Gain-Sparsity and Symmetry-Forced Rigidity in the Plane

Discrete & Computational Geometry, Feb 2016

We consider planar bar-and-joint 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 symmetry-forced 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.

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Gain-Sparsity and Symmetry-Forced Rigidity in the Plane

Discrete Comput Geom Gain-Sparsity and Symmetry-Forced Rigidity in the Plane Tibor Jordán 0 1 2 3 4 5 Viktória E. Kaszanitzky 0 1 2 3 4 5 Shin-ichi Tanigawa 0 1 2 3 4 5 Viktória E. Kaszanitzky 0 1 2 3 4 5 Shin-ichi Tanigawa 0 1 2 3 4 5 0 Department of Operations Research, Eötvös University , Pázmány Péter sétány 1/C, 1117 Budapest , Hungary 1 Department of Operations Research, Eötvös University, and the MTA-ELTE Egerváry Research Group on Combinatorial Optimization, Pázmány Péter sétány 1/C , 1117 Budapest , Hungary 2 Editor in Charge: Günter M. Ziegler 3 Centrum Wiskunde & Informatica (CWI) , Postbus 94079, 1090 GB Amsterdam , The Netherlands 4 Research Institute for Mathematical Sciences, Kyoto University , Sakyo-ku, Kyoto 606-8502 , Japan 5 Department of Mathematics and Statistics, Lancaster University , Lancaster LA1 4YF , UK We consider planar bar-and-joint 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 symmetry-forced rigidity of such structures with rotation symmetry or dihedral symmetry of order 2k with odd k, unifying and extending previous work on this subject. We also explore the matroidal background of our results and show that the matroids induced by the row independence of the orbit matrices of the symmetric frameworks are isomorphic to gain sparsity matroids defined on the quotient graph of the framework, whose edges are labeled by elements of the corresponding symmetry group. The proofs are based on new Henneberg type inductive constructions of the gain graphs that correspond to - the bases of the matroids in question, which can also be seen as symmetry preserving graph operations in the original graph. Mathematics Subject Classification Primary 52C25; Secondary 05B35 · 68R10 1 Introduction A d-dimensional bar-and-joint framework (or, simply, a framework) is a straight-line realization of a finite simple graph G in Euclidean d-space. We think of a bar-and-joint framework as a collection of fixed-length 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 bar-and-joint frameworks with point group symmetry in the symmetry-forced 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 symmetry-forced setting, a framework is said to be symmetry-forced flexible if it has a non-trivial symmetric infinitesimal motion. For the generic frameworks that we consider, this is equivalent to the existence of a non-trivial 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 group-labeled 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 group-labeled quotient graph, in which vertices and edges correspond to vertex and edge orbits, respectively, and which concisely represents the graph structure with the corresponding symmetry. Our main results characterize these matroids in the case of rotation symmetry or dihedral symmetry D2k of order 2k with odd k. If the underlying symmetry is cyclic, the matroid turns out to be a (k, l)-gain-count 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 row-independence of the orbit rigidity matrix. This approach, which has been used in many combinatorial characterizations of rigidity theory, leads to the desired result. In our problems, due to the more complex sparsity conditions and the group labeling, we also need some new operations and extended geometric arguments, to handle the symmetry constraints. The complete answer in the case of dihedral symmetry remains open. However, most of our inductive steps (extending or reducing a symmetric framework or a labeled graph, respectively) are valid also for dihedral groups D2k with even k, and can be used to show that in the even case the irreducible graphs (frameworks), where our reduction operations are not applicable, are very special. Interestingly, the smallest such framework, which is predicted to be rigid by the matroidal count but is flexible is the Bottema mechanism, a well-known 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 non-cyclic 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 symmetric-forced 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 four-regular, is moved to the end of the paper, to Sect. 9. In Sect. 8 we prove additional geometric lemmas and use them, together with the inductive construction of Sect. 7, to prove the second main result, the characterization of rigid frameworks with dihedral symmetry with odd k. We also present frameworks that meet the sparsity requirements but are dependent and flexible when k is even. In Sect. 10 we briefly discuss the algorithmic implications and make some further remarks. In the rest of the introduction, let us introduce notation used throughout the paper. Let E be a finite set. A partition P of E is a family of nonempty subsets of E such that each element of E belongs to exactly one member of P. If E = ∅, the partition of E is defined as the empty set. A subpartition of E is a partition of a subset of E . Let G = (V , E ) be an undirected graph. For v ∈ V , let dG (v) be the degree of v in G and NG (v) be the set of neighbors of v in G. For F ⊆ E , VG (F ) denotes the set of endvertices of edges in F , and let G[F ] = (V (F ), F ), that is, the graph edge-induced by F . If the graph is clear from the context, the subscript G may be dropped. For F ⊆ E and v ∈ V (F ), let dF (v) = dG[F](v). A vertex subset X ⊂ V (G) (resp., an edge subset X ⊂ E (G)) is called a separator (resp., a cut) if the removal of X disconnects G. A separator X with |X | = 1 is called a cut-vertex. G is called k-connected (resp., k-edge-connected) 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 k-connected (resp., essentially k-edge-connected) if the size of any nontrivial separator (resp., any nontrivial cut) is at least k. For simplicity, some properties of edge-induced subgraphs will be associated with the corresponding edge sets as follows. Let F ⊆ E . F is called connected if G[F ] is connected. A connected component of F is the edge set of a connected component of G[F ]. C (F ) denotes the partition of F into connected components of F , and let c(F ) = |C (F )|. F is called a forest if it contains no cycle and called a tree if it is a connected forest. F is called a spanning tree of a graph G = (V , E ) if F is a tree with F ⊆ E and V (F ) = V . Let G = (V , E ) be a directed graph. A walk in G is a sequence W = v0, e1, v1, e2, v2, . . . , vk−1, ek , vk of vertices and edges such that vi−1 and vi are the endvertices of ei for every 1 ≤ i ≤ k. We often denote a walk as a sequence of edges implicitly assuming the incidence at each vertex. For two walks W and W for which the end vertex of W and the starting vertex of W coincide, we denote the concatenation of W and W (that is, the walk W followed by W ) by W ∗ W . A walk is called closed if the starting vertex and the end vertex coincide. It is sometimes convenient to regard the empty set as a subgroup of a group. Let D be a dihedral group. For a cyclic subgroup C of D, C¯ denotes the maximal cyclic subgroup containing C. 2 Gain Graphs In this section we shall review some basic properties of gain graphs. We refer the reader to [5,31,32] for more details. Let G = (V , E ) be a directed graph which may contain multiple edges and loops, and let S be a group. An S-gain graph (G, φ) is a pair, in which each edge is associated with an element of S by a gain function φ : E → S. The orientation of G is, in some sense, arbitrary, and is used only as a reference orientation: the orientation of each edge may be changed, provided that we also modify φ such that if the edge has gain g in one direction then it has gain g−1 in the other direction. Therefore we often do not distinguish between G and the underlying undirected graph and use notations introduced in Sect. 1, implicitly referring to the underlying graph. Let W be a walk in (G, φ). The gain of W is defined as φ (W ) = φ (e1) · φ (e2) · · · φ (ek ) if each edge is oriented in the forward direction through W , and for a backward edge ei we replace φ (ei ) with φ (ei )−1 in the product. Note that φ (W −1) = φ (W )−1. Let (G, φ) be a gain graph. For v ∈ V (G) we denote by π1(G, v) the set of closed walks starting at v. Similarly, for X ⊆ E (G) and v ∈ V (G), π1(X, v) denotes the set of closed walks starting at v and using only edges of X , where π1(X, v) = ∅ if v ∈/ V (X ). Let X ⊆ E (G). The subgroup induced by X relative to v is defined as X φ,v = {φ (W ) | W ∈ π1(X, v)}. The subscript φ of X φ,v is sometimes omitted if it is clear from the context. Note that, for any connected X ⊆ E (G) and two vertices u, v ∈ V (X ), X ψ,u is conjugate to X ψ,v (see, e.g., [5, p. 88] for the proof). 2.1 The Switching Operation For v ∈ V (G) and g ∈ S, a switching operation at v with g changes the gain function φ on E (G) as follows. φ (e) = ⎧ g · φ (e) · g−1 if e is a loop incident with v, ⎪⎪⎪⎨ g · φ (e) if e is a non-loop edge and is directed from v, φ (e) · g−1 if e is a non-loop edge and is directed to v, ⎪ ⎪⎪⎩ φ (e) otherwise. (1) We say that a gain function φ on edge set E (G) is equivalent to another gain function φ on E (G) if φ can be obtained from φ by a sequence of switching operations. The following two facts are fundamental (see, e.g., [5, Sect. 2.5.2] or [31, Sect. 5] for the proofs). Proposition 2.1 Let (G, φ) be a gain graph. Let φ be the gain function obtained from φ by a switching operation. Then, for any X ⊆ E (G) and u ∈ V (G), X φ ,u is conjugate to X φ,u . Proposition 2.2 Let (G, φ) be a gain graph. Then, for any forest F ⊆ E (G), there is a gain function φ equivalent to φ such that φ (e) = id for every e ∈ F . 2.2 Balanced and Cyclic Sets of Edges As we shall see, the subgroup X ψ,v itself will not be important, when we define our matroids induced by gains. We only need to know whether X ψ,v is trivial or not, or whether it is cyclic or not. We now introduce notions to describe these properties. Let (G, φ) be a gain graph. An edge subset F ⊆ E (G) is called balanced if F ψ,v is trivial for every v ∈ V (F ). Note that F is balanced if and only if every cycle in F is balanced. The latter property is the definition of the balancedness given by Zaslavsky [31]. In the same way, an edge subset F ⊆ E (G) is called cyclic if F ψ,v is cyclic for every v ∈ V (F ) (note that the terms balanced and cyclic are not exclusive). A gain graph (G, φ) is called balanced (resp. cyclic) if E (G) is balanced (resp. cyclic), respectively. Proposition 2.2 suggests a simple way to check the above introduced properties of X , in analogy with the fact that the cycle space of a graph is spanned by fundamental cycles. For a connected X ⊆ E (G), take a spanning tree T of the edge induced graph G[X ]. By Proposition 2.2 we can convert the gain function to an equivalent gain function such that φ (e) = id for all e ∈ T . Now consider any closed walk W ∈ π1(X, v), and denote W by W = v1v2, v2v3, . . . , vk vk+1, and let Wi = Pi ∗ {vi vi+1}∗ Pi−+11 for 1 ≤ i < k, where Pi denotes the path from v to vi in T . Then observe φ (W ) = φ (W1)·φ (W2) · · · φ (Wk ). By φ (e) = id for all e ∈ T , we deduce that φ (W ) is a product of elements in {φ (e) : e ∈ X \T }, implying that X φ,v ⊆ φ (e) : e ∈ X \T , where φ (e) : e ∈ X \T is the group generated by {φ (e) : e ∈ X \T }. Conversely, φ (e) is contained in X φ,v for all e ∈ X \T . Thus, X φ,v = φ (e) : e ∈ X \T . In particular, we proved the following. Lemma 2.3 For a connected X ⊆ E (G) and a spanning tree T of G[X ], suppose that φ (e) = id for all e ∈ T . Then, X φ,v = φ (e) : e ∈ X \T . In particular, the following hold. (i) X is unbalanced if and only if there is an edge in X \T whose gain is non-identity. (ii) X is cyclic if and only if all gains of X \T are contained in a cyclic subgroup of S. The following technical lemmas will be used in the proof of our main theorem. Lemma 2.4 Let (G, φ) be an S-gain graph, and X and Y be connected edge subsets such that the graph (V (X ) ∩ V (Y ), X ∩ Y ) is connected. (1) If X and Y are balanced, then X ∪ Y is balanced. (2) If X is balanced and Y is cyclic, then X ∪ Y is cyclic. (3) If X, Y are cyclic and X ∩ Y is unbalanced, then X ∪ Y is cyclic, provided that for every non-trivial cyclic subgroup C of S there is a unique maximal cyclic subgroup C¯ of S containing C. Proof Since the graph (V (X ) ∩ V (Y ), X ∩ Y ) is connected, there is a spanning tree T in G[X ∪ Y ] such that T ∩ X is a spanning tree of G[X ], T ∩ Y is a spanning tree of G[Y ], and T ∩ X ∩ Y is a spanning tree of G[X ∩ Y ]. By Proposition 2.2, there is a gain function φ equivalent to φ such that φ (e) = id for each e ∈ T . If X and Y are balanced, Lemma 2.3 implies that φ (e) = id for all e ∈ X ∪ Y . Thus (1) holds. If X is balanced, then every label in X ∪ Y is contained in Y φ ,v by Lemma 2.3, and hence X ∪ Y is cyclic if Y is cyclic. This implies (2). If X, Y are cyclic and X ∩ Y is unbalanced, then there is an edge e ∈ X ∩ Y for which φ (e) is non-identity. Let C be a cyclic subgroup of S generated by φ (e) and C¯ be the maximal cyclic subgroup containing C. Since X and Y are cyclic, Lemma 2.3 implies that φ (e) ∈ C¯ holds for every e ∈ X and for every e ∈ Y . Therefore X ∪ Y is cyclic. Lemma 2.5 Let (G, φ) be a gain graph, and X and Y be connected balanced edge subsets. If the number of connected components of the graph (V (X ) ∩ V (Y ), X ∩ Y ) is two, then X ∪ Y is cyclic. Proof We take a spanning tree T of G[X ∪ Y ] such that T ∩ X is a spanning tree of G[X ]. Since the number of connected components of (V (X ) ∩ V (Y ), X ∩ Y ) is two, T ∩ Y consists of two connected components, denoted T1 and T2. {V (T1), V (T2)} partitions Y into three subsets {Y1, Y2, Y3} such that Yi = {e ∈ Y : V ({e}) ⊆ V (Ti )} for i = 1, 2 and Y3 = Y \(Y1 ∪ Y2). By Proposition 2.2, we can take a gain function φ equivalent to φ such that φ (e) = id for e ∈ T . Since X and Y are balanced, we have φ (e) = id for e ∈ X ∪ Y1 ∪ Y2. Moreover, assuming that every edge in Y3 is oriented toward V (Y1), we have φ (e) = φ ( f ) for all e, f ∈ Y3, since otherwise T1 ∪ T2 ∪ {e, f } contains an unbalanced cycle, contradicting the fact that Y is balanced. Therefore X ∪ Y is cyclic. 3 Gain Count Matroids 3.1 Matroids Induced by Submodular Functions Let E be a finite set. A function μ : 2E → R is called submodular if μ(X ) + μ(Y ) ≥ μ(X ∪ Y ) + μ(X ∩ Y ) for every X, Y ⊆ E . μ is monotone if μ(X ) ≤ μ(Y ) for any X ⊆ Y . A monotone submodular function μ : 2E → Z induces a matroid on E , where F ⊆ E is independent if and only if |I | ≤ μ(I ) for every nonempty I ⊆ F . See e.g. [4, Sect. 13.4]. This matroid is denoted by M(μ). For a monotone submodular function μ, let ν = μ − 1. Then, ν is monotone submodular and induces the matroid M(ν). This matroid is referred to as the Dilworth truncation of M(μ). Although the details are omitted here, the name of Dilworth truncation is justified from a connection with Dilworth truncation for general matroids, see [4] for more details. Now we consider the union of two matroids induced by monotone submodular functions μ1 and μ2. Since monotonicity and submodularity are both preserved under the sum operation, μ1 + μ2 is monotone and submodular. In general, the union of M(μ1) and M(μ2) is not equal to M(μ1 + μ2). We do have equality in some special cases, for example, when μ1 = μ2 or when both μ1 and μ2 are nonnegative. As an example, consider the union of two copies of the graphic matroid of a graph G = (V , E ). It is the matroid induced by f2,2 defined by f2,2(F ) = 2|V (F )| − 2 on 3.2 Gain-Count Matroids In this paper we shall consider frame matroids on gain graphs. Let S be a group and (G, φ) be an S-gain graph. The frame matroid of (G, φ) is defined such that F ⊆ E is independent if and only if each connected component of F contains no cycle or just one cycle, which is unbalanced if it exists [32]. If we define gS : 2E → Z by where gS (F ) = (|V (Fi )| − 1 + αS (Fi )), 2E , as f2,2/2 induces the graphic matroid on G. The 2-dimensional generic rigidity matroid is the one induced by f2,2 − 1, and hence it is the Dilworth truncation of the union of two copies of the graphic matroid. In general, for a graph G = (V , E ) and two integers k and l with k ≥ 1 and l ≤ 2k − 1, let fk,l (F ) = k|V (F )| − l (F ⊆ E ). G is called (k, l)-sparse if |F | ≤ fk,l (F ) for any nonempty F ⊆ E . The matroid induced by fk,l is called the (k, l)-count matroid on G. If l ≥ 0, M( fk,l ) is indeed the one induced by fk,0, truncated l times. See e.g. [4] for more details. Below we shall apply the same construction to the union of some copies of a frame matroid to define gain-count matroids. (2) (3) (4) then the frame matroid is the matroid induced by gS . We omit the subscript S from αS if it is clear from the context. For an S-gain graph and two positive integers k and l with k ≤ l, we define gk,l : 2E → Z by gk,l (F ) = kgS (F ) − (l − k) (F ⊆ E ). We call the matroid M(gk,l ) induced by gk,l a (k, l)-gain-count matroid or g-count matroid for short. This matroid is the union of k copies of the frame matroid, followed by l − k Dilworth truncations. In this paper, we shall investigate the (2, 3)-g-count matroid and its variants. The independence of M(gk,l ) can be described in a compact form (see [8] for the proof, which is a rather straightforward calculation). Lemma 3.1 Let (G, φ) be an S-gain graph with G = (V , E ). Then E is independent in M(gk,l ) if and only if |F | ≤ k|V (F )| − l + kα(F ) for any nonempty F ⊆ E . In this sense, we may define (k, l)-gain-sparsity as in the case of (k, l)-sparsity of undirected graphs as follows. Definition 3.1 Let k and l be positive integers with k ≤ l and (G, φ) be an S-gain graph with a graph G = (V , E ) and a group S. An edge set X ⊆ E is called (k, l)gain-sparse (or (k, l)-g-sparse for short) if |F | ≤ gk,l (F ) for any nonempty F ⊆ X , i.e., • |F | ≤ k|V (F )| − l for every nonempty balanced F ⊆ X ; • |F | ≤ k|V (F )| − l + k for every nonempty unbalanced F ⊆ X , and it is called (k, l)-gain-tight (or (k, l)-g-tight for short) if it is (k, l)-g-sparse with |X | = gk,l (X ). If E is (k, l)-g-sparse then graph (G, φ) is said to be (k, l)-g-sparse, and (G, φ) is called maximum (k, l)-g-tight if it is (k, l)-g-sparse with |E | = k|V | − l + k. Remark 3.1 Note that the value of gk,l is invariant under switching operations, and thus the induced matroid is uniquely determined up to equivalence of gain functions. Remark 3.2 We can further consider the union of frame matroids of gain graphs (G, φ1) and (G, φ2) with the same underlying graph but distinct gain functions. We should remark that both graphic matroids and bicircular matroids are special cases of frame matroids. The union of copies of graphic, frame and bicircular matroids on an 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, φ), followed by a single Dilworth truncation, E (G) is independent if and only if |F | ≤ 2|V (F )| − 3 for any balanced set F ⊆ E (G) and |F | ≤ 2|V (F )| − 2 for any nonempty F ⊆ E (G). This matroid was used by Ross [17] for characterizing the generic rigidity of bar-and-joint frameworks on a torus. Tanigawa [24] proposed a more general class of matroids extending matroid union operations. 4 Constructive Characterization of Maximum (2, 3)-g-Tight Graphs 4.1 Operations Preserving (2, 3)-g-Sparsity In this section we define three operations, called extensions, that preserve (2, 3)-gsparsity. The first two operations generalize the well-known Henneberg operations [26,28] to gain graphs. Let (G, φ) be an S-gain graph. The 0-extension adds a new vertex v and two new non-loop edges e1 and e2 to G such that the new edges are incident to v and the other endvertices are two not necessarily distinct vertices of V (G). If e1 and e2 are not parallel then their labels can be arbitrary. Otherwise the labels are assigned such that φ (e1) = φ (e2), assuming that e1 and e2 are directed to v. The 1-extension first chooses an edge e and a vertex z, where e may be a loop and z may be an endvertex of e. It subdivides e, with a new vertex v and new edges e1, e2 such that the tail of e1 is the tail of e and the tail of e2 is the head of e. The labels of the new edges are assigned such that φ (e1) · φ (e2)−1 = φ (e). The 1-extension also adds a third edge e3 oriented to v. The label of e3 is assigned so that it is locally unbalanced, i.e., every two-cycle ei e j , if it exists, is unbalanced. The loop 1-extension adds a new vertex v to G and connects it to a vertex z ∈ V (G) by a new edge with any label. It also adds a new loop l incident to v with φ (l) = id (Fig. 1). The 0-extension and the 1-extension were already considered by Ross [17] for Z2gain graphs. In the covering graph each operation can be seen as a graph operation that preserves the underlying symmetry. Some of them can be recognized as performing socalled Henneberg operations [26,28] simultaneously. In the case of three-fold rotation symmetry, these operations are considered by Schulze [21]. Lemma 4.1 Let (G, φ) be a (2,3)-g-sparse graph. Applying the 0-extension, 1-extension or loop 1-extension to (G, φ) results in a (2,3)-g-sparse graph (G , φ ) with |V (G )| = |V (G)| + 1 and |E (G )| = |E (G)| + 2. Proof For a contradiction, suppose that G contains an edge set F ⊆ E (G ) for which |F | > 2|V (F )|−3+2α(F ). Let v be the new vertex added by the extension, and let Ev be the set of edges incident to v. Since E (G )\Ev ⊆ E (G), Ev ∩ F = ∅. In particular, v ∈ V (F ). Also, since the new labeling is assigned to be locally unbalanced, F is not contained in Ev. If G is constructed by a 1-extension then let e be the subdivided edge of G and let e1 and e2 be the resulting two new edges. Let F = F \Ev. If G is constructed by a 1-extension and {e1, e2} ⊆ F , then we further insert e to F . We then have |F | ≥ |F | − 2, |V (F )| = |V (F )| − 1, and α(F ) ≤ α(F ) in each case. These imply |F | ≥ |F | − 2 > 2|V (F )| − 5 + 2α(F ) ≥ 2|V (F )| − 3 + 2α(F ), contradicting the (2, 3)-g-sparsity of G as ∅ = F ⊆ E (G). We shall define the inverse moves of the operations above, which are called reductions. For a vertex v and two incoming non-loop edges e1 = (u, v) and e2 = (w, v), we denote by e1 · e2−1 a new edge from u to w with label φ (e1) · φ (e2)−1 (by extending φ). If u = w then e1 · e2−1 is a loop. Each reduction corresponds to one of the following operations on a gain graph (G, φ). A 0-reduction chooses a degree two vertex and deletes it from G. A 1-reduction chooses a vertex v with d(v) = 3 that is not incident to a loop. Let e1, e2, e3 be the edges incident to v. Without loss of generality we may assume that each ei is oriented to v. The 1-reduction deletes v with the incident edges and adds one of e1 · e2−1, e2 · e−1 and e3 · e1−1 as a new edge. 3 A loop 1-reduction chooses a vertex incident to exactly one loop and one non-loop edge and deletes the chosen vertex with the incident edges. A 1-reduction may destroy the (2, 3)-g-sparsity of a graph. We say that a reduction (at a vertex v) is admissible if the resulting graph is (2, 3)-g-sparse. 4.2 Constructive Characterization Lemma 4.2 Let (G, φ) be a (2,3)-g-sparse graph and v ∈ V (G) a vertex not incident to a loop with d(v) = 3. Then there is an admissible 1-reduction at v. Proof Let E = E (G), G = G − v and E = E (G ). Let e1, e2, e3 be the edges incident to v in G. Without loss of generality we may assume that each ei is oriented to v. For simplicity we put ei, j = ei · e −j1. Suppose for a contradiction that there is no admissible splitting at v, that is, none of E + e1,2, E + e2,3 and E + e3,1 is independent in M(g2,3). Equivalently, e1,2, e2,3, e3,1 ∈ clg(E ), where clg denotes the closure operator of M(g2,3). Let X = {e1, e2, e3, e1,2, e2,3, e3,1}. Claim 4.3 e1 ∈ clg(X − e1). Proof We split the proof into three cases depending on the cardinality of N (v). If |N (v)| = 3 then, by Proposition 2.2, we may assume φ (e1) = φ (e2) = φ (e3) = id. We then have φ (e1,2) = φ (e2,3) = φ (e3,1) = id. Therefore X forms a balanced K4, which is a circuit of M(g2,3). Thus, e1 ∈ clg(X − e1) holds. If |N (v)| = 2 then we may assume that e1 and e2 are parallel. By Proposition 2.2, we may assume that φ (e2) = φ (e3) = id. This implies φ (e1,3) = φ (e1) and φ (e2,3) = id. Since G is (2, 3)-g-sparse, we have φ (e1) = id by φ (e2) = φ (e3) = id, which implies that e1,2 is an unbalanced loop with φ (e1,2) = φ (e1). It can be easily checked, by counting, that X is indeed a circuit in M(g2,3). Thus, e1 ∈ clg(X − e1) holds. If |N (v)| = 1 then let X = {e1, e2, e3, e1,2}. We have |X | = 2|V (X )| and X is a circuit of M(g2,3). Therefore e1 ∈ clg(X − e1) ⊂ clg(X − e1). Since e1,2, e2,3, e3,1 ∈ clg(E ), by Claim 4.3, we have e1 ∈ clg(X −e1) ⊆ clg(E + X − e1) = clg(E + e2 + e3) = clg(E − e1), which contradicts the (2, 3)-g-sparsity of G. The following constructive characterization of maximum (2, 3)-g-tight graphs is a direct consequence of Lemmas 4.1 and 4.2 (see [8] for the concrete proof). Theorem 4.4 An S-gain graph (G, φ) is maximum (2,3)-g-tight if and only if it can be built up from an S-gain graph with one vertex and an unbalanced loop incident to it with a sequence of 0-extensions, 1-extensions, and loop-1-extensions. Remark 4.1 Theorem 4.4 for the case of three-fold rotation symmetry is implicit in [20]. For Z2-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 Symmetry-Forced Rigidity In this section we define the notion of symmetry-forced infinitesimal rigidity, introduced by Schulze and Whiteley [23]. In Sect. 5.1, we first introduce S-symmetric graphs, whose automorphism group has a subgroup isomorphic to S. In Sect. 5.2 we shall review the conventional notion of infinitesimal rigidity. In Sect. 5.3 we introduce symmetry-forced 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 symmetry-forced infinitesimal rigidity in the subsequent sections. In Sect. 5.5 we prove a necessary condition for symmetric frameworks to be symmetry-forced infinitesimally rigid. 5.1 S-Symmetric Graphs Let H be a simple graph. An automorphism of H is a permutation π : V (H ) → V (H ) such that {u, v} ∈ E (H ) if and only if {π(u), π(v)} ∈ E (H ). The set of all automorphisms of H forms a subgroup of the symmetric group of V (H ), known as the automorphism group Aut(H ) of H . Let S be a group. An action of S on H is a group homomorphism ρ : S → Aut(H ). An action ρ is called free if ρ(g)(v) = v for any v ∈ V and any non-identity g ∈ S. We say that a graph H is (S, ρ)-symmetric if S acts on H by ρ. If ρ is clear from the context, we will simply denote ρ(g)(v) by g · v or gv. Note that, for g ∈ S and u, v ∈ V , {u, v} ∈ E (H ) if and only if {gu, gv} ∈ E (H ), and hence there is an induced action of S on E (H ) defined by g · {u, v} = {gu, gv}. Let H be an (S, ρ)-symmetric graph. The quotient graph H/S of H is a multigraph on the set V (H )/S of vertex orbits, together with the set E (H )/S of edge orbits as the edge set. An edge orbit may be represented by a loop in H/S. Figure 2 is an example when S is a dihedral group. Different graphs may have the same quotient graph. However, if we assume that ρ is free, then a gain labeling makes the relation one-to-one. To see this, we arbitrarily choose a vertex v as a representative vertex from each vertex orbit. Then, each orbit is written by Sv = {gv : g ∈ S}. If ρ is a free action, an edge orbit connecting Su and Sv in H/S can be written by {{gu, ghv} : g ∈ S} for a unique h ∈ S. We then orient the edge orbit from Su to Sv in H/S and assign to it the gain h. In this way, we obtain the quotient S-gain graph, denoted (H/S, φ). Conversely, any S-gain graph (G, φ) can be “lifted” as an (S, ρ)-symmetric graph with a free action ρ. To see this, we simply denote the pair (g, v) of g ∈ S and v ∈ V (G) by gv. The covering graph (also known as the derived graph) of (G, φ) is the simple graph with vertex set S × V (G) = {gv : g ∈ S, v ∈ V (G)} and the edge r r id Cπ id set {{gu, gφ (e)v} : e = (u, v) ∈ E (G), g ∈ S}. Clearly, S freely acts on the covering graph, under which the quotient gain graph comes back to (G, φ). For more properties of covering graphs, see e.g. [5]. 5.2 Infinitesimal Rigidity Before we investigate the rigidity theory of symmetric graphs we review the basic notions of the conventional rigidity of graphs. A d-dimensional bar-and-joint framework (or simply a framework) is a pair (H, p) of a simple graph H and a mapping p : V (H ) → Rd , called a joint-configuration. We denote the set { p(v) : v ∈ V (H )} of points by p(H ). Infinitesimal rigidity is concerned with the dimension of the space of infinitesimal motions. An infinitesimal motion of a framework (H, p) is defined as an assignment m : V (H ) → Rd such that m(u) − m(v), p(u) − p(v) = 0 for all {u, v} ∈ E (H ), (5) where ·, · denotes the standard inner product in the d-dimensional Euclidean space. The set of infinitesimal motions forms a linear space, denoted L(H, p). In general, for a set P ⊆ Rd of points, an infinitesimal isometry of P is defined by m : P → Rd such that m(x ) − m(y), x − y = 0 for all x , y ∈ P. The set of infinitesimal isometries forms a linear space, denoted by iso( P). Notice that, for a skew-symmetric matrix S and t ∈ Rd , a mapping m : P → Rd defined by m(x ) = Sx + t (x ∈ P) is an infinitesimal isometry of P. Indeed, it is well-known that any infinitesimal isometry can be described in this form, and dim iso( P) = d(k + 1) − , where k denotes the affine dimension of P. Example 5.1 Let us consider the infinitesimal isometries of a point set P in the plane. According to (6), we have dim iso( P) = 3 if | P| ≥ 2, 2 if | P| = 1. m(v) = Sp(v) + t (v ∈ V (H )) For t ∈ R2, let mt (x ) = t (x ∈ P). Then, mt is an infinitesimal isometry, called a translation. On the other hand, let mr (x ) = Cπ/2x (x ∈ P), where Cπ/2 denotes the 2 × 2 orthogonal matrix representing the four-fold rotation around the origin. Then, mr is also an infinitesimal isometry, which we call an infinitesimal rotation. It is well known that iso( P) is spanned by {mt , mt , mr } for two linearly independent vectors t, t ∈ R2. See Fig. 3 for examples. An infinitesimal motion m : V (H ) → trivial if m can be expressed by Rd of a framework (H, p) is said to be (6) (7) for some skew-symmetric matrix S and t ∈ Rd . The set of all trivial motions forms a linear subspace of L(H, p), denoted by tri(H, p). By definition, tri(H, p) is isomorphic to iso( p(H )), and hence (6) gives the exact dimension of tri(H, p). (H, p) is called infinitesimally rigid if L(H, p) = tri(H, p). 5.3 Symmetric Frameworks and Symmetry-Forced Infinitesimal Rigidity A discrete point group (or simply a point group) is a finite discrete subgroup of O(Rd ), the orthogonal group of dimension d, i.e., the set of d × d orthogonal matrices over R. For d = 2, point groups are classified into two classes, groups Ck of k-fold rotations and dihedral groups D2k of order 2k. For a special case, D2 consists of a mirrorreflection and the identity. In the subsequent discussion of this section, S denotes a point group. Suppose that H is (S, ρ)-symmetric for a point group S. A joint-configuration p is said to be (S, ρ)-symmetric (or, simply, S-symmetric) if gp(v) = p(gv) for all g ∈ S and for all v ∈ V (H ). (8) Such a pair ( H, p) is called an (S, ρ)-symmetric framework (or simply an S-symmetric framework or a symmetric framework). We shall consider “symmetry-preserving” infinitesimal motions of symmetric frameworks. We say that an infinitesimal motion m : V ( H ) → Rd is symmetric if gm(v) = m(gv) for all g ∈ S and for all v ∈ V ( H ). (9) The set of S-symmetric infinitesimal motions and the set of trivial ones form linear subspaces of L ( H, p) and tri( H, p), denoted L S ( H, p) and triS ( H, p), respectively. We say that ( H, p) is symmetry-forced infinitesimally rigid if L S ( H, p) = triS ( H, p). A set P of points is called S-symmetric if g P = {gp : p ∈ P } = P for all g ∈ S. An infinitesimal isometry m : P → Rd of an S-symmetric point set P is called S-symmetric if gm(x ) = m(gx ) for all x ∈ P and g ∈ S. The set of S-symmetric infinitesimal isometries forms a linear subspace of iso( P ), denoted isoS ( P ). Clearly, triS ( H, p) is isomorphic to isoS ( p( H )). Example 5.2 Let us consider point groups in O(R2), which will be mainly discussed in Sects. 6 and 8. Let P be an S-symmetric point set in R2. See Fig. 3 for examples of Ck -symmetric infinitesimal isometries. In general, if | P | > 1, dim isoCk ( P ) = 3 if k = 1, 1 if k ≥ 2, and if P = { } x , dim isoCk ( P ) = 2 if k = 1, 0 if k ≥ 2 (where x should be the origin). Similarly, for the dihedral group D2k of order 2k, dim isoD2k ( P) = 1 if k = 1, 0 if k ≥ 2. A result of Schulze [21] motivates us to look at S-symmetric infinitesimal rigidity, which states that if (H, p) is not symmetry-forced infinitesimally rigid on an S-generic p, then (H, p) has a nontrivial continuous motion that preserves the (S, ρ)-symmetry. 5.4 The Orbit Rigidity Matrix Let (H, p) be an (S, ρ)-symmetric framework in Rd . Due to (9), the system (5) of linear equations (with respect to m) is redundant. Schulze and Whiteley [23] pointed out that the system can be reduced to |E (H )/S| linear equations. To see this, we first define a joint-configuration p˜ of vertex orbits by p˜ : V (H )/S → Rd . By taking a representative vertex v from each vertex orbit Sv, we set p˜(Sv) = p(v) [then, the locations of the other non-representative vertices are uniquely determined by (8)]. In a similar way, we define an infinitesimal motion of (H/S, p˜) by m˜ : V (H )/S → Rd . By using the representative vertices determined above, we fix a one-to-one correspondence between S-symmetric infinitesimal motions of V (H ) and infinitesimal motions of V (H )/S by m˜ (Sv) = m(v) for each vertex orbit Sv. Let (H/S, φ) be the quotient S-gain graph of H . Recall that each (oriented) edge orbit Se connecting Su and Sv with gain he can be written by Se = {{gu, ghev} : g ∈ S}. The system (5) is hence written by m(gu) − m(ghev), p(gu) − p(ghev) = 0 for all {gu, ghev} ∈ Se (10) over all edge orbits Se ∈ E (H )/S. Recall that the transpose of g is g−1 for any g ∈ O(Rd ). By (8) and (9), m(gu) − m(ghev), p(gu) − p(ghev) = m(u) − hem(v), p(u) − he p(v) = m(u), p(u) − he p(v) + m(v), p(v) − he−1 p(u) = m˜ (Su), p˜(Su) − he p˜(Sv) + m˜ (Sv), p˜(Sv) − he−1 p˜(Su) . Therefore, for p˜ : V (H )/S → Rd , a mapping m˜ : H/S → motion of (H/S, p˜) if and only if Rd is an infinitesimal m˜ (Su), p˜(Su) − he p˜(Sv) + m˜ (Sv), p˜(Sv) − he−1 p˜(Su) = 0 (11) for every oriented edge orbit Se with φ (Se) = he. By regarding (11) as a system of linear equations of m˜ , the corresponding |E (H )/S| × d|V (H )/S|-matrix is called the orbit rigidity matrix. In general, for an S-gain graph (G, φ) and p˜ : V → Rd , we shall define the orbit rigidity matrix as an |E (G)| × d|V (G)|-matrix, in which each row corresponds to an edge, each vertex is associated with a d-tuple of columns, and the row corresponding to e = (u, v) ∈ E (G) is written by u v 0 · · · 0 p˜(u) − φ (e) p˜(v) 0 · · · 0 p˜(v) − φ (e)−1 p˜(u) 0 · · · 0 if e is not a loop, and by v 0 · · · 0 (2Id − φ (e) − φ (e)−1) p˜(v) 0 · · · 0 if e is a loop. The orbit rigidity matrix of (G, φ, p˜) is denoted by O(G, φ, p˜). From the above discussion, it follows that the dimension of the S-symmetric infinitesimal motions can be computed from the rank of the orbit rigidity matrix of the corresponding quotient gain graph, which is formally stated as follows: Theorem 5.1 (Schulze and Whiteley [23]). Let (H, p) be an (S, ρ)-symmetric framework with a free action ρ. Then, dim LS (H, p) = d|V (H )/S| − rank O(H/S, φ, p˜), where (H/S, φ) is the quotient S-gain graph and p˜ is a joint-configuration of vertex orbits corresponding to p. 5.5 Necessary Condition for Symmetry-Forced Rigidity Combining some observations given in Sect. 2, we can show a necessary condition for the row independence of orbit rigidity matrices. Lemma 5.2 Let (G, φ) be an S-gain graph with underlying graph G = (V , E ), and let p : V → Rd . If O(G, φ, p) is row independent, then |F | ≤ {d|V (Fi )| − dim iso Fi φ,w ( p(Fi ))} Fi ∈C(F) for all F ⊆ E and w ∈ V (Fi ), where p(Fi ) = {gp(v) : v ∈ V (Fi ), g ∈ S}. Proof Let RF be the linear space spanned by the row vectors associated with F in O(G, φ, p). Observe that each non-zero entry of the row vector associated with e ∈ F is in the columns associated with V (F ). This means that RF is the direct sum of RF for F ∈ C (F ), and hence it suffices to check the statement for a connected F with V (F ) = V . Clearly, dim RF ≤ d|V |. Since |F | ≤ dim RF , we now show that dim R⊥F ≥ dim iso F φ,w ( p(F )), where R⊥F denotes the orthogonal complement of RF . To see this we first check that a switching operation does not change the rank of the orbit rigidity matrix. Let φ be the gain function obtained from φ by a switching operation at v0 with g0 ∈ S. We define p : V → Rd by p (u) = p(u) if u = v0, g0 p(u) if u = v0. Note that p (F ) = {gp (v) : v ∈ V , g ∈ S} = p(F ). We now show rank O(G, φ, p) = rank O(G, φ , p ). Let us consider a non-loop edge e = (u, v0) oriented to v0 in G. The row corresponding to e in O(G, φ , p ) is written by u v0 0 · · · 0 p (u) − φ (e) p (v0) 0 · · · 0 p (v0) − φ (e)−1 p (u) 0 · · · 0 By (1), we have φ (e) = φ (e)g0−1. Thus, by using (12), the row of e becomes u v0 0 · · · 0 p(u) − φ (e) p(v0) 0 · · · 0 g0( p(v0) − φ (e)−1 p(u)) 0 · · · 0 Similarly, for a non-loop edge e = (v0, u) oriented from v0 in G, the row of e becomes exactly the same form as above. By using the same calculation, for a loop e incident to v0 in G, the row of e in O(G , φ , p ) can be written as (12) (13) v0 0 · · · 0 g0(2Id − φ (e) − φ (e)−1) p(v0) 0 · · · 0 By performing column operations within the d columns associated with v0, these are converted to u v0 0 · · · 0 p(u) − φ (e) p(v0) 0 · · · 0 p(v0) − φ (e)−1 p(u) 0 · · · 0 and v0 0 · · · 0 (2Id − φ (e) − φ (e)−1) p(v0) 0 · · · 0 respectively, which implies that rank O(G, φ, p) = rank O(G, φ , p ). Therefore, the row independence of the orbit rigidity matrix is invariant under switching operations. Moreover, since p(F ) = p (F ), dim iso F φ,w ( p(F )) = dim iso F φ ,w ( p (F )). So it suffices to prove the statement for O(G, φ , p ). Let T be a spanning tree of G. Since we can freely apply switching operations, we may assume that φ (e) = id for all e ∈ T . Then, by Lemma 2.3, F φ,w = φ (e) : e ∈ F \T for a vertex w ∈ V (F ). Let us take any m ∈ iso F φ,w ( p(F )) and let m˜ : V → Rd be defined by m˜ (v) = m( p(v)) for v ∈ V . We show that m˜ is in the orthogonal complement of RF . To check it, let us consider any edge e = (u, v) ∈ F with gain h = φ (e). Since m ∈ iso( p(F )), we have p(u) − hp(v), m( p(u)) − m(hp(v)) = 0. Since m is F φ,w-symmetric, we also have m(hp(v)) = hm( p(v)). Therefore, we obtain 0 = p(u) − hp(v), m( p(u)) − m(hp(v)) = p(u) − hp(v), m˜ (u) − hm˜ (v) , implying that m˜ is in the orthogonal complement of RF . Consequently, dim R⊥F ≥ dim iso F φ,w ( p(F )), and hence |F | ≤ dim RF ≤ d|V | − dim iso F φ,w ( p(F )). This, together with Theorem 5.1, directly implies a necessary condition for symmetric frameworks to be symmetry-forced infinitesimally rigid. Recall that S is a finite family of orthogonal matrices. Let QS be the field generated by Q and the entries of all the matrices in S. Since S is finite, almost all numbers in R are transcendental over QS . For a given gain graph (G, φ), a mapping p˜ : V (G) → Rd is called S-generic if the set of coordinates of p˜(v) for all v ∈ V (G) is algebraically independent over QS . Similarly, for a given (S, ρ)-symmetric graph H , an (S, ρ)-symmetric joint-configuration p : V (H ) → Rd is called S-generic if the corresponding joint-configuration p˜ of the vertex orbits is S-generic. An S-symmetric framework is called S-generic if the joint configuration is 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 D2k with odd k, respectively. 6 Combinatorial Characterization of Generic Rigidity with Cyclic Symmetry In this section we shall prove a combinatorial characterization of the symmetry-forced rigidity of 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, φ) be an S-gain graph for a point group S ⊂ O(R2). Let (G , φ ) be an S-gain graph obtained from (G, φ) by a 0-extension, 1-extension, or loop1-extension. If there is a mapping p : V (G) → R2 such that O(G, φ, p) is row independent, then there is a mapping p : V (G ) → R2 such that O(G , φ , p ) is row independent. Proof If there is a p such that O(G, φ, p) is row independent, then O(G, φ, q) is row independent for all S-generic q. Hence, we may assume that p is S-generic. We only show the difficult case where (G , φ ) is constructed from (G, φ) by a 1-extension (see [8] for the easier case where (G , φ ) is constructed from (G, φ) by a 0-extension or a loop-1-extension). Suppose that (G , φ ) is obtained from (G, φ) by a 1-extension, by removing an existing edge e and adding a new vertex v with three new non-loop edges e1, e2, e3 incident to v. We may assume that ei is outgoing from v. Let ui be the other endvertex of ei , and let gi = φ (ei ) and pi = p(ui ) for i = 1, 2, 3. By the definition of 1-extension, we have φ (e) = g1−1g2. Claim 6.2 The three points gi pi (i = 1, 2, 3) do not lie on a line. Proof Since p is S-generic, u1 = u2 = u3 should hold if they lie on a line. Then p1 = p2 = p3. By the definition of 1-extensions, gi = g j if ui = u j . This implies that g1 p1, g2 p2, g3 p3 are three distinct points on a circle. Thus, they do not lie on a line. We take p : V (G ) → R2 such that p (w) = p(w) for all w ∈ V (G), and p (v) is a point on the line through g1 p1 and g2 p2 but is not equal to g1 p1 or g2 p2. O(G , φ , p ) is described as follows: if u1 = u2 e3 e1 e2 E (G) − e v u1 p (v) − g3 p3 ∗ p (v) − g1 p1 p1 − g1−1 p (v) p (v) − g2 p2 0 0 O(G − e, φ, p) u2 where the right-bottom block O(G −e, φ, p) denotes the orbit rigidity matrix obtained from O(G, φ, p) by removing the row of e, whereas, if u1 = u2, We consider the case when u1 = u2 (the case when u1 = u2 is similar). Since p (v) lies on the line through g1 p1 and g2 p2, p (v) − gi p(ui ) is a scalar multiple of g1 p1 − g2 p2 for i = 1, 2. Hence, by multiplying the rows of e1 and e2 by an appropriate scalar, O(G , φ , p ) becomes e3 e1 e2 E (G) − e v u1 p (v) − g3 p3 ∗ g1 p1 − g2 p2 −g1−1(g1 p1 − g2 p2) g1 p1 − g2 p2 0 0 u2 ∗ 0 ∗ 0 −g2−1(g1 p1 − g2 p2) 0 O(G − e, φ, p) C g2p2 g2p2 Let C be the circle whose center is the origin and which passes through g1 p1 (and hence through g3 p1). We split the proof into two cases depending on whether g3g1−1 is the two-fold rotation Cπ or not. (iv-1) Suppose that g3g1−1 = Cπ . Let C be a circle whose center is the origin and the diameter is much larger than that of C . We shall relocate g2 p2 on C such that g2 p2 is on the line through g1 p1 and the origin as shown in Fig. 10a. Then, if L and L are parallel, we have only two possible locations q and q for g4 p2 (as shown in Fig. 10a). Since the diameter of C can be arbitrarily large, D2k has no element that sends g2 p2 to q or q . In other words, if p is generic, L and L are not parallel. (iv-2) Suppose that g3g1−1 = Cπ . Then g3 p1 is the antipodal point of g1 p1 in C as shown in Fig. 10b. Let C be a circle whose center is the origin and the diameter is slightly larger than that of C . We shall relocate g2 p2 on C such that L is the tangent of C at g1 p1 (see Fig. 10b). Then, we have only two possible locations q and q for g4 p2 as L and L are parallel and g4 p2 is on C , where q is the antipodal point of g2 p2 with respect to the origin and q is the reflection of q2 p2 along the line parallel to L and through the origin. When p is generic, L is not parallel to any reflection lines in D2k , implying g4 p2 = q . Hence, g4 p2 = q. This means that g4g2−1 is also the two-fold rotation Cπ . Recall that Cπ is in the center of O(R2), i.e., gCπ = Cπ g for any g ∈ O(R2). Hence, by g3g1−1 = Cπ , we have g1−1g3 = g1−1Cπ g1 = Cπ . Symmetrically, by g4g2−1 = Cπ , we have g2−1g4 = Cπ . This however implies that g1−1g3 = g2−1g4, which contradicts (19). Following the statement of Claim 8.8, we shall split the proof into two cases. (Case 1) Suppose that L and L are not parallel. Let q be the intersection of L and L . By Claim 8.8(i), we have q = gi pi . We define p : V (G ) → R2 by p (w) = p(w) for w ∈ V (G) and p (v) = q for the added vertex v. Then, O(G , φ , p ) can be written as follows: e1 e2 e3 e4 E(G) − e − f where O(G − e − f, φ, p) is the matrix obtained from O(G, φ, p) by removing the rows of e and f . Consider the rows associated with e1 and e2. Since q is on L, q − gi pi is a scalar multiple of g1 p1 − g2 p2, and hence these two rows can be transformed to the following form by row operations: if u1 = u2 Notice that, in each case, the row of e2 is converted to that of e in O(G, φ, p). In a symmetric manner, the rows of e3 and e4 can be converted to the above form, simply by replacing 1 and 2 with 3 and 4, respectively. Thus, O(G , φ , p ) is converted to v e1 g1 p1 − g2 p2 e3 g3 p3 − g4 p4 E(G) 0 ∗ ∗ O(G, φ, p) The right-bottom block O(G, φ, p) is row independent while the left-top block is also row independent since L and L are not parallel. In other words, O(G , φ , p ) is row independent. (Case 2) Suppose that L and L are parallel. By Claim 8.8, L = L , p1 = p2, p3 = p4, and g−1g2 and g3−1g4 are reflections. Let q be any point on L with q = g1 p1 1 and q = g2 p1. We define p : V (G ) → R2 by p (w) = p(w) for w ∈ V (G) and p (v) = q for the new vertex v. Then, the orbit rigidity matrix is described as follows: e1 e2 e3 e4 E(G) − e − f and then to Since g1−1g2 is a reflection, we have g1−1g2 = g2−1g1. Hence, by adding the half of the second row to the first row, we obtain Since q is on the line L, q − gi pi is a scalar multiple of (g1 − g2) p1 for i = 1, 2. Hence, the rows of e1 and e2 can be converted to Next we consider the rows of e3 and e4. By subtracting the row of e3 from that of e4, we obtain Since L and L are parallel, {(g1 − g2) p1, (g3 − g4) p3} is linearly dependent. Thus, by subtracting the row of e1 from that of e4, we have Moreover, since g4−1g3 is a reflection, Lemma 8.3 implies that (I2 − g4−1g3)g3−1q is a scalar multiple of (I2 − g4−1g3) p3, and hence (g3−1 − g4−1)q is a scalar multiple of (I2 − g−1g3) p3. Therefore, by using g3−1g4 = g4−1g3, the row of e4 can be converted 4 by a scalar multiplication to v u1 u2 e4 0 0 (2I2 − g3−1g4 − g4−1g3) p3 0 In total, O(G , φ , p ) is changed to the following form by row-operations: e1 e3 e2 e4 E(G) − e − f v (g1 − g2)p1 q − g3 p3 0 0 0 u1 0 p3 − g3−1q (2I2 − g1−1g2 − g2−1g1)p1 0 u3 V (G) 0 0 0 0 0 0 (2I2 − g−1g4 − g4−1g3)p3 0 3 O(G − e − f, φ, p) The right-bottom block together with the rows of e2 and e4 forms O(G, φ, p), which is row independent. Also, since q is on L, but not on L , {(g1 − g2) p1, q − g3 p3} is linearly independent. Therefore, O(G , φ , p) is row independent. Combining Theorem 7.12, Lemmas 6.1, 8.1, 8.4, 8.5, and 8.7, we can now complete the proof of Theorem 8.2. 8.2 Symmetry-Forced Infinitesimal Motions with D2k-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 D2k with even k: for a generic (D2k , ρ)-symmetric framework (H, p) with even k and a free action ρ, (H, p) is symmetry-forced infinitesimally rigid if the quotient gain graph can be constructed from a disjoint union of base graphs by 0-extensions, 1-extensions, loop-1-extensions, 2-extensions and loop-2-extensions. However, as we have seen in Fig. 9, there are infinitely many gain graphs that cannot be constructed from base graphs. By Theorem 7.9, minimal examples are D2k -sparse double cycles Cn2. Below, we show that some of them indeed have symmetric infinitesimal motions. For Cn2, the vertex set is denoted by {1, . . . , n} and the edges of the 2-cycle between i and i + 1 (mod n) are denoted by ei,1 and ei,2 for i = 1, . . . , n. Theorem 8.9 Let D4 be the dihedral group of order 4, which consists of the identity I2, the two-fold rotation Cπ , and two reflections r and r . Let (G, φ) be a D4-sparse Cn2 such that • φ (ei,1) = id and φ (ei,2) = r for i = 1, . . . , n − 1; • φ (en,1) = Cπ and φ (en,2) = r . Then, for any D4-generic p : V (G) → R2, rank O(G, φ, p) = 2n if and only if n is odd. Proof Let p : i ∈ V (G) → (xi , yi ) ∈ R2 be a D4-generic mapping. Then Cπ p(i ) = (−xi , −yi ), r p(i ) = (−xi , yi ), r p(i ) = (xi , −yi ). The rows of O(G, φ, p) are as follows: i i + 1 i i + 1 ei,1 0 xi − xi+1 xi+1 − xi 0 0 yi − yi+1 yi+1 − yi 0 ei,2 0 xi − xi+1 xi+1 − xi 0 0 yi + yi+1 yi+1 + yi 0 n 1 n 1 en,1 0 xn + x1 x1 + xn 0 0 yn + y1 y1 + yn 0 en,2 0 xn + x1 x1 + xn 0 0 yn − y1 y1 − yn 0 and where the left and the right half sides correspond to x - and y-coordinates, respectively. For each i , we subtract the first row from the second row and then multiply the first row by an appropriate scalar. We then have, for each i = 1, . . . , n − 1, Cπ r r r id id id r r id Cπ r r id (a) (b) and In other words, O (G, φ , p) is converted to the following form: y2 y1 y3 y2 . . . . . . yn yn yn−1 y1 The determinant of this matrix is 2(1 − (−1)n−1) and only if n is even. n i=1 yi , which is equal to zero if See Fig. 11 for examples of frameworks given in Theorem 8.9. For n = 2, the covering graph is K4,4 and the corresponding framework is known as Bottema’s mechanism (see [23, Sect. 7.2.1]). 9 Proof of Theorem 7.8 In this section we prove Theorem 7.8. For simplicity of the description, a D-gain graph is called essential if it is D-sparse, 4-regular, not a base graph, and not a double cycle. Lemma 7.6 shows that 2-extensions and loop-2-extensions preserve D-sparsity, and hence what we have to prove is the following theorem. Theorem 9.1 Any essential D-gain graph (G, φ) has a vertex at which a 2-reduction or a loop-2-reduction is admissible. For simplicity, in the subsequent discussion we omit gain functions φ when referring to gain graphs if it is clear from the context. Also an edge (u, v) from u to v is simply denoted by uv, and a D-tight set is called a tight set. The proof of Theorem 9.1 consists of four parts. In Sect. 9.1, we shall prove useful lemmas for subsequent discussion. In Sect. 9.2, we prove Theorem 9.1 for the following graphs: • graphs consisting of only special vertices (Lemma 9.5), where a vertex is called special if it is incident with a loop or two parallel classes of edges; • graphs that are not 2-connected (Lemma 9.6), • “almost” near-cyclic graphs (Lemma 9.8), defined below, • graphs that are not essentially 4-edge-connected (Lemma 9.9), • graphs having a vertex v with |N (v)| = 2. In Sect. 9.3 we discuss graphs not belonging to the above classes. In Sect. 9.4 we put everything together to complete the proof of Theorem 9.1. 9.1 Preliminary Facts The following fundamental properties of 4-regular graphs will be frequently used. • A 4-regular graph is Eulerian. Hence, a 4-regular connected graph is 2-edgeconnected. • Let G = (V , E ) be a graph with maximum degree at most 4. Then, for any X ⊆ V , iG (X ) ≤ 2|X | − dG (X )/2 , where iG (X ) denotes the number of edges induced by X . In particular, if G is 4-regular, iG (X ) = 2|X | − dG (X )/2. The next lemma asserts that if the maximum degree is at most 4, then D-sparsity is equivalent to the following simpler properties: (C1) |F | ≤ 2|V (F )| − 3 for every nonempty balanced set F ⊆ E ; (C2) G is not cyclic. Lemma 9.2 Let G = (V , E ) be a D-gain graph with maximum vertex degree at most 4. If G is connected, then G is D-sparse if and only if (i) G is not 4-regular and condition (C1) is satisfied, or (ii) G is 4-regular and conditions (C1) and (C2) are satisfied. If G is not connected, then G is D-sparse if and only if each connected component is D-sparse. Proof If the maximum degree is at most 4, |F | ≤ 2|V (F )| for any F ⊆ E . In particular, if G is connected, we have |F | ≤ iG (V (F )) ≤ 2|V (F )|− dG (V (F ))/2 ≤ 2|V (F )| − 1 for any F ⊆ E with V (F ) = V . Therefore, |F | ≥ 2|V (F )| holds if and only if G is 4-regular and F = E . Thus, to prove Theorem 9.1, we shall investigate whether (C1) and (C2) are satisfied after the reductions. The next lemma will be used when (C2) is not satisfied. We say that (G, φ) is almost near-cyclic if there are two incident edges e and f such that G − e − f is cyclic. Lemma 9.3 Let (G, φ) be a connected 4-regular D-sparse graph with G = (V , E ) and v be a vertex in G that is not incident to a loop. Let e1, e2, e3, e4 be the edges incoming to v, and suppose that G − v + e1 · e2−1 + e3 · e4−1 is connected and cyclic. Then, there is an equivalent gain function φ to φ and a cyclic subgroup C of D such that • φ (e) ∈ C for every e ∈ E \{e3, e4}, and • φ (e3) ∈/ C¯ and φ (e4) ∈/ C¯. In particular, G is almost near-cyclic. Proof Let G = G − v + e1 · e2−1 + e3 · e4−1. Since G is connected and cyclic, by Lemma 2.3, there is an equivalent gain function φ to φ and a cyclic subgroup C of D such that φ (e) ∈ C for all e ∈ E (G ). Let a = φ (e1 · e2−1) ∈ C and a = φ (e3 · e4−1) ∈ C. Then, by using some elements b1, b2 ∈ D, we can express φ (ei ) by φ (e1) = ab1, φ (e2) = b1, φ (e3) = a b2, φ (e4) = b2. We further perform a switching operation at v with b1. We consequently have an equivalent gain function φ to φ such that φ (e1) = a, φ (e2) = id, φ (e3) = a b, φ (e4) = b, where b = b2b1−1. Notice that φ (e) ∈ C for all e ∈ E \{e3, e4}. Since G is not cyclic, we must have b ∈/ C¯, implying that φ (e3) ∈/ C¯ and φ (e4) ∈/ C¯. The following technical lemma is one of the key observations. A vertex in a 4regular graph is called special if it is incident with a loop or two parallel classes of edges with |N (v)| = 2. Lemma 9.4 Let (G, φ) be a connected 4-regular D-sparse graph with G = (V , E ) and |V | ≥ 3, v be a vertex in G that is not special, and e1, e2, e3, e4 be the edges incoming to v. If G − e3 − e4 or G − v + e1 · e2−1 + e3 · e4−1 is connected and cyclic, then at least one of the following holds: (a) G is near-cyclic. (b) G − v + e1 · e3−1 + e2 · e4−1 is D-sparse. (c) v is a cut-vertex in G and G − v + e1 · e3−1 + e2 · e4−1 is connected. Proof For simplicity, we denote ei, j = ei · e −j1 for i, j ∈ {1, 2, 3, 4}. We assume that neither (a) nor (b) occur and show that (c) holds. We claim that there are an equivalent gain function φ to φ and a cyclic subgroup C of D such that φ (e) ∈ C holds for e ∈ E \{e3, e4} and φ (e3) ∈/ C¯ and φ (e4) ∈/ C¯. To see this, first observe that if G − v + e1 · e2−1 + e3 · e4−1 is connected and cyclic, then Lemma 9.3 implies the claim. On the other hand, if G − e3 − e4 is connected and cyclic, then by Lemma 2.3, there is an equivalent φ to φ and a cyclic subgroup C of D such that φ (e) ∈ C for e ∈ E \{e3, e4}. Since G is neither cyclic nor near-cyclic, we have φ (e3) ∈/ C¯, and φ (e4) ∈/ C¯. Note that φ (e1,3) ∈/ C¯ and φ (e2,4) ∈/ C¯. Let us consider G − v. Since G − v is cyclic with |E (G − v)| = 2|V (G − v)| − 2, G − v is (2, 3)-g-sparse. Applying Lemma 7.7 with φ (e1,3) ∈/ C¯, we deduce that G − v + e1,3 is D-sparse. Let G = G − v + e1,3 + e2,4. Since (b) does not hold, G is not D-sparse. By Lemma 9.2, G (or a connected component of G ) violates (C1) or (C2). Case 1: If (C1) is violated, then G − v + e1,3 contains a balanced tight set F such that V (F ) contains the endvertices of e2,4 and F + e2,4 is balanced. Let s and t be the endvertices of e2,4, which are possibly the same vertex. By Lemma 7.1, if |F | > 1, F contains a path from s to t that does not pass through e1,3. Recall that the gain of each edge in this path is included in C, and the concatenation of the path and e2,4 forms an unbalanced closed walk in F + e2,4, contradicting that F + e2,4 is balanced. Therefore, |F | = 1 holds; in particular, since s, t ∈ V (F ) and F + e2,4 is balanced, it follows that F = {e1,3} and {e1,3, e2,4} forms a balanced 2-cycle 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 cut-vertex. Note that, since |E (G − v)| = 2|V (G − v)| − 2, G − v contains an unbalanced cycle C , whose gain is included in C. Let s and t be the endvertices of e2,4, which are possibly the same vertex. Since G − v is connected, there is a path P from s to a vertex in V (C ). We consider a closed walk W1 that first passes through P starting at s, then goes around C , and comes back to s through P−1. We then have φ (W1) ∈ C. Also, since G − v is connected, G − v has a path P connecting s and t . The concatenation of P with e2,4 forms a closed walk W2 starting at s with φ (W2) ∈/ C¯. Thus, {φ (W1), φ (W2)} generates a non-cyclic group. Hence, G satisfies (C2), a contradiction. Thus, v is a cut-vertex in G. Suppose that G is not connected. Then, by the 4-regularity of G, G consists of two connected components, denoted G1 and G2 with e1,3 ∈ E (G1) and e2,4 ∈ E (G2). We have already seen that G − v + e1,3 is D-sparse, and hence its subgraph G1 is D-sparse. However, since G1 is 4-regular, G1 is indeed maximum D-tight. By the symmetry between e1,3 and e2,4, G2 is also maximum D-tight, and thus G is maximum D-tight, a contradiction. Thus G should be connected, which implies (c). 9.2 Special Cases Recall that a vertex is said to be special if it is incident with a loop or two parallel classes of edges. A graph which consists of only special vertices is called a special graph. Special graphs are classified into the following three classes C 2, Cn◦ and P2 n n for n ≥ 2 (Fig. 12): As defined in Sect. 7.2, Cn2 is the graph obtained from the cycle of n vertices by replacing each edge by two parallel copies; Cn◦ is the cycle of n vertices, each of which is incident to a loop; Pn2 is the graph obtained from a path of n vertices by replacing each edge by two parallel copies and adding one loop to each endvertex of the path. Lemma 9.5 Let (G, φ) be an essential D-gain graph whose underlying graph G = (V , E ) is special. Then there is a vertex at which a 2-reduction or a loop-2-reduction is admissible. Proof Since (G, φ) is essential, the underlying graph is either Pn2 or C ◦. n Suppose that the underlying graph is P2. We perform the loop-2-reduction at a n vertex incident to a loop l. The resulting graph is P2 n−1 and clearly it satisfies (C1). If it does not satisfy (C2), then the resulting graph is cyclic and there is a cyclic subgroup C of D such that the gain of every cycle in G except for the loop l is in C. This in turn implies that G − l is cyclic, contradicting the assumption that G is essential. Suppose that the underlying graph is C ◦. We may assume n ≥ 3 since C2◦ = P2. n 2 We perform the 2-reduction at a vertex incident to a loop l. The resulting D-gain graph, denoted G , has the underlying graph Cn◦−1. If G does not satisfy (C2), then the gain of each cycle in G except for the loop l is included in a cyclic subgroup C of D, which again contradicts the fact that G is essential. It can be easily observed that G satisfies (C1) if n > 3. For n = 3, (C1) is violated if the 2-cycle of G is balanced, but in such a case the triangle in the original graph G is balanced, and G turns out to be a fancy triangle, contradicting the fact that G is essential. The next lemma solves the case when the graph is not 2-connected. Lemma 9.6 Let G = (V , E ) be a connected essential D-gain graph with |V | ≥ 2. Suppose that G is not 2-connected. Then a 2-reduction is admissible at some vertex. Proof By Lemma 9.5, we may assume that G is not equal to P|2V |. Then G has a cutvertex v which is not special. We show that a 2-reduction at v is admissible. Note that G −v consists of two connected components by the 4-regularity of G. Let e1, e7, e3, e4 be the edges incident to v, all of them are directed to v. From the 2-edge-connectivity of G, we can assume, without loss of generality, that the endvertices of e1 and e3 are included in a connected component of G − v while those of e2 and e4 are included in the other component. Consider the 2-reduction at v through (e1, e2) and (e3, e4). Let G be the resulting graph. Note that G is connected. Let us check that G satisfies (C1). To see this, recall that any balanced tight set consisting of more than one edge is 2-connected by Lemma 7.1. Note also that e3 · e4−1 is not parallel to e1 · e2−1 as v is not special. Since the endvertices of e3 · e4−1 belong to different connected components in G − v and e1 · e2−1 is the bridge in G − v + e1 · e2−1, G − v + e1 · e2−1 has no balanced tight set F such that V (F ) contains both endvertices of e3 · e4−1. This implies that G satisfies (C1). Therefore, if G satisfies (C2), then G is D-sparse by Lemma 9.2, and a 2-reduction 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 2-reduction at v through (e1, e3) and (e2, e4). The resulting graph, denoted by G , is disconnected. Lemma 9.4 thus implies that G is D-sparse (recall that G is not near-cyclic since G is essential). Thus, in the subsequent discussion, we may focus on 2-connected 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 2-connected essential D-gain graph. Suppose that G has a special vertex not incident to a loop. Then, G has a vertex at which a 2-reduction is admissible. Proof Let w be a special vertex not incident to a loop. By definition of special vertices, |N (w)| = 2 and w is incident to two parallel classes of edges. Since G = Cn2, G contains two adjacent vertices u and v such that v is not special and u is special and not incident to a loop (where u is possibly equal to w). Depending on the size of N ({u, v}), we have two possible cases as shown in Fig. 13. Let us denote the edges incident to u by e1, e2, e3, e4, where e1 and e2 are linking from v to u and e3 and e4 are linking from a vertex in V \{u, v} to u. We perform the 2reduction at u through (e1, e2) and (e3, e4). Since both new edges are unbalanced loops and adding unbalanced loops does not violate (C1), the resulting graph G satisfies (C1). Therefore, if the 2-reduction is not admissible at u, then G does not satisfy (C2), and hence G − e1 − e2 is cyclic by Lemma 9.3. Let a, b, c ∈ V such that N (v) = {u, a, b} and N (u) = {v, c}. Since |N (u)| = 2 with v ∈ N (u), without loss of generality we may assume a ∈/ N (u) (where b = c possibly holds). Recall that G − e1 − e2 is connected and cyclic, and hence we can apply Lemma 9.4 to deduce that the 2-reduction at v through (bv, e1) and (av, e2) is admissible. Indeed, since G is not near-cyclic and v is neither a cut-vertex nor a special vertex, Lemma 9.4 implies that this 2-reduction at v is admissible. The next lemma solves the case when G is almost near-cyclic. Lemma 9.8 Let G = (V , E ) be a 2-connected essential D-gain graph with at least two vertices. Suppose that G is almost near-cyclic. Then a 2-reduction or a loop-2reduction is admissible at some vertex in G. Proof Since G is almost near-cyclic, there are two edges e1 and e2 for which e1 and e2 are incident to a vertex v and G − e1 − e2 is cyclic. Suppose that v is not special. Then, since v is not a cut-vertex, a 2-reduction is admissible at v by Lemma 9.4. Therefore, let us consider the case when v is special. If v is not incident to a loop, then Lemma 9.7 directly implies the claim. We can thus assume that v is incident to a loop. Suppose that both e1 and e2 are non-loop edges. By Lemma 2.3, we may assume that the label of each edge in G − e1 − e2 is contained in a cyclic subgroup C of D. By further performing a switching operation at v with φ (e1), φ is converted such that φ (e1) = id and φ (e) ∈ C for all edges e not incident to v. This implies that if we remove e2 and the loop incident to v from G, the resulting graph is cyclic. In other words, it suffices to consider the case when e1 or e2 is a loop. We hence assume that e1 is the loop incident to v. Let e3 be the remaining non-loop edge incident to v, where φ (e3) ∈ C. Observe that the gain of the non-loop edge e2 is not included in C¯, since otherwise G −e1 becomes cyclic, contradicting the assumption that G is essential. Therefore, φ (e2 · e3−1) ∈/ C¯, and the loop-2-reduction at v adds the edge e2 · e3−1 to the cyclic (2, 3)-g-sparse graph G − v. By Lemma 7.7, the resulting gain graph is D-sparse. By using Lemma 9.8, we can now prove an important consequence for graphs that are not essentially 4-edge-connected. Lemma 9.9 Let G = (V , E ) be a 2-connected essential D-gain graph with |V | = n ≥ 4. Suppose that G is not essentially 4-edge-connected. Then, G has a vertex at which a 2-reduction or a loop-2-reduction is admissible. Proof Since G is 2-edge-connected and is not essentially 4-edge-connected, there exists a subset X of V for which |X | > 1, |V \X | > 1 and dG (X ) = 2. Since G is not Cn◦, we can suppose that B(X ) contains a vertex v not incident to a loop, where B(X ) denotes a set of vertices of X adjacent to some vertices of V \X . By the 2-connectivity, v is not a cut-vertex. Hence, denoting the four edges incident to v by e1, . . . , e4, we may assume that e1, e2, e3 are included in the subgraph induced by X while e4 is not. Note that v is a vertex of degree 3 in G − e4, and hence, by Lemma 7.5, a 1-reduction at v is admissible in G − e4. Without loss of generality, we may assume that G − v + e1 · e2−1 (obtained by a 1-reduction at v in G − e4) is D-sparse. We now consider adding e3 · e4−1 to G − v + e1 · e2−1 to complete the 2-reduction at v. Let G = G − v + e1 · e2−1 + e3 · e4−1, and suppose that G does not satisfy (C1). Since any balanced tight set F is 2-edge-connected if |F | > 1, there is no balanced tight set F for which V (F ) contains both endvertices of e3 · e4−1 unless |F | = 1. If G − v + e1 · e2−1 has a balanced set F such that |F | = 1 and V (F ) contains both endvertices of e3 · e4−1, then the edge in F , denoted by f , is incident to e3 and e4 and connects between X and V \X . However, since dG (X ) = 2, |X | > 1 and |V \X | > 1, the vertex incident to e4 and f turns out to be a cut-vertex of G, contradicting the 2-connectivity of G. Thus, G satisfies (C1). If G does not satisfy (C2), it is cyclic. By Lemma 9.3, G is almost near-cyclic, and we can apply Lemma 9.8 to conclude that a 2-reduction or a loop-2-reduction is admissible at some vertex v. The final special case is when G has a vertex v with |N (v)| = 2. Lemma 9.10 Let G = (V , E ) be a 2-connected essential 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 2-reduction 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 4-regularity, if |V | ≥ 4, G is not essentially-4-edge-connected, and thus Lemma 9.9 implies the statement. If |V | = 3, G is equal to the graph (shown in Fig. 15) of three vertices V = {u, v, w}, three parallel edges e1, e2, e3 between u and v, a loop l attached to w, and two remaining edges uw and vw, denoted by f1 and f2, respectively. We may assume φ ( f1) = φ ( f2) = id. Let C be the subgroup generated by φ (l). Since G is not cyclic, there is an unbalanced cycle whose gain is not included in C¯. If a triangle, say e1 f1 f2 has a gain not included in C¯, then the 2-reduction at u through (e1, f1) and (e2, e3) results in a D-sparse P2. Otherwise, removing e2 and e3 results in 2 a cyclic graph. Then G is almost near-cyclic, and Lemma 9.8 implies the statement. 9.3 The Remaining Cases In a graph G, the star of a vertex v means the subgraph of G whose vertex set is N (v) ∪ {v} and the edge set is the set of edges incident to v. A hat subgraph is a balanced subgraph whose underlying graph is a hat. See Fig. 14 for an example. The following claim, together with the previous lemmas, will complete the proof of Theorem 9.1. Theorem 9.11 Let G = (V , E ) be a 2-connected, essentially 2-edge-connected, and essential graph with |V | ≥ 3. Suppose also that G is not almost near-cyclic. Then, for every vertex v ∈ V that is not incident to a loop with |N (v)| ≥ 3, either a 2-reduction 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 2-reduction is not admissible then G has an obstacle around v. We will investigate Fig. 14 A hat subgraph id id id Fig. 15 The special graph given in the proof of Lemma 9.10 intersection properties of obstacles. The corresponding results for the case of |N (v)| = 3 will be given in Sect. 9.3.2. In Sect. 9.3.3, we prove Theorem 9.11 based on the intersection properties of obstacles. In the rest of this section, clD denotes the closure operator of the underlying matroid MD(G, φ). 9.3.1 Obstacles Around a Vertex v with |N (v)| = 4 Throughout Sect. 9.3.1, (G, φ) denotes a 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 Ev denotes the set of edges incident to v. An edge subset F is called sub-tight if |F | = 2|V (F )| − 4 and F is balanced. We first make a simple observation which describes the situation where 2-reductions are not admissible. Lemma 9.12 Suppose that the 2-reduction through (av, vb) and (cv, vd) is not admissible. Then there is an edge subset F ⊆ E \Ev satisfying one of the following properties: (i) F is balanced tight with a, b ∈ V (F ) and av · vb ∈ clD(F ); (ii) F is balanced tight with c, d ∈ V (F ) and cv · vd ∈ clD(F ); (iii) F is sub-tight with a, b, c, d ∈ V (F ), F + av · vb is balanced tight, and cv · vd ∈ clD(F + av · vb). Proof Let us first consider the graph G = G−v+av·vb. If G is not D-sparse, then, by Lemma 9.2, E \Ev has a balanced tight set F with a, b ∈ V (F ) and av · vb ∈ clD(F ), which satisfies property (i). Hence, let us assume that G is D-sparse. If G + cv · vd is cyclic, Lemma 9.3 implies that G is almost near-cyclic, contradicting the assumption that G is not almost near-cyclic. Therefore, G + cv · vd satisfies (C2). By Lemma 9.2, there exists a balanced tight set F ⊆ E \Ev ∪ {av · vb} with c, d ∈ V (F ) and cv · vd ∈ clD(F ). Depending on whether av · vb ∈ F or not, we find a desired subset of the statement; if av · vb ∈/ F then F is the one satisfying property (ii); otherwise F − av · vb satisfies property (iii) (we remark that, in the latter case, V (F − av · vb) contains a, b, c, d since F is 2-edge-connected). Since the first and the second cases of the statement of Lemma 9.12 are symmetric, we basically have two types of obstacles: for a vertex v and N (v) = {a, b, c, d}, F ⊆ E \Ev is called an obstacle of type 1 (for the 2-reduction through (av, vb) and (cv, vd)) if F satisfies (i) or (ii) of Lemma 9.12; F is called an obstacle of type 2 if F satisfies (iii). Lemma 9.13 Suppose that X is an obstacle of type 2 for the 2-reduction through (av, vb) and (cv, vd). Then, the following holds for X : • |X ∪ Ev| = 2|V (X ∪ Ev)| − 2; • There is an equivalent gain function φ to φ such that φ (e) = id for e ∈ X ∪ {va, vb}, and φ (vc) = φ (vd) = id; • X ∪ Ev is cyclic. Proof By definition, |X | = 2|V (X )| − 4, and hence |X ∪ Ev| = 2|V (X ∪ Ev)| − 2 by N (v) ⊆ V (X ). Since cv · vd ∈ clD(X + av · vb) and X + av · vb is balanced, X + av · vb + cv · vd is also balanced. Hence, by Lemma 2.3, there is an equivalent gain function φ to φ such that φ (e) = id for e ∈ X and φ (av · vb) = φ (cv · vd) = id. We thus have φ (av) = φ (bv) = g and φ (cv) = φ (dv) = g for some g, g ∈ D. By performing a switching operation at v with g if necessary, we may assume that φ (av) = φ (bv) = id and φ (cv) = φ (dv) = g g−1. If g g−1 = id, X ∪ Ev becomes a balanced set with |X ∪ Ev| > 2|V (X ∪ Ev)| − 3, contradicting the D-sparsity of G. Thus, φ (cv) = φ (dv) = id, and X ∪ Ev is cyclic. In the same manner we also have the following technical lemma. Lemma 9.14 Let X and Y be obstacles for the 2-reduction through (av, vb) and (cv, vd) and through (av, vc) and (bv, vd), respectively. Suppose that X is type 2 and X ∪ Y is cyclic. Then, X ∪ Y ∪ Ev is cyclic. Proof Since X is balanced and X ∪ Y is cyclic, for some cyclic subgroup C of D, there is an equivalent gain function φ to φ such that φ (e) = id for every e ∈ X and φ (e) ∈ C for every e ∈ Y by Lemma 2.3. Moreover, since X + av · vb and X + av · vb + cv · vd are balanced, we have φ (av · vb) = φ (cv · vd) = id. As in the previous proof, by applying a switching operation at v, we may assume that φ (va) = φ (vb) = id and φ (vc) = φ (vd). By the definition of the obstacles (whether type 1 or type 2), Y + Y + av · vc or Y + bv · vd is connected and balanced. Hence φ (av · vc) ∈ C¯ or φ (bv · vd) ∈ C¯, which implies φ (vc) = φ (vd) ∈ C¯. Thus, every label of X ∪ Y ∪ Ev is included in ¯. C The following lemmas describe different relations among obstacles. Lemma 9.15 Let X and Y be obstacles for the 2-reduction through (av, vb) and (cv, vd) and through (av, vc) and (bv, vd), respectively. If X ∩ Y = ∅, then X ∪ Y is not a balanced set. Proof Suppose for a contradiction that X ∪ Y is a balanced set with X ∩ Y = ∅. (Case 1) If both X and Y are of type 1, X ∪ Y is tight by Lemma 7.2 and hence |X ∪ Y | = 2|V (X ∪ Y )| − 3. Without loss of generality, we may assume that a, b, c ∈ V (X ∪ Y ), av · vb ∈ clD(X ) and av · vc ∈ clD(Y ). Since X ∪ Y is balanced, there is an equivalent gain function φ to φ such that φ (e) = id for e ∈ X ∪ Y . Moreover, since av · vb ∈ clD(X ) and av · vc ∈ clD(Y ), we have φ (av) = φ (bv) = φ (cv). This implies that X ∪ Y ∪ {av, bv, cv} is a balanced set. However, since |X ∪ Y ∪ {av, bv, cv}| > 2|V (X ∪ Y ∪ {av, bv, cv})| − 3, the existence of such a balanced set contradicts the 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 · vc or Y + bv · vd is balanced and 2-edgeconnected. Without loss of generality, we assume that Y + av · vc is balanced and 2-edge-connected. By Lemma 9.13, there exists an equivalent gain function φ to φ such that φ (e) = id for e ∈ X ∪ {va, vb} and φ (vc) = φ (vd) = id. Moreover, since X ∪ Y is balanced, we may assume that φ (e) = id for e ∈ Y . Since φ (av · vc) = id but φ (e) = id for e ∈ Y , Y + av · vc is unbalanced, a contradiction. Lemma 9.16 Let X and Y be obstacles for the 2-reductions through (av, vb) and (cv, vd) and through (av, vc) and (bv, vd), respectively. If |X | > 1 and |Y | > 1, then X ∩ Y = ∅. Proof Without loss of generality, we assume a ∈ V (X ) ∩ V (Y ). Recall that each balanced tight set is 2-connected if the size is more than one. By the 4-regularity of G, each vertex of N (v) has degree three in G − v. Hence, if X and Y are type 1 with |X | > 1 and |Y | > 1, then X ∩ Y contains an edge incident to a. If X is type 2, then X + av · vb is balanced tight with a, b, c, d ∈ V (X + av · vb) by definition. Hence, if Y is type 1, then X ∩ Y contains an edge incident to c or d. If both X and Y are type 2, then X ∩ Y contains an edge incident to d. Lemma 9.17 Let X , Y , Z be obstacles for the 2-reductions through (av, vb) and (cv, vd), through (av, vc) and (bv, vd), and through (av, vd) and (bv, vc), respectively. If there is no hat subgraph containing the star of v, then X ∩ Y = ∅, Y ∩ Z = ∅ or Z ∩ X = ∅ holds. Proof Note that a type 2 obstacle consists of more than one edge. If two of X, Y and Z are not singleton sets, then the lemma follows from Lemma 9.16. Hence we may assume that |Y | = |Z | = 1, and denote Y = {ey } and Z = {ez }. Clearly, ey = ez . (Case 1) Let us first consider the case when X is also a singleton set. Let X = {ex }. Depending on the relative position of ex , ey and ez , we have two situations: (I) ex , ey and ez share a vertex or (II) ex , ey and ez form a triangle. In case (I), the star of v is included in a hat subgraph. Indeed, if denoting without loss of generality ex = ab, ey = ac, and ez = ad, {ex , ey , ez , va, vb, vc, vd} forms a hat if it is balanced. Since X, Y and Z are obstacles, we have φ (ex ) = φ (av · vb), φ (ey ) = φ (av · vc) and φ (ez ) = φ (av · vd), and hence this subgraph is indeed balanced. In case (II), without loss of generality, we assume ex = ab, ey = bc and ez = ca. Then {ex , ey , ez , va, vb, vc} forms K4. Since φ (ex ) = φ (av · vb), φ (ey ) = φ (bv · vc) and φ (ez ) = φ (cv · va), this K4 does not have any unbalanced cycle. Therefore, Case (II) cannot happen because of the D-sparsity of G, as a balanced K4 is not 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 ∪ Ev| = 2|V (X ∪ Ev)| − 2 by Lemma 9.13. Also, by Lemma 9.13, there exists an equivalent gain function φ to φ such that φ (e) = id for e ∈ X ∪ {va, vb} and φ (vc) = φ (vd) = id. Denote φ (vc) by g. Since Y and Z are obstacles, we have φ (ey ) = φ (ez ) = g, which in particular implies ey , ez ∈/ X . By N (v) ⊆ V (X ) and ey = ez , |X ∪ Y ∪ Z ∪ Ev| = 2|V (X ∪ Y ∪ Z ∪ Ev)|, which in turn implies E = X ∪ Y ∪ Z ∪ Ev. Notice that the label of each edge in X ∪ Y ∪ Z ∪ Ev is either the identity or g. In other words, X ∪ Y ∪ Z ∪ Ev is cyclic, contradicting the D-sparsity of G. The remaining case is when X is type 1. Without loss of generality we assume a, b ∈ V (X ). By |X | > 1 and Lemma 7.1, dX (a) ≥ 2 and dX (b) ≥ 2. Since ey is either ac or bd and ez is either ad or bc, it suffices to consider the following two cases by symmetry: (i)(ey , ez ) = (ac, ad), and (ii)(ey , ez ) = (ac, bc). In subcase (i), X ∩ Y or X ∩ Z contains an edge incident to a as dX (a) ≥ 2 and dG−v(a) = 3. In subcase (ii), notice that, {av, bv, cv, ey , ez , av · vb} is a circuit of the underlying D-sparsity matroid since it forms a balanced K4. By av · vb ∈ clD(X ), we have cv ∈ clD(X + av + bv + ey + ez ) ⊆ clD(E − cv), contradicting the independence of E . Therefore, this case does not occur and the proof is complete. 9.3.2 Obstacles Around a Vertex v with |N (v)| = 3 In this subsection we shall investigate obstacles for a 2-reduction at a vertex v with |N (v)| = 3. Most of the arguments are similar to the previous subsection. Throughout Sect. 9.3.2, (G, φ) denotes a D-gain graph satisfying the assumptions of Theorem 9.11, v denotes a vertex with |N (v)| = 3, N (v) = {a, b, c}, and Ev denotes the set of edges incident to v. Without loss of generality, we assume that there are parallel edges e1 and e2 between v and a, and we denote Ev = {e1, e2, vb, vc}. We again have three possible ways for a 2-reduction at v. In each case, there exists an obstacle if the operation is not admissible. The proof of the following claim is identical to that of Lemma 9.12 and hence is omitted. Lemma 9.18 Suppose that the 2-reduction through (e1, vb) and (e2, vc) is not admissible. Then there is an edge subset F ⊆ E \Ev satisfying one of the following properties: (i) F is balanced tight with a, b ∈ V (F ) and e1 · vb ∈ clD(F ); (ii) F is balanced tight with a, c ∈ V (F ) and e2 · vc ∈ clD(F ); (iii) F is sub-tight with a, b, c ∈ V (F ), F + e1 · vb is balanced tight, and e2 · vc ∈ clD(F + e1 · vb). For the 2-reduction through (e1, e2) and (bv, vc), we encounter an even simpler situation. Lemma 9.19 Suppose that the 2-reduction through (e1, e2) and (bv, vc) is not admissible. Then there is a balanced tight set F ⊆ E \Ev with b, c ∈ V (F ) and bv · vc ∈ clD(F ). Proof Note that e1·e2−1 is a loop. G−v+e1·e2−1 is D-sparse by Lemma 9.2 since adding an unbalanced loop does not affect (C1). Note that G −v+e1 ·e2−1 +bv·vc is connected. If G−v+e1·e2−1+bv·vc does not satisfy (C2), then Lemma 9.3 implies that G is almost near-cyclic, which contradicts our assumption on G. If G − v + e1 · e2−1 + bv · vc does not satisfy (C1), then G − v + e1 · e2−1 contains a balanced tight set F with b, c ∈ V (F ) and bv · vc ∈ clD(F ). Since a balanced tight set does not contain a loop by Lemma 7.1, we have F ⊆ E \Ev. According to Lemmas 9.18 and 9.19, we can define the type of an obstacle as in the previous subsection. Lemma 9.19 also says that we only encounter type 1 obstacles for the 2-reduction through (e1, e2) and (bv, vc). The next two lemmas are counterparts of Lemmas 9.14 and 9.15, respectively, with identical proofs, which are omitted. Lemma 9.20 Let X and Y be obstacles for distinct 2-reductions at v. If X is type 2 and X ∪ Y is cyclic, then X ∪ Y ∪ Ev is cyclic. Lemma 9.21 Let X and Y be obstacles for distinct 2-reductions at v. Then, if X ∩ Y = ∅, then X ∪ Y is balanced. To prove the counterpart of Lemma 9.17, we need the following two additional lemmas. Lemma 9.22 Suppose that Z is an obstacle of type 1 for the 2-reduction through (e1, e2) and (bv, vc). Then, there is an equivalent gain function φ to φ such that φ (e) = id for e ∈ Z ∪ {vb, vc}. Proof Z + bv · vc is balanced. Hence, by Lemma 2.3, there is an equivalent gain function φ to φ such that φ (e) = id for e ∈ Z + bv · vc. By performing a switching operation at v with φ (bv) if necessary, we may assume that φ (bv) = φ (vc) = id. Lemma 9.23 Let X be an obstacle of type 2 for the 2-reduction through (e1, vb) and (e2, vc). Suppose further that there is no obstacle of type 1 for the 2-reduction through (e1, vb) and (e2, vc). Then dX (a) + dX (b) + dX (c) ≥ 5 holds. Proof Let X = X +e1·vb. By definition, X is balanced tight with a, b, c ∈ V (X ) and |X | > 1. Such a balanced tight set is 2-connected and essentially 3-edge-connected by Lemma 7.1. We thus have dX (u) ≥ 2 for u ∈ {a, b, c}. Suppose that dX (a) = dX (b) = 2. Since X is essentially 3-edge-connected and e1 ·vb is incident to a and b, X must be a triangle on a, b, c. This means that X contains an edge linking from a to c, denoted by e . Recall that X + e2 · vc is balanced by definition of type 2 obstacles. However, since e and e2 · vc are parallel, for X + e2 · vc to be balanced, {e , e2 · vc} has to be a balanced 2-cycle, that is, {e } is a type 1 obstacle for the 2-reduction through (e1, vb) and (e2, vc), contradicting the assumption of the lemma. Therefore, dX (a) ≥ 3 or dX (b) ≥ 3, implying dX (a) + dX (b) + dX (c) ≥ 7. Since X = X + e1 · vb, we obtain dX (a) + dX (b) + dX (c) ≥ 5. Lemma 9.24 Let X , Y , Z be obstacles for the 2-reductions through (e1, vb) and (e2, vc), through (e1, vc) and (e2, vb), and through (e1, e2) and (bv, vc), respectively. Then, X ∩ Y = ∅, Y ∩ Z = ∅, or Z ∩ X = ∅ holds. ey e1 e2 ex e1 e2 ex ey id v id id c b c b c Proof We split the proof into two cases depending on whether a type 1 obstacle exists for the 2-reduction through (e1, vb) and (e2, vc). (Case 1) Suppose that there is no type 1 obstacle for the 2-reduction through (e1, vb) and (e2, vc). Then, X is type 2. By Lemma 9.23, dX (a) + dX (b) + dX (c) ≥ 5 holds. If dX (a) ≥ 2, then X ∩ Y contains an edge incident to a since dG−v(a) = 2 and dY (a) ≥ 1. If dX (a) = 1, then we have dX (b) ≥ 2 and dX (c) ≥ 2. Since dG−v(b) = dG−v(c) = 3, |Z | = 1 holds if X ∩ Z = ∅. However, in this case, we have dX∪Z (b) = dX∪Z (c) = 3, and thus X ∩ Y or Y ∩ Z contains an edge incident to b or c. In a symmetric manner, we are done in the case when a type 1 obstacle does not exist for the 2-reduction through (e1, vc) and (e2, vb). (Case 2) We now consider the case when both X and Y are type 1. If |X | > 1 or |Y | > 1, then X or Y is 2-connected, and hence X ∩ Y contains an edge incident to a as dG−v(a) = 2. We thus assume |X | = |Y | = 1 and X = Y . Let us denote X = {ex } and Y = {ey }. Without loss of generality, we assume that ex connects from a to b. Also, by Lemma 9.22, we may assume φ (e) = id for e ∈ Z ∪ {vb, vc}. Since e1 · vb ∈ clD(X ), we have φ (ex ) = φ (e1 · vb) = φ (e1). The proof is completed by a further case analysis: (i) ey connects from a to c or (ii) ey connects from a to b (see Fig. 16). In case (i), we have e1 · vc ∈ clD(Y ) by definition. Therefore, φ (ey ) = φ (e1 · vc) = φ (e1). Notice that {e1, vb, vc, ex , ey , bv · vc} forms a K4 without unbalanced cycles by φ (ey ) = φ (e1) = φ (ex ). Moreover, since bv · vc ∈ clD(Z ), we obtain e1 ∈ clD({vb, vc, ex , ey , bv · vc}) ⊆ clD(E − e1). This contradicts the independence of E in the underlying D-sparsity matroid. Let us consider case (ii). If |Z | > 1, then X ∩ Z or Y ∩ Z contains an edge incident to b as Z is type 1 and dZ (b) ≥ 2. Suppose that |Z | = 1, X ∩ Y = ∅, X ∩ Z = ∅ and Y ∩ Z = ∅. Then X ∪ Y ∪ Z ∪ Ev induces a subgraph in which v, a and b have degree four. So, if |V | > 4, then c becomes a cut-vertex, contradicting the 2-connectivity of G. On the other hand, if |V | = 4, then G becomes the graph shown in Fig. 16(ii’). In this case removing e2 and ey results in a cyclic graph (where any cycle except the loop is balanced by φ (e1) = φ (ex )). This means that G is almost near-cyclic, a contradiction. 9.3.3 Proof of Theorem 9.11 Proof of Theorem 9.11 Suppose that no 2-reduction is admissible at v. Then we have three obstacles X , Y and Z for the three possible 2-reductions at v. Suppose further that the star of v is not contained in a hat subgraph. Then, by Lemmas 9.17 and 9.24, we may assume without loss of generality that X ∩ Y = ∅ holds. If |X ∪ Y | ≥ 2|V (X ∪ Y )| − 1, then V (X ∪ Y ) ∪ {v} = V must hold since G is essentially 4-edge-connected. We then have |X ∪ Y ∪ Ev| ≥ 2|V | + 1, contradicting the D-sparsity of G. Therefore we have |X ∪ Y | ≤ 2|V (X ∪ Y )| − 2. To derive a contradiction, we next show that the number of connected components in (V (X ) ∩ V (Y ), X ∩ Y ) is equal to two. To see this, let c0 be the number of trivial connected components (i.e., singleton vertex components) in (V (X ) ∩ V (Y ), X ∩ Y ) and let c1 be the number of nontrivial connected components in it. Then, |X | + |Y | ≥ 2|V (X )| − 4 + 2|V (Y )| − 4 = 2|V (X ∪ Y )| + 2|V (X ∩ Y )| + 2c0 − 8, (21) |X ∩ Y | ≤ 2|V (X ∩ Y )| − 3c1, where the last inequality comes from |F | ≤ 2|V (F )|−3 for any non-empty F ⊆ X ∩Y . From (20–22), we obtain 2c0 + 3c1 ≤ 6. On the other hand by X ∩ Y = ∅ we also have c1 ≥ 1. Hence we get c1 + c2 ≤ 2, and the number of connected components in the graph (V (X ) ∩ V (Y ), X ∩ Y ) is at most two. If the number of connected components in (V (X )∩ V (Y ), X ∩Y ) is one, then, since X and Y are connected and balanced, Lemma 2.4(1) implies that X ∪ Y is balanced, which contradicts Lemmas 9.15 and 9.21. Thus the number of connected components in (V (X ) ∩ V (Y ), X ∩ Y ) is two. Then 2c0 + 3c1 ≥ 5. Hence by (21) and (22) we have |X ∪ Y | ≥ 2|V (X ∪ Y )| − 3. Also by Lemma 2.5 X ∪ Y is cyclic. This implies that X ∪ Y is not tight, as X ∪ Y cannot be cyclic tight by (20). If both X and Y are type 1, then X ∪ Y is tight by Lemma 7.2, which does not happen. Hence X or Y is type 2, and Lemmas 9.14 and 9.20 imply that X ∪ Y ∪ Ev is also cyclic. Also by (23) and N (v) ⊆ X ∪ Y (as X or Y is type 2) we obtain |X ∪ Y ∪ Ev| ≥ 2|V (X ∪ Y ∪ Ev)| − 1. Thus, due to the essential 4-edge-connecitivity of G, |V (X ∪ Y ∪ Ev)| ≥ |V | − 1 must hold. If V (X ∪ Y ∪ Ev) = V , then |X ∪ Y ∪ Ev| = |E | − 1, and hence G is near cyclic, as X ∪ Y ∪ Ev is cyclic. On the other hand, if V (X ∪ Y ∪ Ev) = V − u for some u ∈ V , then u is incident to a loop and two non-loop edges by 4-regularity. Observe that removing this loop and one of the two non-loop edges results in a cyclic graph. This means that G is almost near-cyclic. In both cases G turns out to be almost near-cyclic, which contradicts the assumption on G. This completes the proof. (22) (23) 9.4 Proof of the Main Theorem We are now ready to prove Theorem 9.1, which also completes the proof of Theorem 7.8. Proof of Theorem 9.1 By Lemmas 9.5, 9.6, 9.8 and 9.9, we may assume that G is 2-connected, essentially 4-edge-connected, not special, and not almost near-cyclic. Also, by Lemma 9.10, we may assume that every vertex v with |N (v)| = 2 is incident to a loop. Since G is not special, G has a vertex v that is not incident to a loop. Then |N (v)| ≥ 3. By Theorem 9.11, either the 2-reduction at v is admissible or the star of v is contained in a hat subgraph H . Suppose the latter holds. We denote the vertices of H by a1, a2, b1, b2, b3, and assume that a1 and a2 have degree four in H (and hence a1 or a2 is v). Since H is balanced, we may assume that all labels in H are identity. Moreover, since G is not a fancy hat, we may assume that b1 is not incident to a loop. We prove that some 2-reduction at b1 is admissible. Suppose that no 2-reduction is admissible at b1. Then, by Theorem 9.11, the star of b1 is contained in a hat subgraph H . Note that H is different from H . We claim that H contains a triangle on b1, ai , b j for some i ∈ {1, 2} and j ∈ {2, 3}. To see this first suppose that a1a2 ∈/ E (H ). Then, since each vertex has degree at least 2 in H , we have a1b2 ∈ E (H ) or a1b3 ∈ E (H ) by NG (a1) = {a2, b1, b2, b3} and a1a2 ∈/ E (H ). This also implies b1b2 ∈ E (H ) or b1b3 ∈ E (H ), respectively, as b1 is incident to all the vertices of H . Thus H has a triangle on b1, a1, b j for some j ∈ {2, 3}. If a1a2 ∈ E (H ), then H contains a triangle on b1, a1, a2. In a hat subgraph, two vertices of each triangle have degree four, which implies N (ai ) ⊆ V (H ) for some i ∈ {1, 2}. Therefore, ai b2 ∈ E (H ) and b1b2 ∈ E (H ), and hence b1b2ai forms a triangle. Consequently, without loss of generality, we may assume that H contains a triangle on b1, b2, a1. Recall that a hat subgraph is balanced. Since φ (a1b1) = φ (a1b2) = id, we obtain φ (b1b2) = id as H contains a triangle on a1, b1, b2. Observe then that {a1, a2, b1, b2} induces a K4 in which the label of each edge is the identity. This contradicts the D-sparsity of G. Consequently, the 2-reduction at b1 is admissible. 10 Concluding Remarks The main results of this paper (Theorems 6.3 and 8.2) give rise to efficient algorithms for testing generic symmetric rigidity with rotation symmetry or dihedral symmetry D2k with odd k. This can be done by computing the rank of the quotient graphs in the corresponding matroids M(g2,3) or MD(G, φ). Here we briefly describe the main algorithmic ideas and show that testing independece in these matroids can be done in polynomial time. Let (G, φ) be a gain graph with G = (V , E ). First consider M(g2,3), in which E is independent if and only if (i) G is (2, 1)-sparse and (ii) every nonempty balanced subset F ⊆ E is (2, 3)-sparse, cf. Lemma 3.1. There exist efficient algorithms for testing (k, l)-sparsity for any pair of integers k, l, see e.g. [2,10], so checking (i) is easy. Observe that G satisfies (ii) if and only if every minimally non-(2, 3)-sparse graph (also called a (2, 3)-circuit or an M -circuit) is unbalanced. Suppose that G satisfies (i) and consider one of its M -components, i.e. a subgraph H of G induced by a connected component of the (2, 3)-sparsity matroid of G (see [2, 7] for more details on M -components). Each (2, 3)-circuit is a subgraph of some M -component, so we may deal with them separately. The key observation is that within H the complements of the (2, 3)-circuits are pairwise edge-disjoint. Since the M -components are pairwise edge-disjoint, this shows that the number of (2, 3)-circuits in G is O (n) and they can easily be enumerated. Then it remains to test whether each of these circuits is unbalanced, which can be done by switching and using Lemma 2.3 (similar arguments are given in [1]). Next consider MD(G, φ), in which E is independent if and only if (i) G is (2, 0)sparse and (ii) every cyclic subset F ⊆ E is (2, 1)-sparse, and (iii) every balanced subset F ⊆ E is (2, 3)-sparse. As above, testing (2, 0)-sparsity is easy. We can again observe that G satisfies (ii) if and only if every minimally non-(2, 1)-sparse graph (a (2, 1)-circuit) is non-cyclic. Suppose that G satisfies (i). Then it is easy to see that these circuits are edge-disjoint, 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 non-cyclic. So suppose G satisfies (ii) as well. As above, it remains to check whether every (2, 3)-circuit is unbalanced. Let H be an M -component of G. It is not hard to see that H − e is (2, 1)-sparse for all e ∈ E ( H ). 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Tibor Jordán, Viktória E. Kaszanitzky, Shin-ichi Tanigawa. Gain-Sparsity and Symmetry-Forced Rigidity in the Plane, Discrete & Computational Geometry, 2016, 314-372, DOI: 10.1007/s00454-015-9755-1