
We hence assume that X and Y are cyclic, and they are connected by Lemma 7.1. Take a spanning tree T in G[Y] such that \(X\cap T\) is a spanning tree of G[X]. By Proposition 2.2, there is an equivalent gain function \(\phi '\) to \(\phi \) such that \(\phi '(f)=\mathrm{id}\) for \(f\in T\). By Lemma 2.3, there is a cyclic subgroup \({{{\mathcal {C}}}}\) of \({{{\mathcal {D}}}}\) such that \(\phi '(f)\in {{{\mathcal {C}}}}\) for every \(f\in Y\), where \({{{\mathcal {C}}}}\) is the trivial group if Y is balanced. Since \(f_{{{\mathcal {D}}}}(X)=f_{{{\mathcal {D}}}}(X+e)\) and \(X\subseteq Y\), we have \(\phi '(e)\in \bar{{{\mathcal {C}}}}\), and hence \(f_{{{\mathcal {D}}}}(Y)=f_{{{\mathcal {D}}}}(Y+e)\) holds. \(\square \)
We are ready to prove that the family of \({{{\mathcal {D}}}}\)sparse edge subsets is a family of independent sets of a matroid on groundset E. We shall also characterize the rank function of this matroid.
Theorem 7.4
Proof
For a partition \({{{\mathcal {P}}}}=\{E_1',\ldots , E_t'\}\) of \(E'\subseteq E\), we denote \(\mathrm{val}({{{\mathcal {P}}}})=\sum _{i=1}^tf_{{{\mathcal {D}}}}(E_t')\). We shall check the following independence axiom of matroids: (I1) \(\emptyset \in {{{\mathcal {I}}}}\); (I2) for any \(X, Y\subseteq E\) with \(X\subseteq Y\), \(Y\in {{{\mathcal {I}}}}\) implies \(X\in {{{\mathcal {I}}}}\); (I3) for any \(E'\subseteq E\), maximal subsets of \(E'\) belonging to \({{{\mathcal {I}}}}\) have the same cardinality.
\({{{\mathcal {I}}}}\) obviously satisfies (I1) and (I2). To see (I3), let \(E'\subseteq E\) and let \(F\subseteq E'\) be a maximal subset of \(E'\) in \({\mathcal {I}}\). Since \(F\in {\mathcal {I}}\) we have \(F\le \mathrm{val}({\mathcal {P}})\) for all partitions \({\mathcal {P}}\) of \(E'\). We shall prove that there is a partition \({\mathcal {P}}\) of \(E'\) with \(F=\mathrm{val}({\mathcal {P}})\), from which (I3) follows.
Let \(J=(V,F)\) denote the subgraph with the edge set F. Consider the family \(\{F_{1},F_{2},\ldots ,F_{t}\}\) of all maximal \({{{\mathcal {D}}}}\)tight sets in J. Since each edge \(f\in F\) forms a \({{{\mathcal {D}}}}\)tight set, \(\bigcup _{i=1}^{t}F_i=F\) holds. Since \(F_{i}\cap F_{j}=\emptyset \) for every pair \(1\le i<j\le t\) by Lemma 7.2 and the maximality, \({\mathcal {P}}_F=\{F_{1},F_{2},\ldots ,F_{t}\}\) is a partition of F and \(F=\mathrm{val}({\mathcal {P}}_F)\) follows.
Based on \({{{\mathcal {P}}}}_F\), we construct a partition \({\mathcal {P}}\) of \(E'\) with \(\mathrm{val}({\mathcal {P}})=\mathrm{val}({\mathcal {P}}_F)=F\). Consider an edge \((u,v)=e\in E'F\). Since F is a maximal subset of \(E'\) in \({\mathcal {I}}\) we have \(F+e\not \in {\mathcal {I}}\). Hence there must be a \({{\mathcal {D}}}\)tight set \(X_e\) in J with \(u,v\in V(X_e)\) and \(X_e+e\not \in {\mathcal {I}}\). \(X_e\subseteq F_{i}\) for some \(1\le i\le t\). Choose such an \(F_i\) for every \(e\in E'F\) and define \(E_i=F_i\cup \{e:F_i \hbox { was chosen for }e\}\). Clearly \({\mathcal {P}}=\{E_{1},E_{2},\ldots ,E_{t}\}\) is a partition of \(E'\). By Lemma 7.3, \(f_{{\mathcal {D}}}(F_i)=f_{{\mathcal {D}}}(E_i)\) for every \(1\le i\le t\) and hence \(\mathrm{val}({\mathcal {P}})=\mathrm{val}({\mathcal {P}}_F)=F\). \(\square \)
The matroid which was introduced and denoted by \({{{\mathcal {M}}}}_{{{\mathcal {D}}}}(G,\phi )\) in Theorem 7.4 is called the \({{{\mathcal {D}}}}\)sparsity matroid of \((G,\phi )\).
7.2 Constructive Characterization of Maximum \({{{\mathcal {D}}}}\)Tight Graphs
We now present a constructive characterization of maximum \({{{\mathcal {D}}}}\)tight graphs. Notice that the average vertex degree in a maximum \({{{\mathcal {D}}}}\)tight graph \((G,\phi )\) is four, which means that G has a vertex of degree at most 3 if and only if G is not 4regular. Thus we shall take special care with 4regular \({{{\mathcal {D}}}}\)sparse graphs.
7.2.1 0Extension, 1Extension, and Loop1Extension
Before looking at 4regular graphs and vertices of degree four, we consider the 0extension, 1extension, and loop1extension operations. Recall that the corresponding inverse operations are called reductions. A reduction is admissible if the resulting graph is \({{{\mathcal {D}}}}\)sparse.
Lemma 7.5
Let \((G,\phi )\) be a \({{{\mathcal {D}}}}\)sparse graph with \(G=(V,E)\). Applying a 0extension, 1extension or loop1extension to \((G,\phi )\) results in a \({{{\mathcal {D}}}}\)sparse graph with \(V+1\) vertices and \(E+2\) edges.
Conversely, for any vertex v of degree 2 or 3, the 0reduction, loop1reduction, or some of the 1reductions at v are admissible if \(V\ge 2\).
Proof
The proof of the first claim is exactly the same as the proof of Lemma 4.1 (indeed, we just need to change \(2\alpha (F)\) with \(\beta (F)\) in the proof of Lemma 4.1).
To see that some reduction is admissible at a vertex v of degree three, we just need to observe that each circuit of \({{{\mathcal {M}}}}(g_{2,3})\) appearing in the proof of Claim 4.3 is also a circuit in \({{{\mathcal {M}}}}_{{{\mathcal {D}}}}(G,\phi )\). We can thus apply exactly the same proof as in Lemma 4.2 to conclude that some reduction is admissible at v. \(\square \)
7.2.2 2Extension and Loop2Extension
Besides 0extensions, 1extensions and loop1extensions, we shall introduce 2extensions and loop2extensions for constructing 4regular \({{{\mathcal {D}}}}\)sparse graphs.
The following lemma shows that these operations preserve \({{{\mathcal {D}}}}\)sparsity.
Lemma 7.6
Let \((G,\phi )\) be a \({{{\mathcal {D}}}}\)sparse graph. Then, any \({{{\mathcal {D}}}}\)gain graph \((G',\phi ')\) obtained from G by a 2extension or a loop2extension is \({{{\mathcal {D}}}}\)sparse.
Proof
Suppose that \((G',\phi ')\) is obtained by a 2extension. Let us denote the removed edges by e and f and the new edges by \(e_1,\ldots , e_4\) as above. Suppose that there is \(F\subseteq E(G')\) that violates the \({{{\mathcal {D}}}}\)sparsity condition. Let \(F'=F{\setminus } \{e_1,\ldots , e_4\}\). Since \(\{e_1,\ldots , e_4\}\) satisfies the \({{{\mathcal {D}}}}\)sparsity condition, \(F'\ne \emptyset \). Let us add e to \(F'\) if \(\{e_1,e_2\}\subseteq F\) and add f to \(F'\) if \(\{e_3,e_4\}\subseteq F\). Observe that \(F'\ge F2\), \(V(F)\ge V(F')+1\) and \(\beta (F)\ge \beta (F')\). Since \(F>f_{{{\mathcal {D}}}}(F)\), we obtain \(F'\ge F2>f_{{{\mathcal {D}}}}(F)2=2V(F)3+\beta (F)2\ge 2V(F')3+\beta (F')=f_{{{\mathcal {D}}}}(F')\). This contradicts the \({{{\mathcal {D}}}}\)sparsity of G since \(\emptyset \ne F'\subseteq E(G)\). Therefore \((G',\phi ')\) is \({{{\mathcal {D}}}}\)sparse.
In the same manner, it can be easily checked that a loop2extension also preserves \({{{\mathcal {D}}}}\)sparsity. \(\square \)
We shall define the inverse moves of these operations. Recall that, for a vertex v and two incoming nonloop edges \(e_1=(u,v)\) and \(e_2=(w,v)\), we denote by \(e_1\cdot e_2^{1}\) a new edge from u to w with gain \(\phi (e_1)\cdot \phi (e_2)^{1}\).
Let v be a vertex of degree four, not incident to a loop, and \(e_i=(v_i,v)\) for \(i=1,\ldots , 4\) be the edges incident to v, assuming that all of them are oriented to v. The 2reduction (at v) deletes v and adds one of \(\{e_1\cdot e_2^{1}, e_3\cdot e_4^{1}\}\), \(\{e_1\cdot e_3^{1}, e_2\cdot e_4^{1}\}\) and \(\{e_1\cdot e_4^{1}, e_2\cdot e_3^{1}\}\). We sometimes refer to a specific one: the 2reduction at v through \((e_i, e_j)\) and \((e_k, e_l)\) deletes v and adds \(\{e_i\cdot e_j^{1}, e_k\cdot e_l^{1}\}\).
Let v be a vertex of degree four, incident to a loop l, and \(e_i=(v_i,v)\) for \(i=1,2\) be the nonloop edges incident to v, assuming that all of them are oriented to v. The loop2reduction (at v) deletes v and adds \(e_1\cdot e_2^{1}\).
A 2reduction or a loop2reduction is said to be admissible if the resulting graph is \({{{\mathcal {D}}}}\)sparse.
7.2.3 Base Graphs
Our main theorem asserts that these operations are sufficient to construct all 4regular \({{{\mathcal {D}}}}\)sparse graphs from certain classes of \({{{\mathcal {D}}}}\)sparse graphs. Here, the classes can be categorized into three groups: the first group includes special small graphs as in the conventional constructive characterizations, the second group is a class of graphs, which are obtained from cycles by duplicating each edge, and the third one consists of nearcyclic 4regular graphs.
The first group consists of three types of special \({{{\mathcal {D}}}}\)tight graphs, called trivial graphs, fancy triangles, and fancy hats. A trivial graph is a \({{{\mathcal {D}}}}\)sparse graph with a single vertex and with two loops as shown in Fig. 6a. The gain function is assigned so that the gains of two loops generate a noncyclic group.
A fancy triangle is a \({{{\mathcal {D}}}}\)gain graph whose underlying graph is obtained from a triangle by adding a loop to each vertex, as shown in Fig. 6b. The gain function is assigned so that it is \({{{\mathcal {D}}}}\)sparse and the triangle is balanced.
The third group consists of nearcyclic graphs, which, intuitively speaking, are the \({{{\mathcal {D}}}}\)tight graphs closest to (2, 3)gtight graphs. More precisely, we say that a \({{{\mathcal {D}}}}\)sparse graph \((G,\phi )\) is nearcyclic if there is an edge e such that \((Ge,\phi )\) is cyclic.
The following lemma shows how to construct nearcyclic graphs.
Lemma 7.7
Let \((G,\phi )\) be a (2, 3)gsparse \({{{\mathcal {D}}}}\)gain graph with \(G=(V,E)\), and suppose that there is a cyclic subgroup \({{{\mathcal {C}}}}\) of \({{{\mathcal {D}}}}\) such that \(\phi (e)\in {{{\mathcal {C}}}}\) for all \(e\in E\). If we add a new edge e having a gain in \({{{\mathcal {D}}}}{\setminus } \bar{{{\mathcal {C}}}}\), then \((G+e,\phi )\) is \({{{\mathcal {D}}}}\)sparse.
Proof
Suppose that \((G+e,\phi )\) is not \({{{\mathcal {D}}}}\)sparse. Then there is a subset \(F\subseteq E\) such that \(F+e>f_{{{\mathcal {D}}}}(F+e)\). Since F is cyclic, either (i) \(\beta (F)=0\) or (ii) \(\beta (F)=2\).
By Lemma 7.1(i), F is connected, and clearly the endvertices of e are contained in V(F). Moreover, every cycle in \(F+e\) that passes through e has a gain not contained in \(\bar{{{\mathcal {C}}}}\), as the gain of e is not in \(\bar{{{\mathcal {C}}}}\). Thus, \(F+e\) contains a cycle whose gain is not contained in \(\bar{{{\mathcal {C}}}}\). This means that (i) \(\beta (F)=0\) implies \(\beta (F+e)=1\), and (ii) \(\beta (F)=2\) implies \(\beta (F+e)=3\). In each case, we obtain \(\beta (F+e)\beta (F)\ge 1\). Therefore, \(F=F+e1>f_{{{\mathcal {D}}}}(F+e)1 \ge f_{{{\mathcal {D}}}}(F)\), and this contradicts the \({{{\mathcal {D}}}}\)sparsity of \((G,\phi )\) as \(\emptyset \ne F\subseteq E\). \(\square \)
7.2.4 Constructive Characterizations
We are ready to state our constructive characterization of 4regular \({{{\mathcal {D}}}}\)sparse graphs. We say that a 4regular \({{{\mathcal {D}}}}\)sparse graph is a base graph if it is a trivial graph, a fancy triangle, a fancy hat, or a nearcyclic graph.
Theorem 7.8
Let \((G,\phi )\) be a \({{{\mathcal {D}}}}\)gain graph. Then, \((G,\phi )\) is 4regular and \({{{\mathcal {D}}}}\)sparse if and only if it can be built up from a disjoint union of base graphs and \({{{\mathcal {D}}}}\)sparse double cycles by a sequence of 2extension and loop2extension operations.
We have proved that these operations preserve \({{{\mathcal {D}}}}\)sparsity in Lemma 7.6. The proof of the converse direction will be given in Sect. 9, where we will show that a 2reduction or a loop2reduction is admissible at some vertex if the graph is neither a base graph nor a double cycle.
Combining Theorem 7.8 and Lemma 7.5, we obtain the following:
Theorem 7.9
Let \((G,\phi )\) be a \({{{\mathcal {D}}}}_{2k}\)gain graph with \(k\ge 2\). Then, \((G,\phi )\) is maximum \({{{\mathcal {D}}}}_{2k}\)tight if and only if it can be built up from a disjoint union of base graphs and \({{{\mathcal {D}}}}_{2k}\)sparse double cycles by a sequence of 0extension, 1extension, loop1extension, 2extension and loop2extension operations.
The theorems can be strengthened if k is odd, in which case every \({{{\mathcal {D}}}}_{2k}\)sparse double cycle can be reduced to a trivial graph. To see this, let us prove the following technical lemma.
Lemma 7.10

\(g_1, g_2\) and \(g_3\) are distinct nonidentity elements,

\(\{g_1,g_2,g_3\}\) generates a noncyclic group, and

\(g_4=\mathrm{id}\).
Proof
Since \(\{g_1,g_2,g_3\}\) generates a noncyclic group, we may assume that \(g_1\) is a reflection r along a line. Suppose that \(\{g_1,g_2g_3^{1}\}\) is cyclic. Then, \(g_2g_3^{1}=\mathrm{id}\) or \(g_2g_3^{1}=r\). Since \(g_2\ne g_3\), we have \(g_2=rg_3\).
If \(g_3\) is also a reflection \(r'\), which is different from r, then consider \(\{g_1g_2^{1},g_3\}=\{rr'r^{1}, r'\}\). Clearly \(rr'r^{1}\ne \mathrm{id}\). If \(rr'r^{1}=r'\) or equivalently \((rr')^{2}=\mathrm{id}\) then k has to be even which is a contradiction. Thus \(\{g_1g_2^{1},g_3\}\) generates a noncyclic group.
If \(g_3\) denotes a rotation C, then consider \(\{g_1g_3^{1},g_2\}=\{rC^{1}, rC\}\). Since \(rC^{1}\) and rC are nonidentity and reflections, if they generate a cyclic group, then \(rC^{1}=rC\), implying \(C^{2}=\mathrm{id}\). This contradicts the parity of k. \(\square \)
Lemma 7.11
Let \({{{\mathcal {D}}}}_{2k}\) be a dihedral group with odd \(k\ge 3\), and \((G,\phi )\) be a \({{{\mathcal {D}}}}_{2k}\)sparse double cycle \(C_n^2\) with \(n\ge 2\). Then, a 2reduction is admissible at some vertex.
Proof
Let v be a vertex, and we denote the edges incident to v by \(e_i\) for \(i=1,\ldots , 4\). Without loss of generality, we assume that all of \(e_i\) are oriented to v, \(e_1\) and \(e_2\) are parallel, and \(e_3\) and \(e_4\) are parallel.
We now consider the remaining two possible 2reductions at v. In each reduction, the resulting underlying graph is \(C_{n1}^2\), and it can be easily checked that the 2reduction is admissible if one of the resulting \({{{\mathcal {D}}}}_{2k}\)gain graphs \((G_1,\phi _1)\) and \((G_2,\phi _2)\) is not cyclic.
To see that \((G_1,\phi _1)\) or \((G_2,\phi _2)\) is not cyclic, let \(g_i=\phi (e_i)\) for \(i=1,\ldots , 4\). Observe that \(\{g_1,\ldots , g_4\}\) satisfies the condition of Lemma 7.10. Since \(\{g_1\cdot g_2^{1}, g_3\cdot g_4^{1}\}\) generates a cyclic group, this implies, by Lemma 7.10, that \(\{g_1\cdot g_3^{1}, g_2\cdot g_4^{1}\}\) or \(\{g_1\cdot g_4^{1}, g_2\cdot g_3^{1}\}\) is not cyclic, implying that \((G_1,\phi _1)\) or \((G_2,\phi _2)\) is not cyclic. \(\square \)
Combining Theorem 7.9 and Lemma 7.11, we obtain the following constructive characterization.
Theorem 7.12
Let \({{{\mathcal {D}}}}_{2k}\) be a dihedral group with odd \(k\ge 3\). Then a \({{{\mathcal {D}}}}_{2k}\)gain graph \((G,\phi )\) is maximum \({{{\mathcal {D}}}}_{2k}\)tight if and only if it can be built up from a disjoint union of base graphs by a sequence of 0extension, 1extension, loop1extension, 2extension and loop2extension operations.
8 Combinatorial Characterization of Generic Rigidity with Dihedral Symmetry
In this section we discuss our combinatorial characterization of symmetryforced infinitesimal rigidity with dihedral symmetry. We begin with a necessary condition based on Lemma 5.2.
Lemma 8.1
Let \({{{\mathcal {D}}}}_{2k}\) be a dihedral group with \(k\ge 2\), and (H, p) be a generic \(({{{\mathcal {D}}}}_{2k},\rho )\)symmetric framework with a free action \(\rho \). If (H, p) is symmetryforced infinitesimally rigid, then the quotient gain graph contains a spanning maximum \({{{\mathcal {D}}}}_{2k}\)tight subgraph.
Proof
Let \((H/{{{\mathcal {D}}}}_{2k},\phi )\) be the quotient gain graph of H and \({\tilde{p}}\) be a joint configuration of the vertex orbits \(V(H/{{{\mathcal {D}}}}_{2k})\) corresponding to p. By Theorem 5.1, it suffices to prove that if \(O(H/{{{\mathcal {D}}}}_{2k},\phi ,{\tilde{p}})\) is row independent, then \((H/{{{\mathcal {D}}}}_{2k},\phi )\) is \({{{\mathcal {D}}}}_{2k}\)sparse.
In Sect. 8.1 we shall prove that \({{{\mathcal {D}}}}_{2k}\)sparsity is also sufficient for row independence when \(k\ge 3\) is odd. On the other hand, in Sect. 8.2 we give a family of examples showing that this implication does not always hold when k is even.
8.1 Combinatorial Characterization of SymmetryForced Rigidity with \({{{\mathcal {D}}}}_{2k}\)Symmetry for Odd k
Our goal of this subsection is to prove the following characterization of symmetryforced infinitesimal rigidity.
Theorem 8.2
Let \({{{\mathcal {D}}}}_{2k}\) be a dihedral group with odd \(k\ge 3\), and (H, p) be a generic \(({{{\mathcal {D}}}}_{2k},\rho )\)symmetric framework with a free action \(\rho \). Then (H, p) is symmetryforced infinitesimally rigid if and only if the quotient gain graph contains a spanning maximum \({{{\mathcal {D}}}}_{2k}\)tight subgraph.
Necessity follows from Lemma 8.1. Therefore, by Theorem 5.1, it suffices to prove that, for a maximum \({{{\mathcal {D}}}}_{2k}\)tight graph \((G,\phi )\), there is a mapping \(p:V(G)\rightarrow {\mathbb {R}}^2\) such that \(O(G,\phi ,p)\) is row independent. The proof of this claim is based on the constructive characterization of maximum \({{{\mathcal {D}}}}_{2k}\)tight graphs formulated in Sect. 7.
By Theorem 7.12, \((G,\phi )\) can be constructed from a disjoint union of base graphs by 0extension, 1extension, loop1extension, 2extension, and loop2extension operations. Therefore, what we have to prove is that (i) the orbit rigidity matrix of each base graph is row independent and (ii) each extension preserves the row independence of the orbit rigidity matrix by extending p appropriately. (i) will be solved in Lemma 8.4 whereas (ii) will be solved in Lemmas 8.5 and 8.7. Note that there is no parity condition for k in these lemmas.
In the rest of this section, we identify \({{{\mathcal {D}}}}_{2k}\) with the symmetry group of a regular kgon, which consists of kfold rotations around the origin and reflections along (fixed) lines. For a line L through the origin, we denote by \(L^{\bot }\) the orthogonal complement of L, that is, the line orthogonal to L and through the origin. We first note an elementary fact from geometry.
Lemma 8.3

If g is the reflection along a line L, then \((I_2g)p\in L^{\bot }{\setminus } \{0\}\) for any \(p\in {\mathbb {R}}^2{\setminus } L\).

If g is a rotation, then \((2I_2gg^{1})p\in \mathrm{span}\{p\}{\setminus } \{0\}\) for any \(p\in {\mathbb {R}}^2{\setminus } \{0\}\).
Lemma 8.4
Let \((G,\phi )\) be a base graph. Then, there is a mapping \(p:V(G)\rightarrow {\mathbb {R}}^2\) such that \(O(G,\phi ,p)\) is row independent.
Proof
(Case 1) Suppose that \((G,\phi )\) is a trivial graph. Let v be the vertex. Take \(p:V(G)\rightarrow {\mathbb {R}}^2\) such that p(v) does not lie on reflection lines L in \({{{\mathcal {D}}}}_{2k}\) and their orthogonal complements \(L^{\bot }\). Then, \(O(G,\phi ,p)\) consists of two row vectors, which are linearly independent by Lemma 8.3.
(Case 2) Suppose that \((G,\phi )\) is a fancy triangle or a fancy hat. Since these two gain graphs are small, one can check the statement by choosing p randomly and computing the rank of \(O(G,\phi ,p)\) (see [8] for finding such p deterministically).
(Case 3) Suppose that \((G,\phi )\) is nearcyclic. Then there is an edge e such that \(Ge\) is cyclic. Let \({{{\mathcal {C}}}}=\langle Ee\rangle _v\) for a vertex \(v\in V(G)\), and denote \(g_e=\phi (e)\). We may assume that the labels of the edges in \(Ee\) are all contained in \({{{\mathcal {C}}}}\). Then \(g_e\notin \bar{{{\mathcal {C}}}}\).
By Theorem 6.3, \(O(Ge,\phi ,p)\) is row independent for any \({{{\mathcal {D}}}}_{2k}\)generic joint configuration \(p:V(G)\rightarrow {\mathbb {R}}^2\), and the kernel space of \(O(Ge,\phi ,p)\) is onedimensional.
The next two lemmas show that loop2extensions and 2extensions preserve the independence.
Lemma 8.5
Let \((G,\phi )\) be a maximum \({{{\mathcal {D}}}}_{2k}\)tight graph with \(k\ge 2\) and \((G',\phi ')\) a maximum \({{{\mathcal {D}}}}_{2k}\)tight graph obtained from \((G,\phi )\) by a loop2extension. If there is a mapping \(p:V(G)\rightarrow {\mathbb {R}}^2\) such that \(O(G,\phi ,p)\) is row independent, then there is a mapping \(p':V(G')\rightarrow {\mathbb {R}}^2\) such that \(O(G',\phi ',p')\) is row independent.
Proof
We may assume that p is \({{{\mathcal {D}}}}_{2k}\)generic. Suppose that \(G'\) is obtained from G by a loop2extension, by removing an existing edge e, adding a new vertex v with new nonloop edges \(e_1\) and \(e_2\) and a new loop l incident to v (see Fig. 5). We may assume that \(e_1\) and \(e_2\) are outgoing from v. Let \(u_i\) be the other endvertex of \(e_i\) and let \(g_i=\phi '(e_i)\) for \(i=1,2\). By the definition of loop2extension, \(\phi (e)=g_1^{1}g_2\). Also, denote \(h=\phi '(l)\).
Claim 8.6
 (1)
\(u_1=u_2\) and h is the reflection along the line orthogonal to L with \(h=g_2g_1^{1}\), or
 (2)
\(u_1=u_2\), h is a rotation, and \(g_2g_1^{1}\) is the twofold rotation.
Proof
We split the proof into two cases.
Suppose that h is the reflection along some line R through the origin. By Lemma 8.3, \((2I_2hh^{1})q\) is orthogonal to R. This means that, if \(\{(2I_2hh^{1})q, g_1p_1g_2p_2\}\) is dependent, L is orthogonal to R. Since p is \({{{\mathcal {D}}}}_{2k}\)generic, L cannot be orthogonal to reflection line R if \(p_1\ne p_2\). Thus, \(p_1=p_2\) (and hence \(u_1=u_2\)), and \(h=g_2g_1^{1}\) as p is \({{{\mathcal {D}}}}_{2k}\)generic.
Suppose that h is a rotation. By Lemma 8.3, \((2I_2hh^{1})q\) is a scalar multiple of q. Hence, if \(\{(2I_2hh^{1})q, g_1p_1g_2p_2\}\) is dependent for any \(q\in L{\setminus } \{g_1p_1,g_2p_2\}\), L passes through the origin. Since p is \({{{\mathcal {D}}}}_{2k}\)generic, L passes through the origin if and only if \(p_1=p_2\) (and hence \(u_1=u_2\)) and \(g_2p_1\) is the antipodal point of \(g_1p_1\). Observe that \(g_2p_1\) is the antipodal point of \(g_1p_1\) if and only if \(g_2g_1^{1}\) is the twofold rotation as p is \({{{\mathcal {D}}}}_{2k}\)generic. \(\square \)
By Claim 8.6 we may focus on cases (1) and (2) of Claim 8.6.
Case (1). Suppose that \(u_1=u_2\) and h is the reflection along R with \(h=g_2g_1^{1}\), where R is the line orthogonal to L and through the origin. Note that \(\phi (e)=g_1^{1}g_2\) is a reflection since \(g_2g_1^{1}\) is a reflection (a conjugate of a reflection is also a reflection).
To avoid the situation where \(p'(v)=0\), we continuously perturb \(p'(v)\) in the direction of d. To see the perturbation more precisely, for each \(t\in {\mathbb {R}}\), let us define \(p_t':V\cup \{v\}\rightarrow {\mathbb {R}}^2\) by \(p_t'(v)=td\) and \(p_t'(u)=p'(u)\) for each \(u\in V\). Then, observe that for all \(t\in {\mathbb {R}}{\setminus } \{0\}\) the row of l in \(O(G',\phi ',p_t')\) is a nonzero scalar multiple of that of l in \({\bar{O}}(G',\phi ',p')\) by Lemma 8.3. Therefore, \(\mathrm{rank} \ O(G',\phi ',p_t')=\mathrm{rank}\ {\bar{O}}(G',\phi ',p_t')\) for all \(t\in {\mathbb {R}}{\setminus }\{0\}\). Since \({\bar{O}}(G',\phi ',p_0')={\bar{O}}(G',\phi ',p')\) and the latter matrix is row independent, it follows that \({\bar{O}}(G',\phi ',p_t')\) is row independent for almost all t. This in turn implies that \(O(G',\phi ',p_t')\) is row independent for almost all \(t\in {\mathbb {R}}{\setminus } \{0\}\).
This completes the proof of the lemma. \(\square \)
Lemma 8.7
Let \((G,\phi )\) be a maximum \({{{\mathcal {D}}}}_{2k}\)tight graph with \(k\ge 2\) and \((G',\phi ')\) a maximum \({{{\mathcal {D}}}}_{2k}\)tight graph obtained from \((G,\phi )\) by a 2extension. If there is a mapping \(p:V(G)\rightarrow {\mathbb {R}}^2\) such that \(O(G,\phi ,p)\) is row independent, then there is a mapping \(p':V(G')\rightarrow {\mathbb {R}}^2\) such that \(O(G',\phi ',p')\) is row independent.
Proof
We may assume that p is \({{{\mathcal {D}}}}_{2k}\)generic. Suppose that \(G'\) is obtained from G by a 2extension, by removing two existing edges e and f and adding a new vertex v with new nonloop edges \(e_1, e_2,e_3,e_4\) incident to v (see Fig. 4). We may assume that \(e_i\) is outgoing from v, and \(e=e_1^{1}\cdot e_2\) and \(f=e_3^{1}\cdot e_4\). Let \(u_i\) be the other endvertex of \(e_i\) and let \(g_i=\phi '(e_i)\). We then have \(\phi (e)=g_1^{1}g_2\) and \(\phi (f)=g_3^{1} g_4\).
Let \(p_i=p(u_i)\) for \(i=1,\ldots , 4\), L be the line through \(g_1p_1\) and \(g_2p_2\), and \(L'\) be the line through \(g_3p_3\) and \(g_4p_4\). We have the following elementary geometric observation.
Claim 8.8
(i) No three points among \(\{g_ip_i:i=1,\ldots , 4\}\) are collinear.

\(L\ne L'\),

\(u_1=u_2\) and \(u_3=u_4\), and

\(g_2g_1^{1}\) is the reflection along \(L^{\bot }\) with \(g_2g_1^{1}=g_4g_3^{1}\).
Proof
The first claim follows from the proof of Claim 6.2.
For the second claim, suppose that L and \(L'\) are parallel. Without loss of generality, we have the following four cases: (i) \(p_1\notin \{ p_2,p_3,p_4\}\), (ii) \(p_1=p_2=p_3=p_4\), (iii) \(p_1=p_2\ne p_3=p_4\), and (iv) \(p_1=p_3\ne p_2=p_4\).
In case (i), the \({{{\mathcal {D}}}}_{2k}\)genericity of p implies that \(g_1p_1\) has no relation to the other three points, and hence L and \(L'\) intersect at a point.
In case (ii), \(g_1p_1,\ldots , g_4p_4\) lie on a circle C. Moreover, since \(u_1=u_2=u_3=u_4\), e and f are loops attached to a vertex (i.e., the 2extension is type (g) of Fig. 4). This implies that the group generated by \(\{g_1^{1}g_2, g_3^{1}g_4\}\) is not cyclic by the \({{{\mathcal {D}}}}_{2k}\)sparsity of G.
Now, L is the line through \(g_1p_1\) and \(g_2p_1\) while \(L'\) is the line through \(g_3p_1\) and \(g_4p_1\). We have two subcases depending on whether \(g_2g_1^{1}\) is a reflection or a rotation.
(ii1) If \(g_2g_1^{1}\) is a reflection, then it is the reflection along the bisector \(L^{\bot }\) of \(g_1p_1\) and \(g_2p_1\). If L and \(L'\) are parallel, then this reflection also sends \(g_3p_3\) to \(g_4p_4\). This means that \(g_2g_1^{1}\) is the reflection along \(L^{\bot }\) with \(g_2g_1^{1}=g_4g_3^{1}\), which implies the statement.
(ii2) If \(g_2g_1^{1}\) is a rotation, \(g_1p_1\) and \(g_2p_1\) are vertices of a regular kgon inscribing C. Since p is generic, if \(L'\) is parallel to L, \(g_3p_1\) and \(g_4p_1\) are also vertices of this regular kgon, and hence \(g_4g_3^{1}\) is also a rotation. Since a conjugate of a rotation is also a rotation, we deduce that \(g_1^{1}g_2\) and \(g_3^{1}g_4\) are rotations as well. This however contradicts the fact that \(\langle g_1^{1}g_2,g_3^{1}g_4\rangle \) is not cyclic.
In case (iii), L is the line through \(g_1p_1\) and \(g_2p_1\) while \(L'\) is the line through \(g_3p_3\) and \(g_4p_3\). Observe that, if \(g_2g_1^{1}\) is a rotation, the line L can have any slope, by moving \(p_1\). Therefore, if L and \(L'\) are parallel for generic p, then \(g_2g_1^{1}\) and \(g_4g_3^{1}\) are both reflections. When \(g_2g_1^{1}\) is a reflection, it is the reflection along the bisector \(L^{\bot }\) of \(g_1p_1\) and \(g_2p_1\). As \(g_4g_3^{1}\) is a reflection and L and \(L'\) are parallel, \(g_4g_3^{1}\) is the reflection along \(L^{\bot }\), which implies the statement.
Let C be the circle whose center is the origin and which passes through \(g_1p_1\) (and hence through \(g_3p_1\)). We split the proof into two cases depending on whether \(g_3g_1^{1}\) is the twofold rotation \(C_{\pi }\) or not.
(iv1) Suppose that \(g_3g_1^{1}\ne C_{\pi }\). Let \(C'\) be a circle whose center is the origin and the diameter is much larger than that of C. We shall relocate \(g_2p_2\) on \(C'\) such that \(g_2p_2\) is on the line through \(g_1p_1\) 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 \(g_4p_2\) (as shown in Fig. 10a). Since the diameter of \(C'\) can be arbitrarily large, \({{{\mathcal {D}}}}_{2k}\) has no element that sends \(g_2p_2\) to q or \(q'\). In other words, if p is generic, L and \(L'\) are not parallel.
(iv2) Suppose that \(g_3g_1^{1}=C_{\pi }\). Then \(g_3p_1\) is the antipodal point of \(g_1p_1\) 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 \(g_2p_2\) on \(C'\) such that L is the tangent of C at \(g_1p_1\) (see Fig. 10b). Then, we have only two possible locations q and \(q'\) for \(g_4p_2\) as L and \(L'\) are parallel and \(g_4p_2\) is on \(C'\), where q is the antipodal point of \(g_2p_2\) with respect to the origin and \(q'\) is the reflection of \(q_2p_2\) along the line parallel to L and through the origin. When p is generic, L is not parallel to any reflection lines in \({{{\mathcal {D}}}}_{2k}\), implying \(g_4p_2\ne q'\). Hence, \(g_4p_2=q\). This means that \(g_4g_2^{1}\) is also the twofold rotation \(C_{\pi }\).
Following the statement of Claim 8.8, we shall split the proof into two cases.
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 \({{{\mathcal {D}}}}_{2k}\)Symmetry 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 \({{{\mathcal {D}}}}_{2k}\) with even k: for a generic \(({{{\mathcal {D}}}}_{2k},\rho )\)symmetric framework (H, p) with even k and a free action \(\rho \), (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 \({{{\mathcal {D}}}}_{2k}\)sparse double cycles \(C_{n}^2\). Below, we show that some of them indeed have symmetric infinitesimal motions.
For \(C_n^2\), the vertex set is denoted by \(\{1,\ldots , n\}\) and the edges of the 2cycle between i and \(i+1\ (\mathrm{mod}\ n)\) are denoted by \(e_{i,1}\) and \(e_{i,2}\) for \(i=1,\ldots , n\).
Theorem 8.9

\(\phi (e_{i,1})=\mathrm{id}\) and \(\phi (e_{i,2})=r'\) for \(i=1,\ldots , n1\);

\(\phi (e_{n,1})=C_{\pi }\) and \(\phi (e_{n,2})=r\).
Proof
9 Proof of Theorem 7.8
In this section we prove Theorem 7.8. For simplicity of the description, a \({{{\mathcal {D}}}}\)gain graph is called essential if it is \({{{\mathcal {D}}}}\)sparse, 4regular, not a base graph, and not a double cycle. Lemma 7.6 shows that 2extensions and loop2extensions preserve \({{{\mathcal {D}}}}\)sparsity, and hence what we have to prove is the following theorem.
Theorem 9.1
Any essential \({{{\mathcal {D}}}}\)gain graph \((G,\phi )\) has a vertex at which a 2reduction or a loop2reduction is admissible.
For simplicity, in the subsequent discussion we omit gain functions \(\phi \) 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 \({{{\mathcal {D}}}}\)tight set is called a tight set.

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 \(N(v)=2\).
9.1 Preliminary Facts

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 \(X\subseteq V\), \(i_G(X)\le 2X\lfloor d_G(X)/2\rfloor \), where \(i_G(X)\) denotes the number of edges induced by X. In particular, if G is 4regular, \(i_G(X)=2Xd_G(X)/2\).
 (C1)

\(F\le 2V(F)3\) for every nonempty balanced set \(F\subseteq E\);
 (C2)

G is not cyclic.
Lemma 9.2
 (i)
G is not 4regular and condition (C1) is satisfied, or
 (ii)
G is 4regular and conditions (C1) and (C2) are satisfied.
Proof
If the maximum degree is at most 4, \(F\le 2V(F)\) for any \(F\subseteq E\). In particular, if G is connected, we have \(F\le i_G(V(F)) \le 2V(F)\lfloor d_G(V(F))/2 \rfloor \le 2V(F)1\) for any \(F\subseteq E\) with \(V(F)\ne V\). Therefore, \(F\ge 2V(F)\) holds if and only if G is 4regular and \(F=E\). \(\square \)
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 \((G,\phi )\) is almost nearcyclic if there are two incident edges e and f such that \(Gef\) is cyclic.
Lemma 9.3

\(\phi '(e)\in {{{\mathcal {C}}}}\) for every \(e\in E{\setminus } \{e_3,e_4\}\), and

\(\phi '(e_3)\notin \bar{{{\mathcal {C}}}}\) and \(\phi '(e_4)\notin \bar{{{\mathcal {C}}}}\).
Proof
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
 (a)
G is nearcyclic.
 (b)
\(Gv+e_1\cdot e_3^{1}+e_2\cdot e_4^{1}\) is \({{{\mathcal {D}}}}\)sparse.
 (c)
v is a cutvertex in G and \(Gv+e_1\cdot e_3^{1}+e_2\cdot e_4^{1}\) is connected.
Proof
For simplicity, we denote \(e_{i,j}=e_i\cdot e_j^{1}\) for \(i,j\in \{1,2,3,4\}\). We assume that neither (a) nor (b) occur and show that (c) holds.
We claim that there are an equivalent gain function \(\phi '\) to \(\phi \) and a cyclic subgroup \({{{\mathcal {C}}}}\) of \({{{\mathcal {D}}}}\) such that \(\phi '(e)\in {{{\mathcal {C}}}}\) holds for \(e\in E{\setminus } \{e_3,e_4\}\) and \(\phi '(e_3)\notin \bar{{{\mathcal {C}}}}\) and \(\phi '(e_4)\notin \bar{{{\mathcal {C}}}}\).
To see this, first observe that if \(Gv+e_1\cdot e_2^{1}+e_3\cdot e_4^{1}\) is connected and cyclic, then Lemma 9.3 implies the claim. On the other hand, if \(Ge_3e_4\) is connected and cyclic, then by Lemma 2.3, there is an equivalent \(\phi '\) to \(\phi \) and a cyclic subgroup \({{{\mathcal {C}}}}\) of \({{{\mathcal {D}}}}\) such that \(\phi '(e)\in {{{\mathcal {C}}}}\) for \(e\in E{\setminus } \{e_3,e_4\}\). Since G is neither cyclic nor nearcyclic, we have \(\phi '(e_3)\notin \bar{{{\mathcal {C}}}}\), and \(\phi '(e_4)\notin \bar{{{\mathcal {C}}}}\).
Note that \(\phi '(e_{1,3})\notin \bar{{{\mathcal {C}}}}\) and \(\phi '(e_{2,4})\notin \bar{{{\mathcal {C}}}}\).
Let us consider \(Gv\). Since \(Gv\) is cyclic with \(E(Gv)=2V(Gv)2\), \(Gv\) is (2, 3)gsparse. Applying Lemma 7.7 with \(\phi '(e_{1,3})\notin \bar{{{\mathcal {C}}}}\), we deduce that \(Gv+e_{1,3}\) is \({{{\mathcal {D}}}}\)sparse. Let \(G'=Gv+e_{1,3}+e_{2,4}\). Since (b) does not hold, \(G'\) is not \({{{\mathcal {D}}}}\)sparse. By Lemma 9.2, \(G'\) (or a connected component of \(G'\)) violates (C1) or (C2).
Case 1: If (C1) is violated, then \(Gv+e_{1,3}\) contains a balanced tight set F such that V(F) contains the endvertices of \(e_{2,4}\) and \(F+e_{2,4}\) is balanced. Let s and t be the endvertices of \(e_{2,4}\), 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 \(e_{1,3}\). Recall that the gain of each edge in this path is included in \({{{\mathcal {C}}}}\), and the concatenation of the path and \(e_{2,4}\) forms an unbalanced closed walk in \(F+e_{2,4}\), contradicting that \(F+e_{2,4}\) is balanced. Therefore, \(F=1\) holds; in particular, since \(s,t\in V(F)\) and \(F+e_{2,4}\) is balanced, it follows that \(F=\{e_{1,3}\}\) and \(\{e_{1,3},e_{2,4}\}\) 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 \(E(Gv)=2V(Gv)2\), \(Gv\) contains an unbalanced cycle C, whose gain is included in \({{{\mathcal {C}}}}\). Let s and t be the endvertices of \(e_{2,4}\), which are possibly the same vertex. Since \(Gv\) is connected, there is a path P from s to a vertex in V(C). We consider a closed walk \(W_1\) that first passes through P starting at s, then goes around C, and comes back to s through \(P^{1}\). We then have \(\phi '(W_1)\in {{{\mathcal {C}}}}\). Also, since \(Gv\) is connected, \(Gv\) has a path \(P'\) connecting s and t. The concatenation of \(P'\) with \(e_{2,4}\) forms a closed walk \(W_2\) starting at s with \(\phi (W_2)\notin \bar{{{\mathcal {C}}}}\). Thus, \(\{\phi '(W_1), \phi '(W_2)\}\) 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 \(G_1'\) and \(G_2'\) with \(e_{1,3}\in E(G_1')\) and \(e_{2,4}\in E(G_2')\). We have already seen that \(Gv+e_{1,3}\) is \({{{\mathcal {D}}}}\)sparse, and hence its subgraph \(G_1'\) is \({{{\mathcal {D}}}}\)sparse. However, since \(G_1'\) is 4regular, \(G_1'\) is indeed maximum \({{{\mathcal {D}}}}\)tight. By the symmetry between \(e_{1,3}\) and \(e_{2,4}\), \(G_2'\) is also maximum \({{{\mathcal {D}}}}\)tight, and thus \(G'\) is maximum \({{{\mathcal {D}}}}\)tight, a contradiction. Thus \(G'\) should be connected, which implies (c). \(\square \)
9.2 Special Cases
Lemma 9.5
Let \((G,\phi )\) be an essential \({{{\mathcal {D}}}}\)gain 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 \((G,\phi )\) is essential, the underlying graph is either \(P_n^2\) or \(C_n^{\circ }\).
Suppose that the underlying graph is \(P_n^{2}\). We perform the loop2reduction at a vertex incident to a loop l. The resulting graph is \(P_{n1}^{2}\) and clearly it satisfies (C1). If it does not satisfy (C2), then the resulting graph is cyclic and there is a cyclic subgroup \({{{\mathcal {C}}}}\) of \({{{\mathcal {D}}}}\) such that the gain of every cycle in G except for the loop l is in \({{{\mathcal {C}}}}\). This in turn implies that \(Gl\) is cyclic, contradicting the assumption that G is essential.
Suppose that the underlying graph is \(C_n^{\circ }\). We may assume \(n\ge 3\) since \(C_2^{\circ }=P_2^{2}\). We perform the 2reduction at a vertex incident to a loop l. The resulting \({{{\mathcal {D}}}}\)gain graph, denoted \(G'\), has the underlying graph \(C_{n1}^{\circ }\).
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 \({{{\mathcal {C}}}}\) of \({{{\mathcal {D}}}}\), 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. \(\square \)
The next lemma solves the case when the graph is not 2connected.
Lemma 9.6
Let \(G=(V,E)\) be a connected essential \({{{\mathcal {D}}}}\)gain graph with \(V\ge 2\). 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 \(P_{V}^{2}\). Then G has a cutvertex v which is not special. We show that a 2reduction at v is admissible. Note that \(Gv\) consists of two connected components by the 4regularity of G. Let \(e_1,e_7,e_3,e_4\) 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 \(e_1\) and \(e_3\) are included in a connected component of \(Gv\) while those of \(e_2\) and \(e_4\) are included in the other component.
Consider the 2reduction at v through \((e_1,e_2)\) and \((e_3,e_4)\). 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 \(e_3\cdot e_4^{1}\) is not parallel to \(e_1\cdot e_2^{1}\) as v is not special. Since the endvertices of \(e_3\cdot e_4^{1}\) belong to different connected components in \(Gv\) and \(e_1\cdot e_2^{1}\) is the bridge in \(Gv+e_1\cdot e_2^{1}\), \(Gv+e_1\cdot e_2^{1}\) has no balanced tight set F such that V(F) contains both endvertices of \(e_3\cdot e_4^{1}\). This implies that \(G'\) satisfies (C1).
Therefore, if \(G'\) satisfies (C2), then \(G'\) is \({{{\mathcal {D}}}}\)sparse 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 \((e_1,e_3)\) and \((e_2,e_4)\). The resulting graph, denoted by \(G''\), is disconnected. Lemma 9.4 thus implies that \(G''\) is \({{{\mathcal {D}}}}\)sparse (recall that G is not nearcyclic since G is essential). \(\square \)
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 \({{{\mathcal {D}}}}\)gain 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 us denote the edges incident to u by \(e_1, e_2,e_3,e_4\), where \(e_1\) and \(e_2\) are linking from v to u and \(e_3\) and \(e_4\) are linking from a vertex in \(V{\setminus } \{u,v\}\) to u. We perform the 2reduction at u through \((e_1,e_2)\) and \((e_3,e_4)\). 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 \(Ge_1e_2\) is cyclic by Lemma 9.3.
Let \(a,b,c\in V\) such that \(N(v)=\{u,a,b\}\) and \(N(u)=\{v,c\}\). Since \(N(u)=2\) with \(v\in N(u)\), without loss of generality we may assume \(a\notin N(u)\) (where \(b=c\) possibly holds). Recall that \(Ge_1e_2\) is connected and cyclic, and hence we can apply Lemma 9.4 to deduce that the 2reduction at v through \((bv, e_1)\) and \((av, e_2)\) 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. \(\square \)
The next lemma solves the case when G is almost nearcyclic.
Lemma 9.8
Let \(G=(V,E)\) be a 2connected essential \({{{\mathcal {D}}}}\)gain 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 \(e_1\) and \(e_2\) for which \(e_1\) and \(e_2\) are incident to a vertex v and \(Ge_1e_2\) 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 \(e_1\) and \(e_2\) are nonloop edges. By Lemma 2.3, we may assume that the label of each edge in \(Ge_1e_2\) is contained in a cyclic subgroup \({{{\mathcal {C}}}}\) of \({{{\mathcal {D}}}}\). By further performing a switching operation at v with \(\phi (e_1)\), \(\phi \) is converted such that \(\phi (e_1)=\mathrm{id}\) and \(\phi (e)\in {{{\mathcal {C}}}}\) for all edges e not incident to v. This implies that if we remove \(e_2\) and the loop incident to v from G, the resulting graph is cyclic. In other words, it suffices to consider the case when \(e_1\) or \(e_2\) is a loop.
We hence assume that \(e_1\) is the loop incident to v. Let \(e_3\) be the remaining nonloop edge incident to v, where \(\phi (e_3)\in {{{\mathcal {C}}}}\). Observe that the gain of the nonloop edge \(e_2\) is not included in \(\bar{{{\mathcal {C}}}}\), since otherwise \(Ge_1\) becomes cyclic, contradicting the assumption that G is essential. Therefore, \(\phi (e_2\cdot e_3^{1})\notin \bar{{{\mathcal {C}}}}\), and the loop2reduction at v adds the edge \(e_2\cdot e_3^{1}\) to the cyclic (2, 3)gsparse graph \(Gv\). By Lemma 7.7, the resulting gain graph is \({{{\mathcal {D}}}}\)sparse. \(\square \)
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 \({{{\mathcal {D}}}}\)gain graph with \(V=n\ge 4\). 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 \(X>1\), \(V{\setminus } X>1\) and \(d_G(X)=2\). Since G is not \(C_n^{\circ }\), 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 \(V{\setminus } X\). By the 2connectivity, v is not a cutvertex. Hence, denoting the four edges incident to v by \(e_1,\ldots ,e_4\), we may assume that \(e_1,e_2,e_3\) are included in the subgraph induced by X while \(e_4\) is not.
Note that v is a vertex of degree 3 in \(Ge_4\), and hence, by Lemma 7.5, a 1reduction at v is admissible in \(Ge_4\). Without loss of generality, we may assume that \(Gv+e_1\cdot e_2^{1}\) (obtained by a 1reduction at v in \(Ge_4\)) is \({{{\mathcal {D}}}}\)sparse.
We now consider adding \(e_3\cdot e_4^{1}\) to \(Gv+e_1\cdot e_2^{1}\) to complete the 2reduction at v. Let \(G'=Gv+e_1\cdot e_2^{1}+e_3\cdot e_4^{1}\), 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 \(e_3\cdot e_{4}^{1}\) unless \(F=1\). If \(Gv+e_1\cdot e_2^{1}\) has a balanced set F such that \(F=1\) and V(F) contains both endvertices of \(e_3\cdot e_{4}^{1}\), then the edge in F, denoted by f, is incident to \(e_3\) and \(e_4\) and connects between X and \(V{\setminus } X\). However, since \(d_G(X)=2\), \(X>1\) and \(V{\setminus } X>1\), the vertex incident to \(e_4\) 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. \(\square \)
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 \({{{\mathcal {D}}}}\)gain 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 \(V\ge 4\), 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 \(e_1,e_2,e_3\) between u and v, a loop l attached to w, and two remaining edges uw and vw, denoted by \(f_1\) and \(f_2\), respectively. We may assume \(\phi (f_1)=\phi (f_2)=\mathrm{id}\). Let \({{{\mathcal {C}}}}\) be the subgroup generated by \(\phi (l)\). Since G is not cyclic, there is an unbalanced cycle whose gain is not included in \(\bar{{{\mathcal {C}}}}\).
If a triangle, say \(e_1f_1f_2\) has a gain not included in \(\bar{{{\mathcal {C}}}}\), then the 2reduction at u through \((e_1,f_1)\) and \((e_2,e_3)\) results in a \({{{\mathcal {D}}}}\)sparse \(P_2^{2}\). Otherwise, removing \(e_2\) and \(e_3\) results in a cyclic graph. Then G is almost nearcyclic, and Lemma 9.8 implies the statement. \(\square \)
9.3 The Remaining Cases
In a graph G, the star of a vertex v means the subgraph of G whose vertex set is \(N(v)\cup \{v\}\) 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 \(V\ge 3\). Suppose also that G is not almost nearcyclic. Then, for every vertex \(v\in V\) that is not incident to a loop with \(N(v)\ge 3\), 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 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.
9.3.1 Obstacles Around a Vertex v with \(N(v)=4\)
Throughout Sect. 9.3.1, \((G,\phi )\) denotes a \({{{\mathcal {D}}}}\)gain graph satisfying the assumptions of Theorem 9.11, v denotes a vertex with \(N(v)=4\), \(N(v)=\{a,b,c,d\}\), and \(E_v\) 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
 (i)
F is balanced tight with \(a,b\in V(F)\) and \(av\cdot vb\in \mathrm {cl}_{{{\mathcal {D}}}}(F)\);
 (ii)
F is balanced tight with \(c,d\in V(F)\) and \(cv\cdot vd\in \mathrm {cl}_{{{\mathcal {D}}}}(F)\);
 (iii)
F is subtight with \(a,b,c,d\in V(F)\), \(F+av\cdot vb\) is balanced tight, and \(cv\cdot vd\in \mathrm {cl}_{{{\mathcal {D}}}}(F+av\cdot vb)\).
Proof
Let us first consider the graph \(G'=Gv+av\cdot vb\). If \(G'\) is not \({{{\mathcal {D}}}}\)sparse, then, by Lemma 9.2, \(E{\setminus } E_v\) has a balanced tight set F with \(a,b\in V(F)\) and \(av\cdot vb\in \mathrm {cl}_{{{\mathcal {D}}}}(F)\), which satisfies property (i).
Hence, let us assume that \(G'\) is \({{{\mathcal {D}}}}\)sparse. If \(G'+cv\cdot vd\) is cyclic, Lemma 9.3 implies that G is almost nearcyclic, contradicting the assumption that G is not almost nearcyclic. Therefore, \(G'+cv\cdot vd\) satisfies (C2). By Lemma 9.2, there exists a balanced tight set \(F'\subseteq E{\setminus } E_v\cup \{av\cdot vb\}\) with \(c,d\in V(F')\) and \(cv\cdot vd\in \mathrm {cl}_{{{\mathcal {D}}}}(F')\). Depending on whether \(av\cdot vb\in F'\) or not, we find a desired subset of the statement; if \(av\cdot vb\not \in F'\) then \(F'\) is the one satisfying property (ii); otherwise \(F'av\cdot vb\) satisfies property (iii) (we remark that, in the latter case, \(V(F'av\cdot vb)\) contains a, b, c, d since \(F'\) is 2edgeconnected). \(\square \)
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 \(N(v)=\{a,b,c,d\}\), \(F\subseteq E{\setminus } E_v\) 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).
As noted above, we have three possible ways for a 2reduction at v, through (av, vb) and (cv, vd), through (av, vc) and (bv, vd), and through (av, vd) and (bv, vc). By Lemma 9.12, if none of them are admissible, \(E{\setminus } E_v\) contains three corresponding obstacles X, Y, Z. We now investigate properties of these obstacles.
We begin with a property of type 2 obstacles.
Lemma 9.13

\(X\cup E_v=2V(X\cup E_v)2\);

There is an equivalent gain function \(\phi '\) to \(\phi \) such that \(\phi '(e)=\mathrm{id}\) for \(e\in X\cup \{va,vb\}\), and \(\phi '(vc)=\phi '(vd)\ne \mathrm{id}\);

\(X\cup E_v\) is cyclic.
Proof
By definition, \(X=2V(X)4\), and hence \(X\cup E_v=2V(X\cup E_v)2\) by \(N(v)\subseteq V(X)\).
Since \(cv\cdot vd\in \mathrm{cl}_{{{\mathcal {D}}}}(X+av\cdot vb)\) and \(X+av\cdot vb\) is balanced, \(X+av\cdot vb+cv\cdot vd\) is also balanced. Hence, by Lemma 2.3, there is an equivalent gain function \(\phi '\) to \(\phi \) such that \(\phi '(e)=\mathrm{id}\) for \(e\in X\) and \(\phi '(av\cdot vb)=\phi '(cv\cdot vd)=\mathrm{id}\). We thus have \(\phi '(av)=\phi '(bv)=g\) and \(\phi '(cv)=\phi '(dv)=g'\) for some \(g,g'\in {{{\mathcal {D}}}}\). By performing a switching operation at v with g if necessary, we may assume that \(\phi '(av)=\phi '(bv)=\mathrm{id}\) and \(\phi '(cv)=\phi '(dv)=g'g^{1}\). If \(g'g^{1}=\mathrm{id}\), \(X\cup E_v\) becomes a balanced set with \(X\cup E_v>2V(X\cup E_v)3\), contradicting the \({{{\mathcal {D}}}}\)sparsity of G. Thus, \(\phi '(cv)=\phi '(dv)\ne \mathrm{id}\), and \(X\cup E_v\) is cyclic. \(\square \)
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 \(X\cup Y\) is cyclic. Then, \(X\cup Y\cup E_v\) is cyclic.
Proof
Since X is balanced and \(X\cup Y\) is cyclic, for some cyclic subgroup \({{{\mathcal {C}}}}\) of \({{{\mathcal {D}}}}\), there is an equivalent gain function \(\phi '\) to \(\phi \) such that \(\phi '(e)=\mathrm{id}\) for every \(e\in X\) and \(\phi '(e)\in {{{\mathcal {C}}}}\) for every \(e\in Y\) by Lemma 2.3. Moreover, since \(X+av\cdot vb\) and \(X+av\cdot vb+cv\cdot vd\) are balanced, we have \(\phi '(av\cdot vb)=\phi '(cv\cdot vd)=\mathrm{id}\). As in the previous proof, by applying a switching operation at v, we may assume that \(\phi '(va)=\phi '(vb)=\mathrm{id}\) and \(\phi '(vc)=\phi '(vd)\).
By the definition of the obstacles (whether type 1 or type 2), \(Y+\) \(Y+av\cdot vc\) or \(Y+bv\cdot vd\) is connected and balanced. Hence \(\phi '(av\cdot vc)\in \bar{{{\mathcal {C}}}}\) or \(\phi '(bv\cdot vd)\in \bar{{{\mathcal {C}}}}\), which implies \(\phi '(vc)=\phi '(vd)\in \bar{{{\mathcal {C}}}}\). Thus, every label of \(X\cup Y\cup E_v\) is included in \(\bar{{{\mathcal {C}}}}\). \(\square \)
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 \(X\cap Y\ne \emptyset \), then \(X\cup Y\) is not a balanced set.
Proof
Suppose for a contradiction that \(X\cup Y\) is a balanced set with \(X\cap Y\ne \emptyset \).
(Case 1) If both X and Y are of type 1, \(X\cup Y\) is tight by Lemma 7.2 and hence \(X\cup Y=2V(X\cup Y)3\). Without loss of generality, we may assume that \(a,b,c\in V(X\cup Y)\), \(av\cdot vb\in \mathrm {cl}_{{{\mathcal {D}}}}(X)\) and \(av\cdot vc\in \mathrm {cl}_{{{\mathcal {D}}}}(Y)\). Since \(X\cup Y\) is balanced, there is an equivalent gain function \(\phi '\) to \(\phi \) such that \(\phi '(e)=\mathrm{id}\) for \(e\in X\cup Y\). Moreover, since \(av\cdot vb\in \mathrm {cl}_{{{\mathcal {D}}}}(X)\) and \(av\cdot vc\in \mathrm {cl}_{{{\mathcal {D}}}}(Y)\), we have \(\phi '(av)=\phi '(bv)=\phi '(cv)\). This implies that \(X\cup Y\cup \{av,bv,cv\}\) is a balanced set. However, since \(X\cup Y\cup \{av,bv,cv\}>2V(X\cup Y\cup \{av,bv,cv\})3\), the existence of such a balanced set contradicts the \({{{\mathcal {D}}}}\)sparsity of G.
(Case 2) Let us consider the case when X is type 2. By definition of obstacles (whether type 1 or type 2), \(Y+av\cdot vc\) or \(Y+bv\cdot vd\) is balanced and 2edgeconnected. Without loss of generality, we assume that \(Y+av\cdot vc\) is balanced and 2edgeconnected. By Lemma 9.13, there exists an equivalent gain function \(\phi '\) to \(\phi \) such that \(\phi '(e)=\mathrm{id}\) for \(e\in X\cup \{va, vb\}\) and \(\phi '(vc)=\phi '(vd)\ne \mathrm{id}\). Moreover, since \(X\cup Y\) is balanced, we may assume that \(\phi '(e)=\mathrm{id}\) for \(e\in Y\). Since \(\phi '(av\cdot vc)\ne \mathrm{id}\) but \(\phi '(e)=\mathrm{id}\) for \(e\in Y\), \(Y+av\cdot vc\) is unbalanced, a contradiction. \(\square \)
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 \(X\cap Y\ne \emptyset \).
Proof
Without loss of generality, we assume \(a\in V(X)\cap V(Y)\). 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 \(Gv\). Hence, if X and Y are type 1 with \(X>1\) and \(Y>1\), then \(X\cap Y\) contains an edge incident to a.
If X is type 2, then \(X+av\cdot vb\) is balanced tight with \(a,b,c,d \in V(X+av\cdot vb)\) by definition. Hence, if Y is type 1, then \(X\cap Y\) contains an edge incident to c or d.
If both X and Y are type 2, then \(X\cap Y\) contains an edge incident to d. \(\square \)
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 \(X\cap Y\ne \emptyset \), \(Y\cap Z\ne \emptyset \) or \(Z\cap X\ne \emptyset \) 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 \(Y=\{e_y\}\) and \(Z=\{e_z\}\). Clearly, \(e_y\ne e_z\).
(Case 1) Let us first consider the case when X is also a singleton set. Let \(X=\{e_x\}\). Depending on the relative position of \(e_x, e_y\) and \(e_z\), we have two situations: (I) \(e_x, e_y\) and \(e_z\) share a vertex or (II) \(e_x, e_y\) and \(e_z\) form a triangle.
In case (I), the star of v is included in a hat subgraph. Indeed, if denoting without loss of generality \(e_x=ab\), \(e_y=ac\), and \(e_z=ad\), \(\{e_x,e_y,e_z,va,vb,vc,vd\}\) forms a hat if it is balanced. Since X, Y and Z are obstacles, we have \(\phi (e_x)=\phi (av \cdot vb)\), \(\phi (e_y)=\phi (av \cdot vc)\) and \(\phi (e_z)=\phi (av\cdot vd)\), and hence this subgraph is indeed balanced.
In case (II), without loss of generality, we assume \(e_x=ab, e_y=bc\) and \(e_z=ca\). Then \(\{e_x,e_y,e_z,va,vb,vc\}\) forms \(K_4\). Since \(\phi (e_x)=\phi (av \cdot vb)\), \(\phi (e_y)=\phi (bv \cdot vc)\) and \(\phi (e_z)=\phi (cv \cdot va)\), this \(K_4\) does not have any unbalanced cycle. Therefore, Case (II) cannot happen because of the \({{{\mathcal {D}}}}\)sparsity of G, as a balanced \(K_4\) is not \({{{\mathcal {D}}}}\)sparse.
(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 \(X\cup E_v=2V(X\cup E_v)2\) by Lemma 9.13. Also, by Lemma 9.13, there exists an equivalent gain function \(\phi '\) to \(\phi \) such that \(\phi '(e)=\mathrm{id}\) for \(e\in X\cup \{va, vb\}\) and \(\phi '(vc)=\phi '(vd)\ne \mathrm{id}\). Denote \(\phi '(vc)\) by g. Since Y and Z are obstacles, we have \(\phi '(e_y)=\phi '(e_z)=g\), which in particular implies \(e_y,e_z\not \in X\). By \(N(v)\subseteq V(X)\) and \(e_y\ne e_z\), \(X\cup Y\cup Z\cup E_v=2V(X\cup Y\cup Z\cup E_v)\), which in turn implies \(E=X\cup Y\cup Z\cup E_v\). Notice that the label of each edge in \(X\cup Y\cup Z\cup E_v\) is either the identity or g. In other words, \(X\cup Y\cup Z\cup E_v\) is cyclic, contradicting the \({{{\mathcal {D}}}}\)sparsity of G.
The remaining case is when X is type 1. Without loss of generality we assume \(a,b\in V(X)\). By \(X>1\) and Lemma 7.1, \(d_X(a)\ge 2\) and \(d_X(b)\ge 2\). Since \(e_y\) is either ac or bd and \(e_z\) is either ad or bc, it suffices to consider the following two cases by symmetry: (i)\((e_y, e_z)=(ac,ad)\), and (ii)\((e_y,e_z)=(ac,bc)\).
In subcase (i), \(X\cap Y\) or \(X\cap Z\) contains an edge incident to a as \(d_X(a)\ge 2\) and \(d_{Gv}(a)=3\).
In subcase (ii), notice that, \(\{av,bv,cv, e_y,e_z, av\cdot vb\}\) is a circuit of the underlying \({{{\mathcal {D}}}}\)sparsity matroid since it forms a balanced \(K_4\). By \(av\cdot vb\in \mathrm {cl}_{{{\mathcal {D}}}}(X)\), we have \(cv\in \mathrm {cl}_{{{\mathcal {D}}}}(X+av+bv+e_y+e_z)\subseteq \mathrm {cl}_{{{\mathcal {D}}}}(Ecv)\), contradicting the independence of E. Therefore, this case does not occur and the proof is complete. \(\square \)
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, \((G,\phi )\) denotes a \({{{\mathcal {D}}}}\)gain graph satisfying the assumptions of Theorem 9.11, v denotes a vertex with \(N(v)=3\), \(N(v)=\{a,b,c\}\), and \(E_v\) denotes the set of edges incident to v. Without loss of generality, we assume that there are parallel edges \(e_1\) and \(e_2\) between v and a, and we denote \(E_v=\{e_1,e_2,vb,vc\}\).
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
 (i)
F is balanced tight with \(a,b\in V(F)\) and \(e_1\cdot vb\in \mathrm {cl}_{{{\mathcal {D}}}}(F)\);
 (ii)
F is balanced tight with \(a,c\in V(F)\) and \(e_2\cdot vc\in \mathrm {cl}_{{{\mathcal {D}}}}(F)\);
 (iii)
F is subtight with \(a,b,c\in V(F)\), \(F+e_1\cdot vb\) is balanced tight, and \(e_2\cdot vc\in \mathrm {cl}_{{{\mathcal {D}}}}(F+e_1\cdot vb)\).
For the 2reduction through \((e_1,e_2)\) and (bv, vc), we encounter an even simpler situation.
Lemma 9.19
Suppose that the 2reduction through \((e_1,e_2)\) and (bv, vc) is not admissible. Then there is a balanced tight set \(F\subseteq E{\setminus } E_v\) with \(b,c\in V(F)\) and \(bv\cdot vc\in \mathrm {cl}_{{{\mathcal {D}}}}(F)\).
Proof
Note that \(e_1\cdot e_2^{1}\) is a loop. \(Gv+e_1\cdot e_2^{1}\) is \({{{\mathcal {D}}}}\)sparse by Lemma 9.2 since adding an unbalanced loop does not affect (C1). Note that \(Gv+e_1\cdot e_2^{1}+bv\cdot vc\) is connected. If \(Gv+e_1\cdot e_2^{1}+bv\cdot vc\) does not satisfy (C2), then Lemma 9.3 implies that G is almost nearcyclic, which contradicts our assumption on G. If \(Gv+e_1\cdot e_2^{1}+bv\cdot vc\) does not satisfy (C1), then \(Gv+e_1\cdot e_2^{1}\) contains a balanced tight set F with \(b,c\in V(F)\) and \(bv\cdot vc\in \mathrm{cl}_{{{\mathcal {D}}}}(F)\). Since a balanced tight set does not contain a loop by Lemma 7.1, we have \(F\subseteq E{\setminus } E_v\). \(\square \)
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 \((e_1,e_2)\) 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 \(X\cup Y\) is cyclic, then \(X\cup Y\cup E_v\) is cyclic.
Lemma 9.21
Let X and Y be obstacles for distinct 2reductions at v. Then, if \(X\cap Y\ne \emptyset \), then \(X\cup Y\) 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 \((e_1,e_2)\) and (bv, vc). Then, there is an equivalent gain function \(\phi '\) to \(\phi \) such that \(\phi '(e)=\mathrm{id}\) for \(e\in Z\cup \{vb,vc\}\).
Proof
\(Z+bv\cdot vc\) is balanced. Hence, by Lemma 2.3, there is an equivalent gain function \(\phi '\) to \(\phi \) such that \(\phi '(e)=\mathrm{id}\) for \(e\in Z+bv\cdot vc\). By performing a switching operation at v with \(\phi '(bv)\) if necessary, we may assume that \(\phi '(bv)=\phi '(vc)=\mathrm{id}\). \(\square \)
Lemma 9.23
Let X be an obstacle of type 2 for the 2reduction through \((e_1,vb)\) and \((e_2,vc)\). Suppose further that there is no obstacle of type 1 for the 2reduction through \((e_1,vb)\) and \((e_2,vc)\). Then \(d_X(a)+d_X(b)+d_X(c)\ge 5\) holds.
Proof
Let \(X'=X+e_1\cdot vb\). By definition, \(X'\) is balanced tight with \(a,b,c\in V(X')\) and \(X'>1\). Such a balanced tight set is 2connected and essentially 3edgeconnected by Lemma 7.1. We thus have \(d_{X'}(u)\ge 2\) for \(u\in \{a,b,c\}\).
Suppose that \(d_{X'}(a)=d_{X'}(b)=2\). Since \(X'\) is essentially 3edgeconnected and \(e_1\cdot vb\) 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 \(X'+e_2\cdot vc\) is balanced by definition of type 2 obstacles. However, since \(e'\) and \(e_2\cdot vc\) are parallel, for \(X'+e_2\cdot vc\) to be balanced, \(\{e',e_2\cdot vc\}\) has to be a balanced 2cycle, that is, \(\{e'\}\) is a type 1 obstacle for the 2reduction through \((e_1,vb)\) and \((e_2,vc)\), contradicting the assumption of the lemma.
Therefore, \(d_{X'}(a)\ge 3\) or \(d_{X'}(b)\ge 3\), implying \(d_{X'}(a)+d_{X'}(b)+d_{X'}(c)\ge 7\). Since \(X'=X+e_1\cdot vb\), we obtain \(d_X(a)+d_X(b)+d_X(c)\ge 5\). \(\square \)
Lemma 9.24
Let X, Y, Z be obstacles for the 2reductions through \((e_1,vb)\) and \((e_2,vc)\), through \((e_1,vc)\) and \((e_2,vb)\), and through \((e_1,e_2)\) and (bv, vc), respectively. Then, \(X\cap Y\ne \emptyset \), \(Y\cap Z\ne \emptyset \), or \(Z\cap X\ne \emptyset \) holds.
Proof
We split the proof into two cases depending on whether a type 1 obstacle exists for the 2reduction through \((e_1,vb)\) and \((e_2,vc)\).
(Case 1) Suppose that there is no type 1 obstacle for the 2reduction through \((e_1,vb)\) and \((e_2,vc)\). Then, X is type 2. By Lemma 9.23, \(d_X(a)+d_X(b)+d_X(c)\ge 5\) holds. If \(d_X(a)\ge 2\), then \(X\cap Y\) contains an edge incident to a since \(d_{Gv}(a)=2\) and \(d_Y(a)\ge 1\). If \(d_X(a)=1\), then we have \(d_X(b)\ge 2\) and \(d_X(c)\ge 2\). Since \(d_{Gv}(b)=d_{Gv}(c)=3\), \(Z=1\) holds if \(X\cap Z= \emptyset \). However, in this case, we have \(d_{X\cup Z}(b)=d_{X\cup Z}(c)=3\), and thus \(X\cap Y\) or \(Y\cap Z\) 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 \((e_1,vc)\) and \((e_2,vb)\).
In case (i), we have \(e_1\cdot vc\in \mathrm {cl}_{{{\mathcal {D}}}}(Y)\) by definition. Therefore, \(\phi (e_y)=\phi (e_1\cdot vc)=\phi (e_1)\). Notice that \(\{e_1,vb,vc,e_x,e_y, bv\cdot vc\}\) forms a \(K_4\) without unbalanced cycles by \(\phi (e_y)=\phi (e_1)=\phi (e_x)\). Moreover, since \(bv\cdot vc\in \mathrm {cl}_{{{\mathcal {D}}}}(Z)\), we obtain \(e_1\in \mathrm {cl}_{{{\mathcal {D}}}}(\{vb,vc,e_x,e_y,bv\cdot vc\})\subseteq \mathrm {cl}_{{{\mathcal {D}}}}(Ee_1)\). This contradicts the independence of E in the underlying \({{{\mathcal {D}}}}\)sparsity matroid.
Let us consider case (ii). If \(Z>1\), then \(X\cap Z\) or \(Y\cap Z\) contains an edge incident to b as Z is type 1 and \(d_Z(b)\ge 2\). Suppose that \(Z=1\), \(X\cap Y=\emptyset \), \(X\cap Z=\emptyset \) and \(Y\cap Z=\emptyset \). Then \(X\cup Y\cup Z\cup E_v\) 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 \(e_2\) and \(e_y\) results in a cyclic graph (where any cycle except the loop is balanced by \(\phi (e_1)=\phi (e_x)\)). This means that G is almost nearcyclic, a contradiction. \(\square \)
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 \(X\cap Y\ne \emptyset \) holds.
If \(X\cup Y\ge 2V(X\cup Y)1\), then \(V(X\cup Y)\cup \{v\}=V\) must hold since G is essentially 4edgeconnected. We then have \(X\cup Y\cup E_v\ge 2V+1\), contradicting the \({{{\mathcal {D}}}}\)sparsity of G.
If the number of connected components in \((V(X)\cap V(Y),X\cap Y)\) is one, then, since X and Y are connected and balanced, Lemma 2.4(1) implies that \(X\cup Y\) is balanced, which contradicts Lemmas 9.15 and 9.21.
If both X and Y are type 1, then \(X\cup Y\) 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 \(X\cup Y\cup E_v\) is also cyclic. Also by (23) and \(N(v)\subseteq X\cup Y\) (as X or Y is type 2) we obtain \(X\cup Y\cup E_v\ge 2V(X\cup Y\cup E_v)1\). Thus, due to the essential 4edgeconnecitivity of G, \(V(X\cup Y\cup E_v)\ge V1\) must hold.
If \(V(X\cup Y\cup E_v)=V\), then \(X\cup Y\cup E_v=E1\), and hence G is near cyclic, as \(X\cup Y\cup E_v\) is cyclic. On the other hand, if \(V(X\cup Y\cup E_v)=Vu\) for some \(u\in V\), 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. \(\square \)
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 \(N(v)\ge 3\). 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 \(a_1, a_2, b_1,b_2,b_3\), and assume that \(a_1\) and \(a_2\) have degree four in H (and hence \(a_1\) or \(a_2\) 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 \(b_1\) is not incident to a loop.
We prove that some 2reduction at \(b_1\) is admissible. Suppose that no 2reduction is admissible at \(b_1\). Then, by Theorem 9.11, the star of \(b_1\) is contained in a hat subgraph \(H'\). Note that \(H'\) is different from H.
We claim that \(H'\) contains a triangle on \(b_1,a_i,b_j\) for some \(i\in \{1,2\}\) and \(j\in \{2,3\}\). To see this first suppose that \(a_1a_2\notin E(H')\). Then, since each vertex has degree at least 2 in \(H'\), we have \(a_1b_2\in E(H')\) or \(a_1b_3\in E(H')\) by \(N_G(a_1)=\{a_2,b_1,b_2,b_3\}\) and \(a_1a_2\not \in E(H')\). This also implies \(b_1b_2\in E(H')\) or \(b_1b_3\in E(H')\), respectively, as \(b_1\) is incident to all the vertices of \(H'\). Thus \(H'\) has a triangle on \(b_1,a_1,b_j\) for some \(j\in \{2,3\}\).
If \(a_1a_2\in E(H')\), then \(H'\) contains a triangle on \(b_1,a_1,a_2\). In a hat subgraph, two vertices of each triangle have degree four, which implies \(N(a_i)\subseteq V(H')\) for some \(i\in \{1,2\}\). Therefore, \(a_ib_2\in E(H')\) and \(b_1b_2\in E(H')\), and hence \(b_1b_2a_i\) forms a triangle.
Consequently, without loss of generality, we may assume that \(H'\) contains a triangle on \(b_1, b_2, a_1\). Recall that a hat subgraph is balanced. Since \(\phi (a_1b_1)=\phi (a_1b_2)=\mathrm{id}\), we obtain \(\phi (b_1b_2)=\mathrm{id}\) as \(H'\) contains a triangle on \(a_1, b_1, b_2\). Observe then that \(\{a_1,a_2,b_1,b_2\}\) induces a \(K_4\) in which the label of each edge is the identity. This contradicts the \({{{\mathcal {D}}}}\)sparsity of G. Consequently, the 2reduction at \(b_1\) is admissible. \(\square \)
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 \({{{\mathcal {D}}}}_{2k}\) with odd k. This can be done by computing the rank of the quotient graphs in the corresponding matroids \({{{\mathcal {M}}}}(g_{2,3})\) or \({{{\mathcal {M}}}}_{{{\mathcal {D}}}}(G,\phi )\). Here we briefly describe the main algorithmic ideas and show that testing independece in these matroids can be done in polynomial time.
Let \((G,\phi )\) be a gain graph with \(G=(V,E)\). First consider \({{{\mathcal {M}}}}(g_{2,3})\), in which E is independent if and only if (i) G is (2, 1)sparse and (ii) every nonempty balanced subset \(F\subseteq E\) 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 Mcircuit) is unbalanced. Suppose that G satisfies (i) and consider one of its Mcomponents, 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 Mcomponents). Each (2, 3)circuit is a subgraph of some Mcomponent, 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 Mcomponents 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 \({{{\mathcal {M}}}}_{{{\mathcal {D}}}}(G,\phi )\), in which E is independent if and only if (i) G is (2, 0)sparse and (ii) every cyclic subset \(F\subseteq E\) is (2, 1)sparse, and (iii) every balanced subset \(F\subseteq E\) 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 Mcomponent of G. It is not hard to see that \(He\) is (2, 1)sparse for all \(e\in E(H)\). Thus, by using the arguments above, it follows that we have \(O(n^2)\) circuits to enumerate and test, which can also be done efficiently by the same techniques.
Notes
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.
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