aa r X i v : . [ m a t h . C O ] A p r Generic Rigidity Matroids with Dilworth Truncations
Shin-ichi Tanigawa ∗ June 6, 2018
Abstract
We prove that the linear matroid that defines the generic rigidity of d -dimensional body-rod-bar frameworks (i.e., structures consisting of disjoint bodies and rods mutually linkedby bars) can be obtained from the union of (cid:0) d +12 (cid:1) copies of a graphic matroid by applyingvariants of Dilworth truncation operations n r times, where n r denotes the number of rods.This result leads to an alternative proof of Tay’s combinatorial characterizations of thegeneric rigidity of rod-bar frameworks and that of identified body-hinge frameworks. One of the main topics in rigidity theory is to reveal a combinatorial characterization of thegeneric rigidity of frameworks. Celebrated Laman’s theorem [18] asserts that a 2-dimensional bar-joint framework (Fig. 1(a)) is minimally rigid on a generic joint-configuration if and onlyif the graph G = ( V, E ) obtained by regarding each joint as a vertex and each bar as an edgesatisfies the following counting condition: | E | = 2 | V |− | F | ≤ | V ( F ) |− F ⊆ E , where V ( F ) denotes the set of vertices spanned by F . However, in spite of exhaustingefforts so far, the 3-dimensional counterpart has not been obtained yet (see, e.g.,[13, 41, 42]).A common strategy to deal with a difficult problem in graph theory is to restrict a graphclass, and several partial results are also known for the problem of characterizing 3-dimensionalgeneric rigidity, for, e.g., triangulations [9, 41], bipartite graphs [38], sparse graphs [13], someminor closed classes [25], the squares of graphs [16]. In rigidity theory, it is also reasonable toconsider special types of structural models. Tay [30] considered a body-bar framework (Fig. 1(b))that consists of rigid bodies linked by bars. He proved that, if we represent the underlying graphby identifying each vertex with each body and each edge with each bar, a body-bar frameworkis generically rigid in R if and only if the underlying graph contains six edge-disjoint spanningtrees. Tay [31, 32] and Whiteley [39] independently proved that, even for the body-hinge models(Fig. 1(c)), the same combinatorial characterization is true. Specifically, a body-hinge frameworkis a structure consisting of rigid bodies connected by hinges. Its underlying graph is representedby identifying each body with a vertex and each hinge with an edge. In this setting, Tay-Whiteley’s theorem asserts that a body-hinge framework is generically rigid in R if and onlyif the graph obtained by duplicating each edge by five parallel copies contains six edge-disjointspanning trees. Jackson and Jord´an [14] further discuss the relation of generic rigidity of the body-bar-hinge model to the forest-packing problem in undirected graphs.Although it is barely mentioned, Tay’s work was actually done in more general setting. An identified body-hinge framework is a body-hinge framework in which each hinge allows to connect ∗ Research Institute for Mathematical Sciences, Kyoto University. E-mail: [email protected] rod-bar frameworks .A rod-bar framework is a structure consisting of disjoint rods linked by bars in R (Fig. 1(d)).Each bar connects between two rods, and each rod is allowed to be incident to several distinctbars. This structural model naturally comes up from body-bar frameworks by regarding eachrod as a degenerated 1-dimensional body.Unfortunately, Tay’s proof is based on a Henneberg-type graph construction with intricateand long analysis (the combinatorial part now follows from the recent result by Frank andSzeg¨o [6]), and the combinatorics behind rigidity of rod-bar frameworks has not been under-stood well. To shed light on Tay’s result, this paper provides a new proof of the combinatorialcharacterization of rod-bar frameworks.We actually cope with a more general structural model, body-rod-bar frameworks , and provethat the linear matroid defining its generic rigidity is equal to a counting matroid defined onthe underlying graphs (Theorem 4.9 and Corollary 4.14). Our proof technique is inspired by theidea of Lov´asz and Yemini given in [21]. They proved, as a new proof of Laman’s theorem, thatthe linear matroid that defines the generic rigidity of 2-dimensional bar-joint frameworks can beobtained from the union of two copies of a graphic matroid by Dilworth truncation. Roughlyspeaking, Dilworth truncation is an operation to construct a new linear matroid from old one,by restricting the domain of entries of each vector to a generic hyperplane (see Subsection 2.4for the definition). The main difference between our situation and that of Lov´asz and Yemini isthat we need to apply such truncation operations more than once (while they used it only once).Indeed, it is not trivial to keep up the representation of the resulting matroid when applyingDilworth truncation operations several times, as each hyperplane must be inserted in “generic”position relative to the preceding hyperplanes. We will overcome the difficulty by extending anidea of Lov´asz [20] so that each truncation is performed within a designated subspace.A bar-joint framework can be considered as a body-bar framework consisting of 0-dimensionalbodies. As combinatorial properties of body-bar frameworks with 3-dimensional bodies arewell understood [30, 37, 39] in R , it is then natural to consider body-bar frameworks with 1-dimensional bodies (i.e., rods) towards a combinatorial characterization of bar-joint frameworks.Our proof explicitly describes how each 3-dimensional body can be replaced by a 1-dimensionalbody by the use of truncations.The paper is organized as follows. In Section 2, we first review (poly)matroids induced bysubmodular functions, and then review two classical techniques proposed by Lov´asz [20]: thefirst one shows how to obtain a maximum matroid from a polymatroid defined by a family of2ats in projective space, and the second one is Dilworth truncation. In Section 3, we providea proof of a combinatorial characterization of body-bar frameworks by Tay [30] from the viewpoint of matroids of flat families (discussed in Section 2). Our main result is Section 4, where weprove a combinatorial characterization of body-rod-bar frameworks. In Section 5, we will discussidentified body-hinge frameworks and several unsolved problems. As another application of theDilworth truncation, in Section 6, we provide a direct proof of the combinatorial characterizationof d -dimensional direction-rigidity given by Whiteley [41, Theorem 8.2.2]. We believe that ourproof technique is so powerful that it can be applied to more wide range of truncated matroidsappeared in combinatorial geometry (see, e.g., [41]).We conclude introduction by listing some notation used throughout the paper. For a vectorspace W = R k , let P ( W ) denote the projective space P k − associated with W . For a vector v =( v , . . . , v k ) ∈ W , the projective point associated with v is denoted by [ v ] = [ v , . . . , v k ] ∈ P ( W ).For a flat A in P ( W ), the rank of A is defined by rank( A ) = dim W ′ , where W ′ is the linearsubspace of W associated with A . For a finite family A of flats, the span of A is denoted by A . A is called disconnected if there is a partition {A , A } of A into nonempty subsets such thatrank( A ) = P i =1 , rank( A i ) (equivalently, A ∩ A = ∅ ). Otherwise A is said to be connected .(Note that a singleton set is connected.)We consider a finite graph G = ( V, E ) that may contain parallel edges but no loop. If G hasneither parallel edges nor a loop, G is said to be simple . We sometimes use notation V ( G ) and E ( G ) to denote the sets of vertices and edges of G , respectively. For v ∈ V , let δ G ( v ) be the setof edges incident to v in G . We say that F ⊆ E spans v ∈ V if v is incident to some edge of F .For F ⊆ E , V ( F ) denotes the set of vertices spanned by F . Let E be a finite set. A function µ : 2 E → R is called submodular if µ ( X ) + µ ( Y ) ≥ µ ( X ∪ Y ) + µ ( X ∩ Y ) for every X, Y ⊆ E . µ is called monotone if µ ( X ) ≤ µ ( Y ) for every X ⊆ Y .Suppose µ : 2 E → Z is an integer-valued function on E satisfying µ ( ∅ ) = 0. The pair ( E, µ )is called a polymatroid if µ is monotone and submodular, and µ is called the rank function of( E, µ ). It is particularly called a matroid if µ further satisfies µ ( e ) ≤ e ∈ E . F ⊆ E is called independent if | F | = µ ( F ), and a maximal independent set and a minimal dependentset are called a base and a circuit , respectively. An element e ∈ E is called a coloop if every basecontains e . Suppose µ : 2 E → Z is a monotone submodular function such that µ ( F ) ≥ F ⊆ E (but f ( ∅ ) < µ : 2 E → Z byˆ µ ( F ) = min { P ki =1 µ ( F i ) } ( F ⊆ E ) (1)where the minimum is taken over all partitions { F , . . . , F k } of F into nonempty subsets. It isknown that ˆ µ is a monotone submodular function satisfying ˆ µ ( ∅ ) = 0 (see, e.g.,[28, Chapter 48]or [7]), and hence the pair ( E, ˆ µ ) forms a polymatroid. It is also known that ˆ µ is the uniquelargest among all monotone submodular functions µ ′ satisfying 0 ≤ µ ′ ( F ) ≤ µ ( F ) for each F ⊆ E . 3dmonds and Rota [3] observed that a monotone submodular function µ : 2 E → Z induces a matroid ( E, r µ ) on E , where F ⊆ E is independent if and only if | F ′ | ≤ µ ( F ′ ) for everynonempty F ′ ⊆ F (see also [27]). Observe that this matroid takes the maximum rank amongthose satisfying r µ ( F ) ≤ min { µ ( F ) , | F |} for every nonempty F ⊆ E , and indeed the rankfunction r µ can be written as r µ ( F ) = min F ⊆ F {| F | + ˆ µ ( F \ F ) } ( F ⊆ E ) (2)(see, e.g.,[28, Section 44.6a]). Namely, r µ ( F ) = min {| F | + P ki =1 µ ( F i ) } ( F ⊆ E ) (3)where the minimum is taken over all partitions { F , F , . . . , F k } of F such that F , . . . , F k arenonempty (and F = ∅ is allowed). Geometric interpretations of these results will be discussedin the next two subsections. More detailed descriptions on general (poly)matroids can be foundin, e.g., [7, 26, 28]. Let E be a finite set. We associate each element e ∈ E with a flat A e in a real projective space,and let A = { A e : e ∈ E } . Also, for F ⊆ E , we denote { A e ∈ A : e ∈ F } by A F . If we define arank function rank A : 2 E → Z by rank A ( F ) = rank( A F ) for F ⊆ E , the pair ( E, rank A ) formsa linear polymatroid, which is denoted by PM ( A ). A polymatroid turns out to be a matroidby bounding the rank of each element by one. Below, we review a geometric method for gettinga maximum linear matroid from the linear polymatroid PM ( A ).We shall associate a representative point x e ∈ A e with each A e ∈ A . Let us denote { x e : e ∈ E } by X . The set X of representative points is said to be in generic position if, for every X ′ ⊆ X and for every x e ∈ X ′ , x e ∈ X ′ − x e ⇒ A e ⊆ X ′ − x e . (4)It is not difficult to see that, for any finite flat family A , the set X of representative points canbe taken to be in generic position; for any x e ∈ X , A e \ S { X ′ : X ′ ⊆ X − x e with A e X ′ } forms a dense open subset of A e ; hence, if x e ∈ X ′ for some X ′ ⊆ X − x e with A e X ′ , then bycontinuously (and slightly) moving x e on A e it can avoid X ′ without creating a new violationfor generic position.For F ⊆ E , the dimension of the linear subspace spanned by { x e : e ∈ F } is definedas the rank of F (with respect to X ), and we denote it by rank X ( F ), i.e., rank X ( F ) =rank( { x e : e ∈ F } ). The linear matroid ( E, rank X ) is called a matroid associated with A . Theorem 2.1 (Lov´asz [20]) . Let A = { A e : e ∈ E } be a finite family of flats, and X be a set ofrepresentative points of A in generic position. Then, rank X ( E ) = min F ⊆ E {| E \ F | + rank( A F ) } . (5)By restricting the argument to F ⊆ E , we also have rank X ( F ) = min F ′ ⊆ F {| F \ F ′ | +rank( A F ′ ) } . The rank of the linear matroid associated with A does depend on the choice of X .However, Theorem 2.1 implies that it attains the maximum and is invariant when X is in genericposition. (Notice that “ ≤ ” direction of (5) holds even though X is not in generic position; Forany F ⊆ E , rank X ( E ) ≤ rank X ( E \ F ) + rank X ( F ) ≤ | E \ F | + rank( { A e ∈ A : e ∈ F } ).) Thismotivates us to define the generic matroid. The generic matroid associated with A , denoted M ( A ), is defined to be M ( A ) = ( E, rank X ) with X in generic position.4 .4 Dilworth truncation Let A be a finite set of flats. We now consider restricting flats of A to a generic hyperplane.A hyperplane H is called generic relative to A if it satisfies the following condition ; for any A , A ∈ A and any F ⊆ { A ∩ H : A ∈ A} ,( A ∩ H ) ∪ F ∩ ( A ∩ H ) ∪ F 6 = F ⇒ A ∪ F ∩ A ∪ F 6⊂ H. (6)Although the detail is omitted, it can be verified that almost all hyperplanes are generic relativeto A . For a family A of flats and a hyperplane H , we shall abbreviate { A ∩ H : A ∈ A} as A ∩ H . The following result is also done by Lov´asz [20]. Theorem 2.2 (Lov´asz [20]) . Let A be a finite family of flats in a real projective space, and H be a generic hyperplane relative to A . Then, rank( A ∩ H ) = min { P ki =1 (rank( A i ) − } , (7) where the minimum is taken over all partitions {A , . . . , A k } of A into nonempty subsets. This operation (of restricting flats to a generic hyperplane) is referred to as
Dilworth trunca-tion . Indeed, as noted in [28], Theorem 2.1 and Theorem 2.2 provide geometric interpretationsof the formulae (1) and (3) for linear polymatroids.The same result was also obtained by Mason [22, 23] from the view point of combinatorialgeometry (projective matroids). The papers of Mason [22, 23] include examples of Dilworthtruncation. M -connectivity and P -connectivity Let M = ( E, r ) be a matroid on a finite set E with the rank function r . A subset F ⊆ E is called M -connected if, for any pair e, e ′ ∈ F , F has a circuit of M that contains e and e ′ .For simplicity of the description, a singleton { e } is also considered as an M -connected set. Amaximal M -connected set is called an M -connected component . It is well know that the union oftwo M -connected sets is M -connected if their intersection is nonempty, and thus E is uniquelypartitioned into f -connected components E , . . . , E k (see, e.g.,[26, Chapter 4]). Since there isno circuit intersecting two components, we have r ( E ) = P ki =1 r ( E i ). Alternatively, we can useit for the definition of M -connectivity: F ⊆ E is M -connected if and only if there no partition { F , . . . , F k } of F into at least two nonempty subsets such that r ( F ) = P ki =1 r ( F i ).The concept of connectivity can be extended to polymatroids. Let PM = ( E, µ ) be apolymatroid on a finite set E . Then, F ⊆ E is said to be P -connected if there is no partition { F , . . . , F k } of F into at least two nonempty subsets such that µ ( F ) = P ki =1 µ ( F i ). A maximal P -connected set is called a P -connected component . The union of two P -connected sets is P -connected if their intersection is nonempty, and thus E is uniquely partitioned into P -connected Lov´asz claimed Theorem 2.2 with a much weaker assumption; he defined that a hyperplane H is generic if, forany subsets X, Y and Z of A satisfying ( X ∩ H ) ∪ Y ∩ ( X ∩ H ) ∪ Z ⊆ H , we have ( X ∩ H ) ∪ Y ∩ ( X ∩ H ) ∪ Z ⊆ X ∩ H . Theorem 2.2 however fails in this setting. For example, suppose the underlying projective space is 3-dimensional, and A consists of three distinct hyperplanes { A , A , A } such that A ∩ A = A ∩ A = A ∩ A isa line l . If we take H as a hyperplane distinct from A i but containing l , H satisfies the condition to be generic.However, the left hand side of (7) is rank( { A ∩ H : A ∈ A} ) = rank( l ) = 2 while the right hand side is equal torank( { A , A , A } ) − P -connectivity coincides withthe connectivity of flats we introduced in the introduction.A P -connected set (and similarly, an M -connected set) is called trivial if it is singleton;otherwise nontrivial. A body-bar framework is a structure consisting of rigid bodies linked by bars (Figure 1(b)). Thegeneric rigidity of body-bar frameworks is characterized by Tay [30] (and a simpler proof wasgiven by Whiteley [39]). In this section, we shall present a proof of this characterization fromthe viewpoint of matroids of flat families. In the subsequent sections, d denotes the dimensionof frameworks, and let D = (cid:0) d +12 (cid:1) . We first review the union of copies of graphic matroid to which Tay related the generic rigiditymatroid in the body-bar model.
Let G = ( V, E ) be a finite undirected graph. We denote the graphic matroid of G by G ( G ), thatis, the matroid induced by the monotone submodular function g : 2 E → Z defined by g ( F ) = | V ( F ) | − F ⊆ E . Namely, F ⊆ E is independent in G ( G ) if and only if | F | ≤ | V ( F ) | − F ⊆ E , and equivalently F is a forest.Let I ( G ) = [ a ij ] be the incidence matrix of a digraph obtained from G by arbitrary assigninga direction to each edge, i.e, a ij = v j is the tail of arc e i − v j is the head of arc e i . It is well known that G ( G ) is linear as it is represented by the row matroid of I ( G ). For a matroid M = ( E, I ) with a collection I of independent sets, the union of D independentsets, i.e., { I ∪ · · · ∪ I D : I i ∈ I , i = 1 , . . . , D } , again forms the collection of independent setsof a matroid. This matroid is called the union of D copies of M . In the union of D copies ofthe graphic matroid, denoted D G ( G ), F ⊆ E is independent if and only if F can be partitionedinto D edge-disjoint forests. D G ( G ) is indeed the matroid induced by the monotone submodularfunction Dg := D ( | V ( · ) | −
1) defined on E [24]. This implies that E can be partitioned into D edge-disjoint spanning trees if and only if | E | = D ( | V | −
1) and | F | ≤ D ( | V ( F ) | −
1) for anynonempty F ⊆ E .It is also known that D G ( G ) can be represented as a row vector matroid by introducingindeterminates. For each integer k with 1 ≤ k ≤ D , let I k = [ a kij ] be a | E | × | V | -matrix defined6y a kij = α ke i if vertex v j is the tail of arc e i − α ke i if vertex v j is the head of arc e i , where α ke ’s are algebraically independent indeterminates over Q . Denote the | E | × D | V | -matrix[ I | I | . . . | I D ] by DI ( G ). Then, D G ( G ) is represented by DI ( G ) (see, e.g., [23, 39]).This representation gives us another way to look at D G ( G ). We associate a D -dimensionalvector space V u = R D with each vertex u in the subsequent discussion, and V V denotes the directproduct of V u for all u ∈ V . In DI ( G ), the row associated with an edge e = uv is representedby (0 , ··· · · · , , u z }| { α e , . . . , α De , , ··· · · · , , v z }| { − α e , . . . , − α De , , ··· · · · , , (8)where we changed the column ordering so that the entries associated with each vertex forma block (and throughout the subsequent discussions we will refer to this ordering). Whenlooking α e , . . . , α De as independent parameters in R , the space spanned by vectors (8) form a D -dimensional vector space contained in V u × V v . We can identify this D -dimensional vector spacewith a ( D − P ( V V ). We denote this flat by A e and let A := { A e : e ∈ E } .Then, D G ( G ) can be considered as the generic matroid M ( A ) associated with A . Throughout the paper, let W = R d +1 . For simplicity, we shall use the standard basis e , . . . , e d +1 of W = R d +1 and use the dot product as an inner product. Also W is identified with its dual.Recall that the exterior product V k W of degree k is a (cid:0) d +1 k (cid:1) -dimensional vector space andcan be naturally identified with R ( d +12 ) by associating e i ∧ · · · ∧ e i k with an element of thestandard basis of R ( d +12 ) for each 1 ≤ i < · · · < i k < d + 1. In particular, V W = R D .The collection of k -dimensional subspaces in W is called the Grassmannian , denoted Gr ( k, W ).The Pl¨ucker embedding p : Gr ( k, W ) → P ( V k W ) is a bijection between k -dimensional vectorspaces X ∈ Gr ( k, W ) and projective equivalence classes [ v ∧ · · · ∧ v k ] ∈ P ( V k W ) of decom-posable elements, where { v , . . . , v k } is a basis of X . In the subsequent discussions, we shallidentify Gr ( k, W ) and its image of the Pl¨ucker embedding, and regard Gr ( k, W ) as a subset of P ( V k W ).It is well-known that each point of Gr ( k, W ) can be coordinatized by the so-called Pl¨uckercoordinate once we fix a basis of W . If a basis { v , . . . , v k } of X ∈ Gr ( k, W ) is represented by v i = P d +1 j =1 p ij e j with the k × ( d + 1)-matrix P = [ p ij ], then we have v ∧ · · · ∧ v k = P i < ···
0] : [ α ] ∈ P D − } ⊆ P ( V V ) . (12)In order to prove B ( G ) = M ( A ), it is sufficient to show that the representative point x e of A e (that defines M ( A )) can be taken to be in general position fromˆ A e = { [0 , ··· · · · , , u α , , ··· · · · , , v − α , , ··· · · · ,
0] : [ α ] ∈ Gr (2 , W ) } ⊆ P ( V V ) . (13)9pecifically, we need to show that there exists X = { x e ∈ ˆ A e : e ∈ E } such that, for each X ′ ⊆ X and x e ∈ X ′ , x e ∈ X ′ − x e ⇒ A e ⊆ X ′ − x e , (c.f. (4)). Let us consider the case d = 3 (and D = 6). For e = uv ∈ E , let us pick a point x e = [0 , ··· · · · , , u z }| { x e , . . . , x e , , ··· · · · , , v z }| { − x e , . . . , − x e , , ··· · · · , ∈ A e . Then, x e ∈ ˆ A e if and only if x e x e − x e x e + x e x e = 0. We now focus on a 5-dimensional affine space A by setting x e = 1. Note that Gr (2 , W ) ∩ A is a smooth 4-dimensional manifold parameterizedby x e , x e , x e , x e since x e = − x e x e + x e x e .Let us take x e so that the set of parameters x e , x e , x e , x e for all e ∈ E is algebraicallyindependent over Q . Suppose, for a contradiction, that x e ∈ X ′ − x e but A e X ′ − x e for some e = uv . Let us consider a hyperplane H of P ( V V ) that contains X ′ − x e but does not contain A e . We can take such a hyperplane H so that each coefficient is written as a polynomial of { x e ′ , x e ′ , x e ′ , x e ′ : e ′ ∈ E − e } over Q . Moreover, H ∩ ˆ A e is a lower-dimensional subspace of ˆ A e since Gr (2 , W ) is quadratic and irreducible. In particular, H does not contain ˆ A e . Therefore, if x e ∈ H , then { x e , x e , x e , x e : e ∈ E } satisfies a nontrivial algebraic relation over Q , contradictingthe choice of x e .The general d -dimensional case follows in the same way based on the following fact. If Gr (2 , W ) is restricted to a ( D − A by fixing one coordinate, then Gr (2 , W ) ∩ A is known to be a smooth 2( d − Gr (2 , W ) ∩ A is written as a rational function of 2( d −
1) parameterswith coefficients in Q . Thus, we can apply the exactly same argument. We now provide our main result on the generic rigidity of body-rod-bar frameworks. We firstintroduce a counting matroid defined on graphs in Subsection 4.1, and then in Subsection 4.2 weshow that generic rigidity of body-rod-bar frameworks can be characterized by the combinatorialmatroid.
Let G = ( V, E ) be a graph with an (ordered) partition P = { B, R } of V into two subsets (where B and R will represent a set of bodies and a set of rods, respectively, in the next subsection).We define an integer-valued function f on E defined by f ( F ) = D ( | V ( F ) | − − | R ( F ) | ( F ⊆ E ) , (14)where R ( F ) denotes the set of vertices in R spanned by F , and D = (cid:0) d +12 (cid:1) as in Section 3.Then, f is a monotone submodular function on E , since f ( F ) = D | B ( F ) | + ( D − | R ( F ) | − D and | B ( · ) | and | R ( · ) | are both monotone and submodular. Thus, f induces the matroid ( E, r f )on E , denoted M f ( G, P ). If the bipartition P is clear from the context, we abbreviate it andsimply denote M f ( G ). This matroid is a special case of so-called count matroids on undirectedgraphs, see e.g., [4, Section 13.5] for more detail.10 f ◦ G Figure 2: Example of f ◦ G for D = 3, where circles and squares represent vertices of R and B ,respectively.We denote by f ◦ G the graph obtained from G by replacing each edge e by f ( e ) parallelcopies of e (see Figure 2). Also, f ◦ e denotes the set of corresponding copies of e , and let f ◦ F = S e ∈ F f ◦ e . We can naturally extend f to that on f ◦ E by setting f ( F ) = D | V ( F ) | − D − | R ( F ) | for F ⊆ f ◦ E .Let us consider ˆ f : 2 E → Z defined by (1), i.e., for F ⊆ E ,ˆ f ( F ) = min { P ki =1 ( D ( | V ( F i ) | − − | R ( F i ) | ) : a partition { F , . . . , F k } of F } . (15)As mentioned in Subsection 2.1, ˆ f is a monotone submodular function satisfying f ( ∅ ) = 0,and thus ( E, ˆ f ) forms a polymatroid, denoted by PM f ( G, P ) (or simply by PM f ( G )). Thefollowing lemma implies that PM f ( G ) is essentially the same as M f ( f ◦ G ). Lemma 4.1.
For any F ⊆ E , ˆ f ( F ) = r f ( f ◦ F ) . Namely, the rank of F ⊆ E in PM f ( G ) isequal to the rank of f ◦ F in M f ( f ◦ G ) .Proof. Recall that, for any F ⊆ E , r f ( f ◦ F ) is written as r f ( f ◦ F ) = min {| F | + P ki =1 f ( F i ) } ,where the minimum is taken over partitions { F , F , . . . , F k } of f ◦ F such that F , . . . , F k = ∅ (see(3)). Let { F ∗ , F ∗ , . . . , F ∗ k } be a partition of f ◦ F that attains that minimum. Since | f ◦ e | = f ( e )for every e ∈ E , we may assume F ∗ = ∅ . Also, since f ( f ◦ F ) = f ( F ) for any F ⊆ E , we mayassume that each F ∗ i ( ⊆ f ◦ F ) is written as F ∗ i = f ◦ F ′ i for some F ′ i ⊆ F . Thus, r f ( f ◦ F ) isactually written as r f ( f ◦ F ) = min { P ki =1 f ( f ◦ F ′ i ) } = min { P ki =1 f ( F ′ i ) } , where the minimumis taken over all partitions { F ′ , . . . , F ′ k } of F . This is exactly the definition of ˆ f ( F ).A reduction technique of general polymatroids to matroids can be found in, e.g.,[28, Sec-tion 44.6b]. M f We now show several properties of M f ( G, P ) for a graph G = ( V, E ) with a bipartition P = { B, R } of V . (These lemmas are generally known for count matroids. We provide proofs for thecompleteness.) Lemma 4.2.
Let C be a circuit of M f ( G ) . Then, r f ( C ) = f ( C ) .Proof. Since C is a minimal dependent set, | C | > f ( C ) and | C | − | C − e | ≤ f ( C − e ) ≤ f ( C )for any e ∈ C . This implies | C | = f ( C ) + 1. Thus, r f ( C ) = | C | − f ( C ). Lemma 4.3.
Let F ⊆ E be a nontrivial M -connected set in M f ( G ) . Then, r f ( F ) = f ( F ) . roof. Suppose r f ( F ) < f ( F ). Then, there are u, v ∈ V ( F ) with uv / ∈ F such that r f ( F + uv ) = r f ( F ) + 1. Let us take two distinct edges e and e ′ of F incident to u and v , respectively. (It iseasy to see that such two edges exist since F is M -connected.) Since F is M -connected, thereis a circuit C ⊆ F that contains e and e ′ . Then, by Lemma 4.2 and by f ( C + uv ) = f ( C ), weobtain r f ( C + uv ) ≤ f ( C + uv ) = f ( C ) = r f ( C ), implying r f ( C + uv ) = r f ( C ). In other words, uv is contained in the closure of C . This contradicts r f ( F + uv ) = r f ( F ) + 1. Lemma 4.4.
Let F ⊆ E be a nontrivial M -connected set in M f ( G ) . Then, the closure of F ,that is, { e ∈ E ( G ) : r f ( F + e ) = r f ( F ) } , is the set of edges induced by V ( F ) . In particular, if F is an M -connected component, then ( V ( F ) , F ) is an induced subgraph.Proof. Since f ( F + e ) = f ( F ) holds for any edge e induced by V ( F ), the claim follows fromLemma 4.3. PM f Let us consider M f ( f ◦ G ) for a graph G = ( V, E ) with a bipartition P . By Lemma 4.4, an M -connected component C of M f ( f ◦ G ) is either trivial or of the form C = f ◦ F for some F ⊆ E with | F | ≥
2. The M -connected component decomposition of M f ( f ◦ G ) thus induces a uniquepartition { C , . . . , C k } of E such that C i is singleton or f ◦ C i is an M -connected componentin M f ( f ◦ G ). The following lemma says that this partition coincides with the P -connectedcomponent decomposition of PM f ( G ). Lemma 4.5.
For a graph G = ( V, E ) with a bipartition P = { B, R } of V , the following holds: (i) Any nontrivial M -connected component X of M f ( f ◦ G ) can be written as X = f ◦ F forsome nontrivial P -connected component F ⊆ E . (ii) If F ⊆ E is a nontrivial P -connected set in PM f ( G ) , then f ◦ F is M -connected in M f ( G ) . (iii) The P -connected component decomposition { C , . . . , C k } of PM f ( G ) is a minimizer of theright hand side of (15).Proof. (i) and (ii) are direct consequences of Lemma 4.1.For the last claim, Lemma 4.1, Lemma 4.4 and (ii) imply ˆ f ( E ) = P ki =1 ˆ f ( C i ) = P ki =1 r f ( f ◦ C i ) = P ki =1 f ( f ◦ C i ) = P ki =1 f ( C i ).For a simple graph G = ( V, E ), it is sometimes useful to introduce the underlying completesimple graph K ( V ) on V that contains G , and extend PM f ( G, P ) to PM f ( K ( V ) , P ). We shalldenote by cl the closure operator of PM f ( K ( V ) , P ), i.e., cl( F ) = { uv ∈ K ( V ) : ˆ f ( F + uv ) =ˆ f ( F ) } for F ⊆ E . Then, by Lemma 4.5, cl( F ) forms the complete graph on V ( F ) if F is P -connected.The following lemmas are key observations used in the proof of main theorem (Theorem 4.12). Lemma 4.6.
Let G = ( V, E ) be a connected simple graph with a bipartition P = { B, R } of V . Suppose D ≥ . Then G has (i) three vertices each of which is spanned by exactly two P -connected components of PM f ( G ) or (ii) a vertex that is spanned by only one P -connectedcomponent. roof. Let { C , . . . , C k } be the P -connected component decomposition of PM f ( G ). Note then,since G is simple, any nontrivial P -connected component C i satisfies | V ( C i ) | ≥ C i , we consider the following graph operation on G , called the simpli-fication of C i ; remove C i , insert a new vertex v c to B , and connect each vertex of V ( C i ) with v c . Namely, we replace the induced subgraph ( V ( C i ) , C i ) by the star ( V ( C i ) ∪ { v c } , S ) with thecentered new vertex v c and the set S of edges between v c and V ( C i ) (see Figure 3). v c Figure 3: Simplification.
Claim 4.7.
Let C be a nontrivial P -connected component of PM f ( G, P ) . Let G ′ be the graphobtained by the simplification of C , where we denote V ( G ′ ) = V ∪ { v c } and E ( G ′ ) = ( E \ C ) ∪ S ,with the bipartition P ′ = { B ∪ { v c } , R } of V ( G ′ ) . Then, each new edge e ∈ f ◦ S is a coloop in M f ( f ◦ G ′ , P ′ ) .Proof. From the definition of f , it is easy to check that f ◦ S is independent in M f ( f ◦ G ′ , P ′ ).Since C is a P -connected component, we have cl( S ) ∩ cl( C ′ ) = ∅ for any other P -connectedcomponent C ′ of PM f ( G, P ). This implies that there is no circuit of M f ( f ◦ G ′ , P ′ ) thatintersects both f ◦ ( E \ C ) and f ◦ S . Since f ◦ S is independent, there is also no circuit within f ◦ S and thus no circuit that contains e ∈ f ◦ S in M f ( f ◦ G ′ , P ′ ).Claim 4.7 implies that, if we apply the simplification of the P -connected component C i ,then no new nontrivial P -connected component appears, and C , . . . , C i − , C i +1 , . . . , C k are allnontrivial P -connected components in the resulting polymatroid. Hence, we may apply thesimplifications for all C , . . . , C k simultaneously. Let G ′′ be the resulting graph with the corre-sponding bipartition P ′′ of V ( G ′′ ) after the simplifications. Notice that the degree of each vertex v ∈ V ( G ) in G ′′ corresponds to the number of P -connected components among C , . . . , C k thatspan v in G . We also remark that each vertex of V ( G ′′ ) \ V ( G ) has degree at least three since | V ( C i ) | ≥
3. Thus, to complete the proof, it is sufficient to show that G ′′ has at least threevertices of degree 2 or a vertex of degree 1. To see this, observe that f ◦ E ( G ′′ ) is indepen-dent in M f ( f ◦ G ′′ , P ′′ ) by Claim 4.7. So, we have | f ◦ E ( G ′′ ) | = r f ( f ◦ E ( G ′′ )). This implies( D − | E ( G ′′ ) | ≤ P e ∈ E ( G ′′ ) f ( e ) = | f ◦ E ( G ′′ ) | = r f ( f ◦ E ( G ′′ )) ≤ D | V ( G ′′ ) | − D . Let d avg bethe average degree of G ′′ . Then, we have d avg = | E ( G ′′ ) || V ( G ′′ ) | ≤ DD − (cid:16) − | V ( G ′′ ) | (cid:17) . Suppose there is no vertex of degree 1. Denoting the set of vertices of degree 2 in G ′′ by V , wehave d avg ≥ − | V || V ( G ′′ ) | . Putting them together, we obtain | V | ≥ DD − + D − D − | V ( G ′′ ) | ≥ DD − + D − D − = 3 . (where we used D ≥ | V ( G ′′ ) | ≥ emark. Lemma 4.6 does not hold for d = 2 and D = 3. For example, in the cube graph,all P -connected components are trivial and hence each vertex is spanned by three P -connectedcomponents since each vertex has degree 3. Lemma 4.8.
Let G = ( V, E ) be a simple graph for which E is P -connected in PM f ( G, P ) .Suppose further that there are two disjoint nonempty P -connected sets C and C both of whichspan a vertex u ∈ V . Then, G contains a P -connected set C such that C ⊆ C ⊆ E \ C and uv ∈ cl( C ) ∩ cl( E \ C ) for some uv ∈ K ( V ) .Proof. Let us take an inclusion-wise maximal P -connected set C such that C ⊆ C ⊆ E \ C .Since E is P -connected, we have cl( C ) ∩ cl( E \ C ) = ∅ , and hence there is an edge vw ∈ K ( V )such that vw ∈ cl( C ) ∩ cl( E \ C ). If either v = u or w = u , then C satisfies the required property.Thus, suppose contrary that every edge in cl( C ) ∩ cl( E \ C ) is not incident to u . Let C ′ be a P -connected set in E \ C with vw ∈ cl( C ′ ). Since vw ∈ cl( E \ C ), such C ′ exist. ( C ′ = { vw } may hold if vw ∈ E \ C .)If C ∩ C ′ = ∅ , then C ∪ C ′ is P -connected, and hence cl( C ∪ C ′ ) forms the complete graphon V ( C ∪ C ′ ). Since u ∈ V ( C ) and v ∈ V ( C ′ ), we obtain uv ∈ cl( C ∪ C ′ ) ⊆ cl( E \ C ). On theother hand, since C is P -connected with u, v ∈ V ( C ), we also have uv ∈ cl( C ). These howevercontradicts that every edge in cl( C ) ∩ cl( E \ C ) is not incident to u If C ∩ C ′ = ∅ , then C ∪ C ′ is P -connected since cl( C ) ∩ cl( C ′ ) is nonempty, and thus C ∪ C ′ is P -connected with C ⊆ C ∪ C ′ ⊆ E \ C and is larger than C , contradicting the choice of C . A body-rod-bar framework is a body-bar framework in which some of bodies are degenerateas 1-dimensional bodies in the case of d = 3. In general dimensional case, a body-rod-barframework can be defined as a structure consisting of d -dimensional subspaces (bodies) and( d − d = 3.) We thus define a body-rod-bar framework as( G, q , r ), where • G = ( V, E ) is a graph with a bipartition P = { B, R } of V ; • r is a mapping called a rod-configuration : r : R → Gr ( d − , W ) ⊆ P ( V d − W ) v [ r v ] = [ r v , . . . , r Dv ] • q is a bar-configuration : q : R → Gr (2 , W ) ⊆ P ( V W ) e [ q e ] = [ q e , . . . , q De ]satisfying the incidence condition : q e · r v = 0 if e ∈ E is incident to v ∈ R. (16)14amely, r ( v ) represents a rod associated with v ∈ R , and [ r v ] denotes the Pl¨ucker coordinate ofthe rod. Recall that, for [ q ] ∈ Gr (2 , W ) and [ r ] ∈ Gr ( d − , W ), q · r = 0 holds if and only if thecorresponding linear subspaces have nonzero intersection (equivalently, the corresponding flatshave a nonempty intersection). Thus, the system (16) describes incidence constraints betweenrods and bars. Throughout the subsequent discussions, we also impose an additional conditionthat all rods are distinct, i.e., r ( u ) = r ( v ) for any u, v ∈ R with u = v .As in the case of body-bar frameworks, an infinitesimal motion of ( G, q , r ) is defined as m : V → V d − W satisfying (10), and m is called trivial if m ( u ) = m ( v ) for all u, v ∈ V .For each v ∈ R , define m v : V → V d − W by m v ( v ) = r v and m v ( u ) = 0 for u ∈ V \ { v } .Then, by incidence condition (16), m v always satisfies (10), and m v is an infinitesimal motionof ( G, q , r ). Conventionally, we also include m v in the set of trivial motions. The set of alltrivial motions thus forms a ( D + | R | )-dimensional vector space. If every motion of ( G, q , r ) istrivial, it is said to be infinitesimally rigid . As defined in the body-bar matroid, the body-rod-bar matroid BR ( G, q , r ) is defined as that on E whose rank is the maximum size of independent linear equations in (10) (for unknown m ).From the definition, ( G, q , r ) is infinitesimally rigid if and only if the rank of BR ( G, q , r ) is D | V | − ( D + | R | ). The following theorem is our main result. Theorem 4.9.
Let G = ( V, E ) be a graph with a bipartition P = { B, R } of V and f be thefunction defined by (14). Suppose d ≥ . Then, for almost all bar-configurations q and almost allrod-configurations r , BR ( G, q , r ) = M f ( G, P ) . Namely, I ⊆ E is independent in BR ( G, q , r ) if and only if | F | ≤ D | V ( F ) | − D − | R ( F ) | for any nonempty F ⊆ I . We need to introduce a notation for the proof. Let r : R → Gr ( d − , W ) be a rod-configuration. For each v ∈ R , let H r ( v ) be the dual hyperplane to the point [ r v ] in P ( V W ),i.e., H r ( v ) = { [ p ] ∈ P ( V W ) : p · r v = 0 } . For easiness of the description, we also define H r ( v )for v ∈ B to be H r ( v ) = P ( V W ). Notice that, due to the incidence condition (16), the spaceof q uv is restricted to Gr (2 , W ) ∩ H r ( u ) ∩ H r ( v ) for uv ∈ E . We hence define two subspacesassociated with e = uv ∈ E as follows: A e ( r ) = { [0 , · · · , , u α , , · · · , , v − α , , · · · ,
0] : [ α ] ∈ P ( V W ) ∩ H r ( u ) ∩ H r ( v ) } , (17)ˆ A e ( r ) = { [0 , · · · , , u α , , · · · , , v − α , , · · · ,
0] : [ α ] ∈ Gr (2 , W ) ∩ H r ( u ) ∩ H r ( v ) } . (18)Also, let A ( r ) = { A e ( r ) : e ∈ E } , and as before let A F ( r ) = { A e ( r ) : e ∈ F } for F ⊆ E .The proof of Theorem 4.9 proceeds as follows: we first show that BR ( G, q , r ) is equal tothe linear matroid M ( A ( r )) associated with the flat family A ( r ) for almost all configurations(Theorem 4.10). We then provide an explicit formula of the rank of A ( r ) in terms of theunderlying graph G (Theorem 4.12) and finally show that M ( A ( r )) is indeed equal to M f ( G, P )(Corollary 4.13). Theorem 4.10.
Let G = ( V, E ) be a graph with a bipartition P = { B, R } . Then, for almost allrod-configurations r and bar-configurations q , BR ( G, q , r ) = M ( A ( r )) .Proof. The proof is basically the same as that of Theorem 3.1. Recall that BR ( G, q , r ) is alinear matroid on E in which each element e = uv ∈ E is represented by(0 , ··· · · · , , u q e , , ··· · · · , , v − q e , , ··· · · · , , q e ] is restricted to Gr (2 , W ) ∩ H r ( u ) ∩ H r ( v ) in the case of body-rod-bar frameworks.Hence, to prove BR ( G, q , r ) = M ( A ( r )), it is sufficient to show that a representative point x e = [0 , . . . , , x e , . . . , x De , , . . . , , − x e , . . . , − x De , , . . . ,
0] of A e ( r ) can be taken from ˆ A e ( r ) sothat X = { x e : e ∈ E } is in generic position (in the sense of definition (4)).Let us consider the case d = 3. Let us take r so that r ( u ) = r ( v ) for each u, v ∈ V with u = v .Then, for each e = uv ∈ E , A e ( r ) is isomorphic to P ( V W ) ∩ H r ( u ) ∩ H r ( v ) = P k , where k = 3if u, v ∈ R ; k = 4 if either u ∈ R or v ∈ R ; otherwise k = 5. Recall that the quadratic variety Gr (2 , W ) ∩ H r ( u ) ∩ H r ( v ) is singular if the associated matrix is singular. Since the determinantof the associated matrix is a polynomial of entries of r ( u ) and r ( v ), Gr (2 , W ) ∩ H r ( u ) ∩ H r ( v )becomes a non-singular quadratic variety of P k for almost all rod-configurations r . Then, bysetting x e = 1, it can be easily checked that Gr (2 , W ) ∩ H r ( u ) ∩ H r ( v ) can be parameterized by x e and x e such that the rest of coordinates x e , . . . , x e are described as rational functions of x e and x e with coefficients in Q . If we take x e so that { x e , x e : e ∈ E } is algebraically independentover Q , X = { x e : e ∈ E } is in generic position by the same reason as the proof of Theorem 3.1.The general d -dimensional case follows in the same way, as each coordinate of a point in Gr (2 , W ) ∩ A is written as a rational function of 2( d −
1) parameters among x i,je (1 ≤ i < j ≤ d +1),if Gr (2 , W ) is restricted to a ( D − A (see, e.g., [10]).As noted above, BR ( G, q , r ) takes the rank at most D | V | − D − | R | since the correspondingframework ( G, q , r ) always has D + | R | trivial motions. The same argument can be applied toshow the following fact. Lemma 4.11.
Let G = ( V, E ) be a graph with a bipartition P = { B, R } of V . Then, for anyrod-configuration r such that r ( u ) = r ( v ) for u, v ∈ R with u = v , rank( A ( r )) ≤ D | V | − D − | R | . The following is a key result for proving Theorem 4.9.
Theorem 4.12.
Let G = ( V, E ) be a graph with a bipartition P = { B, R } of V . If d ≥ , thenfor almost all rod-configurations r , rank( A ( r )) = min { P ki =1 ( D | V ( E i ) | − D − | R ( E i ) | ) } , (19) where the minimum is taken over all partitions { E , . . . , E k } of E into nonempty subsets.Namely, the linear polymatroid PM ( A ( r )) defined by A ( r ) is equal to the combinatorial poly-matroid PM f ( G, P ) for almost all rod-configurations r . One direction of Theorem 4.12 is straightforward from Lemma 4.11; For any partition { E , . . . , E k } of E , we have rank( A ( r )) ≤ P ki =1 rank( A E i ( r )) ≤ P ki =1 ( D | V ( E i ) | − D − | R ( E i ) | ).Since the proof is not short, the converse direction is left to the next subsection. Corollary 4.13.
Let G = ( V, E ) be a graph with a bipartition P = { B, R } of V . If d ≥ , then M ( A ( r )) = M f ( G, P ) for almost all rod-configurations r .Proof. This directly follows from Theorem 4.12 and general results on polymatroids reviewed inSection 2. Indeed, by Theorem 2.1 and Theorem 4.12, the rank of F ⊆ E in M ( A ( r )) is writtenas min {| F | + P ki =1 ( D | V ( F i ) | − D − | R ( F i ) | ) } where the minimum is taken over all partitions { F , F , . . . , F k } of F such that F , . . . , F k = ∅ .This is exactly the rank formula (3) of the matroid induced by f .16ombining Theorem 4.10 and Corollary 4.13, we conclude the proof of Theorem 4.9. Remark.
Due to the absence of Lemma 4.6, the proof of Theorem 4.12 (given in the nextsubsection) could not be applied to the 2-dimensional case. Although Theorem 4.12 can beproved even for the 2-dimensional case with a slightly different manner, we would not go intothe detail as there are already many simpler proofs for this case [21, 35, 39, 41].Theorem 4.9 is restated in terms of rigidity as follows.
Corollary 4.14.
Let G = ( V, E ) be a graph with a bipartition P = { B, R } of V . Then,there exists a bar-configuration q and a rod-configuration r such that the body-rod-bar framework ( G, q , r ) is minimally infinitesimally rigid (i.e., removing any bar results in a flexible framework)in R d if and only if G satisfies the following counting conditions: • | E | = D | B | + ( D − | R | − D ; • | F | ≤ D | B ( F ) | + ( D − | R ( F ) | − D for any nonempty F ⊆ E . Tay’s combinatorial characterization of rod-bar frameworks is an easy consequence.
Corollary 4.15 (Tay[31, 32]) . Let G = ( V, E ) be a graph. Then, there exists a bar-configuration q and a rod-configuration r such that the rod-bar framework ( G, q , r ) is minimally infinitesimallyrigid in R d if and only if G satisfies the following counting conditions: • | E | = ( D − | V | − D ; • | F | ≤ ( D − | V ( F ) | − D for any nonempty F ⊆ E .Proof. The rod-bar framework ( G, q , r ) is a body-rod-bar framework with R = V and B = ∅ . Inthis case D ( | V ( F ) | − − | R ( F ) | = ( D − | V ( F ) | − D for each F ⊆ E . Therefore, the statementfollows from Corollary 4.14. Proof.
We have already seen “ ≤ ” direction of (19). The converse direction is proved by inductionon the lexicographical ordering of the triples ( | V | , | R | , | E | ). Since the base case is trivial, let usconsider the general case. Since A e ( r ) = A e ′ ( r ) for any parallel e and e ′ , we may assume that G is simple throughout the proof.We split the proof into two cases depending on whether B = ∅ or not. B = ∅ Let us first consider the easier case where there is a vertex u ∈ B . Let N ( u ) = { v , . . . , v t } bethe neighbors of u in G . We remove u and insert the edge set K ( N ( u )), that is, the edge setof the complete graph on N ( u ). Let H = ( V − u, E \ δ G ( u ) ∪ K ( N ( u ))) be the resulting graphwith the bipartition { B − u, R } of V − u .Let { E ∗ , . . . , E ∗ k } be the P -connected component decomposition of E ( H ) in PM f ( H ). ByLemma 4.5, { E ∗ , . . . , E ∗ k } is a minimizer of the right hand side of (19) for E ( H ). By induction,we have rank( { A e ( r ) : e ∈ E ( H ) } ) = P ki =1 f ( E ∗ i ) (20)for almost all rod-configurations r : R → Gr ( d − , W ).17f N ( u ) = { v } for some v ∈ V , then E = E ( H )+ uv . It is easy to see A ( r ) = { A e ( r ) : e ∈ E ( H ) }⊕ A uv ( r ), and hence rank( A ( r )) = rank( { A e ( r ) : e ∈ E ( H ) } ) + rank( A uv ( r )) = P ki =1 f ( E ∗ i ) + f ( { uv } ), implying “ ≥ ” direction of (19) since { E ∗ , . . . , E ∗ k , { uv }} is a partition of E .Thus, let us assume | N ( u ) | ≥
2. Since K ( N ( u )) is a clique in H , it is straightforward tocheck that K ( N ( u )) is P -connected in PM f ( H ), and hence a P -connected component, say E ∗ k ,contains K ( N ( u )) as a subset. This implies f ( E ∗ k \ K ( N ( u )) ∪ δ G ( u )) = f ( E ∗ k ) + D. (21)Observe that, for any vw ∈ K ( N ( u )), we have A vw ( r ) ⊆ A vu ( r ) ∪ A uw ( r ) . (22)Indeed, any element of A vw ( r ) is written as[0 , ··· · · · , , v α , , ··· · · · , , w − α , , ··· · · · , , for some α ∈ P ( V W ) ∩ H r ( v ) ∩ H r ( w ). This can be decomposed as[0 , ··· · · · , , v α , , ··· · · · , , u − α , , ··· · · · ,
0] + [0 , ··· · · · , , u α , , ··· · · · , , w − α , , ··· · · · , , where these two terms are contained in A vu ( r ) and A uw ( r ), respectively, because H r ( u ) = P ( V W ) by u ∈ B .(22) implies { A e ( r ) : e ∈ E ( H ) } ⊆ A ( r ). Moreover, we can always take independent D points p , . . . , p D from { A e ( r ) : e ∈ δ ( u ) } since u ∈ B and | N ( u ) | ≥
2. Note that they alwayssatisfy { p , . . . , p D } ∩ { A e ( r ) : e ∈ E ( H ) } = ∅ since u / ∈ V ( H ). We thus obtainrank( A ( r )) ≥ rank( { A e ( r ) : e ∈ E ( H ) } ) + D. (23)Combining (21), (20), and (23), we obtain rank( A ( r )) ≥ P k − i =1 f ( E ∗ i ) + f ( E ∗ k \ K ( N ( u )) ∪ δ ( u )),implying “ ≥ ” direction of (19) since { E ∗ , . . . , E ∗ k − , E ∗ k \ K ( N ( u )) ∪ δ ( u ) } is a partition of E .This completes the proof for case B = ∅ . B = ∅ For any u ∈ V , let P u = { B + u, R − u } . Note that, by induction, the linear polymatroid PM ( A ( r ′ )) is equal to PM f ( G, P u ) for almost all rod-configurations r ′ on R − u . Our proofis based on this inductive relation. Intuitively speaking, we will replace a body associated with u by a rod r ( u ). This operation corresponds with restricting P ( V u ) = P ( V W ) to a hyperplane H r ( u ) of P ( V W ), which is the dual of the point r ( u ). This operation is equivalent to therestriction of A ( r ′ ) to a special hyperplane H in P ( V V ) such that H ∩ P ( V u ) = H r ( u ) and P ( V v ) ⊂ H for all v ∈ V − u . This hyperplane H is not generic within P ( V V ) (and hence thisoperation is not Dilworth truncation), but we may take H so that H ∩ P ( V u ) is generic within P ( V u ). We will show that the naturally extended rank formula of Dilworth truncation holds forthis operation for some u ∈ V .The proof consists of sequence of lemmas. We first define a generic hyperplane within P ( V u )for a vertex u ∈ V and show the existence of generic hyperplanes in Lemma 4.16. We then discussabout an extension of a rod-configuration r ′ : R − u → Gr ( d − , W ) to r : R → Gr ( d − , W ),where r is said to be an extension of r ′ if r ( v ) = r ′ ( v ) for all v ∈ V − u . We shall define a generic18xtension of a rod-configuration based on a generic hyperplane in P ( V u ). Then in Lemma 4.18we shall show an existence of a vertex u ∈ V having special properties and finally perform avariant of Dilworth truncation at u in Lemma 4.19.For a flat A of P ( V V ) and a vertex u ∈ V , proj u ( A ) denotes the orthogonal projection of A onto P ( V u ). A hyperplane H u of P ( V u ) is called generic relative to a finite set A of flats in P ( V V )if it satisfies the following property; for every A , A ⊆ A with proj u ( A ∩ A ) = ∅ (where weallow A = A ), rank(proj u ( A ∩ A ) ∩ H u ) = rank(proj u ( A ∩ A )) − . (24)The next lemma shows the existence of generic hyperplanes. Lemma 4.16.
Let u ∈ V and A be a finite set of flats in P ( V V ) . Suppose Gr ( d − , W ) ⊆ P ( V u ) (by identifying V u with V d − W ). Then, for almost all points [ r u ] ∈ Gr ( d − , W ) , the hyperplane H u of P ( V u ) dual to [ r u ] is generic relative to A .Proof. Take any A , A ⊆ A with proj u ( A ∩ A ) = ∅ , and let us denote A = A ∩ A forsimplicity. It is clear that rank(proj u ( A ) ∩ H u ) ≥ rank(proj u ( A )) − H u . Let usconsider the “ ≤ ” direction. If proj u ( A ) = P ( V u ), this relation clearly holds. Otherwise proj u ( A )is a linear subspace of P ( V u ), and hence rank(proj u ( A ) ∩ H u ) ≤ rank(proj u ( A )) − r u ] ∈ Gr ( d − , W ) so that [ r u ] is not contained in the dual of proj u ( A ) in P ( V u ). Sincethe intersection of the dual of proj u ( A ) with Gr ( d − , W ) is a lower dimensional subvariety of Gr ( d − , W ), almost all [ r u ] satisfy this property.Since there are a finite number of possible A = A ∩ A , almost all hyperplanes H u of P ( V u )are indeed generic.We now define a generic extension of a rod-configuration r ′ : R − u → Gr ( d − , W ) asfollows: a rod-configuration r : R → Gr ( d − , W ) is a generic extension of r ′ if (Condition for extension): r ( v ) = r ′ ( v ) for v ∈ V − u ; (Condition for genericity): r ( u ) satisfies the property that the dual hyperplane of r ( u ) in P ( V u ) is generic relative to A ( r ′ ).By Lemma 4.16, almost all extensions are generic.Once we pick out a generic extension r of r ′ , the unique hyperplane H of P ( V V ) is determinedin such a way that H ∩ P ( V u ) is the dual hyperplane of r ( u ) in P ( V u ) and P ( V v ) ⊂ H for all v ∈ V − u . Such a unique hyperplane H is called the hyperplane associated with the genericextension .It is important to observe A e ( r ) = A e ( r ′ ) ∩ H for every e ∈ E. (25)Also, if we define χ u by χ u ( A ) = ( u ( A ) = ∅ More precisely, let W ′ be the linear subspace of V V satisfying A = P ( W ′ ), and let proj u ( W ′ ) be the orthogonalprojection of W ′ onto V u . We define proj u ( A ) by P (proj u ( W ′ )). A ⊂ P ( V V ), then we have the following from the genericity (24): for every A , A ⊆A ( r ′ ), rank(( A ∩ A ) ∩ H ) = rank( A ∩ A ) − χ u ( A ∩ A ) . (26)Note that, setting A = A , (26) implies, for every A ⊆ A ( r ′ )rank( A ∩ H ) = rank( A ) − χ u ( A ) . (27)In particular, for any A e ( r ′ ) ∈ A ( r ′ ),rank( A e ( r ′ ) ∩ H ) = rank( A e ( r ′ )) − χ u ( A e ( r ′ )) . (28)By (25), our goal is now to extend Theorem 2.2 to the case of our special hyperplane H .Such an extension will be given in Lemma 4.19 by performing a truncation at a vertex u shownin the following lemma (Lemma 4.18). Before that, we need an easy observation. Lemma 4.17.
Let C be a P -connected set in PM ( G, P ) with C = E . Then, for almost allrod-configurations r on R , A C ( r ) is connected.Proof. Let us consider the restriction to C , i.e., consider G ′ = ( V ( C ) , C ), P ′ = { B ∩ V ( C ) , R ∩ V ( C ) } . Note that | V ( C ) | ≤ | V | , | R ∩ V ( C ) | ≤ | R | and | C | < | E | by C ⊆ E − e . Hence,by induction, the linear polymatroid PM ( A C ( r )) is equal to PM f ( G ′ , P ′ ) for almost all rod-configurations r on R . Since C is P -connected in PM f ( G ′ , P ′ ), A C ( r ) is connected. Lemma 4.18.
There exists a vertex u satisfying one of the following two properties: For almostall rod-configurations r ′ on R − u and almost all extension r , (A) G has an edge subset C with δ G ( u ) ⊂ C ( E such that A C ( r ) is connected; or (B) G has disjoint edge subsets C and C ′ with δ G ( u ) ⊂ C ∪ C ′ such that both A C ( r ) and A C ′ ( r ) are connected. Furthermore, if A ( r ′ ) is connected, then proj u ( A C ( r ′ ) ∩ A E \ C ( r ′ )) = ∅ .Proof. Take any edge e ∈ E , and consider G − e . By Lemma 4.6, G − e has (i) three verticeseach of which is spanned by two P -connected components of PM ( G − e, P ), or (ii) a vertexspanned by exactly one P -connected component of PM ( G − e, P ). Since any P -connectedset of PM ( G − e, P ) is also P -connected in PM ( G, P ), these P -connected components are P -connected in PM ( G, P ).We define C, C ′ ⊆ E − e as follows: If (i) occurs, then take a vertex u that is not an endpointof e and is spanned by two P -connected components in PM ( G − e, P ). Let C and C ′ besuch components. If (ii) occurs, then we have a vertex u spanned by exactly one P -connectedcomponent in PM ( G − e, P ). Let C be that component. Furthermore, if u is an endpoint of e ,let C ′ = { e } .Consequently, one of the followings holds: (i’) C is P -connected set with δ G ( u ) ⊂ C ( E or(ii’) C and C ′ are disjoint P -connected sets (that may be trivial) with δ G ( u ) ⊂ C ∪ C ′ . Notethat, both C and C ′ are proper subsets of E , and thus Lemma 4.17 implies that A C ( r ) and A C ′ ( r ) are connected for almost all rod-configurations r .The remaining thing is to prove the last property of (B) when (ii’) occurs. Recall P u = { B + u, R − u } , and the linear polymatroid PM ( { A e ( r ′ ) : e ∈ K ( V ) } ) is equal to PM f ( K ( V ) , P u )by induction on the lexicographical order of ( | V | , | R | , | E | ). Since A ( r ′ ) is connected, E is P -connected in PM f ( G, P u ). Thus, applying Lemma 4.8, we may assume that there is a vertex v ∈ V − u with uv ∈ cl( C ) ∩ cl( E \ C ) for the closure operator of PM f ( G, P u ). This implies A uv ( r ′ ) ⊂ A C ( r ′ ) ∩ A E \ C ( r ′ )), and thus proj u ( A C ( r ′ ) ∩ A E \ C ( r ′ )) = ∅ .20e are now ready to extend Theorem 2.2 to our nongeneric hyperplane. Recall that, for afamily A of flats and a hyperplane H , we abbreviate { A ∩ H : A ∈ A} as A ∩ H . Note that( A ) ∩ H implies { A : A ∈ A} ∩ H , which may not be equal to A ∩ H = { A ∩ H : A ∈ A} . Lemma 4.19.
Let u be a vertex shown in Lemma 4.18 and r ′ be a generic rod-configuration on R − u . Then, for the hyperplane H of P ( V V ) associated with a generic extension of r ′ , rank( A ( r ′ ) ∩ H ) = min { P ki =1 (rank( A E i ( r ′ )) − χ u ( A E i ( r ′ ))) } , (29) where the minimum is taken over all partitions { E , . . . , E k } of E into nonempty subsets.Proof. For simplicity, we abbreviate A e ( r ′ ) as A ′ e and A ( r ′ ) as A ′ , respectively. Consider theconnected component decomposition of A ′ ∩ H (that is, the P -connected component decompo-sition of the linear polymatroid PM ( A ′ ∩ H )). To see the equality of (29), we show (29) foreach connected component of A ′ ∩ H Thus, by induction, we may assume A ′ ∩ H is connectedand it is sufficient to show rank( A ′ ∩ H ) = rank( A ′ ) − . (30)From the choice of u , (A) or (B) of Lemma 4.18 holds. Let C and C ′ be subsets of E satisfying properties of Lemma 4.18, where C ′ = ∅ if (A) holds (otherwise we may assume C ′ = ∅ ). Namely, if C ′ = ∅ , A ′ C ∩ H is connected with δ G ( u ) ⊆ C . (Note that, in the currentsituation, A ′ C ∩ H corresponds to A C ( r ) of the statement of Lemma 4.18).) If C ′ = ∅ , A ′ C ∩ H and A ′ C ′ ∩ H are connected with δ G ( u ) ⊆ C ∪ C ′ . We may further assume δ G ( u ) ∩ C = ∅ and δ G ( u ) ∩ C ′ = ∅ , since otherwise we have the former case.We now calculate the rank of A ′ C ∩ H , A ′ C ′ ∩ H , and A ′ E \ C ∩ H . The connectivity of A ′ C ∩ H and δ G ( u ) ∩ C imply rank( A ′ C ∩ H ) = rank( A ′ C ) − C ′ = ∅ , the connectivity of A ′ C ′ ∩ H and δ G ( u ) ∩ C ′ implyrank( A ′ C ′ ∩ H ) = rank( A ′ C ′ ) − . (32)Also, since all flats of A ′ E \ ( C ∪ C ′ ) are contained in H by δ G ( u ) ⊂ C ∪ C ′ , we have A ′ E \ ( C ∪ C ′ ) ∩ H = A ′ E \ ( C ∪ C ′ ) . (33)Suppose C ′ = ∅ , and let us take an edge e ∈ δ G ( u ) ∩ C ′ and a point x ∈ A ′ e \ H . (Note that,by (28), A ′ e \ H = ∅ .) Then, clearly rank(( A ′ C ′ ∩ H ) ∪ { x } ) = rank( A ′ C ′ ∩ H ) + 1. Combinedwith (32), we have A ′ C ′ = ( A ′ C ′ ∩ H ) ∪ { x } . (34)By (33) and (34), A ′ E \ C ∩ H = A ′ E \ ( C ∪ C ′ ) ∪ A ′ C ′ ∩ H = ( A ′ E \ ( C ∪ C ′ ) ∩ H ) ∪ ( A ′ C ′ ∩ H ) ∪ { x } ∩ H = ( A ′ E \ ( C ∪ C ′ ) ∩ H ) ∪ ( A ′ C ′ ∩ H ) = A ′ E \ C ∩ H. (35)Thus, applying (27), we obtainrank( A ′ E \ C ∩ H ) = rank( A ′ E \ C ∩ H ) = rank( A ′ E \ C ) − C ′ = ∅ . In total, combining (33) and (36),rank( A ′ E \ C ∩ H ) = ( rank( A ′ E \ C ) − C ′ = ∅ )rank( A ′ E \ C ) (if C ′ = ∅ ) . (37)We then compute the rank of ( A ′ C ∩ H ) ∩ ( A ′ E \ C ∩ H ). Since rank(( A ′ C ) ∩ H ) = rank( A ′ C ) − A ′ C ∩ H = A ′ C ∩ H. (38)By (33), (35) and (38), we obtain( A ′ C ∩ H ) ∩ ( A ′ E \ C ∩ H ) = ( A ′ C ) ∩ ( A ′ E \ C ) ∩ H. (39)Therefore, applying (39) and then (26), we obtainrank(( A ′ C ∩ H ) ∩ ( A ′ E \ C ∩ H )) = rank(( A ′ C ) ∩ ( A ′ E \ C ) ∩ H )= rank( A ′ C ∩ A ′ E \ C ) − χ u ( A ′ C ∩ A ′ E \ C ) . (40)We show that χ u ( A ′ C ∩ A ′ E \ C ) takes distinct values depending on whether C ′ = ∅ or not. If C ′ = ∅ , then no edge in E \ C is incident to u by δ G ( u ) ⊆ C , and proj u ( A ′ E \ C ) = ∅ . Thus, χ u ( A ′ C ∩ A ′ E \ C ) = 0. On the other hand, if C ′ = ∅ , then property (B) of Lemma 4.18 impliesproj u ( A ′ C ∩ A E \ C ) = ∅ since A ′ is connected from the connectivity of A ′ ∩ H . Therefore, χ u ( A ′ C ∩ A ′ E \ C ) = 1 if C ′ = ∅ . In total, (40) can be rewritten byrank(( A ′ C ∩ H ) ∩ ( A ′ E \ C ∩ H )) = ( rank( A ′ C ∩ A ′ E \ C ) − C ′ = ∅ )rank( A ′ C ∩ A ′ E \ C ) (if C ′ = ∅ ) . (41)By (31), (37), (41), and the modularity of rank( · ),rank( A ′ ∩ H ) = rank(( A ′ C ∩ H ) ∪ ( A ′ E \ C ∩ H ))= rank( A ′ C ∩ H ) + rank( A ′ E \ C ∩ H ) − rank(( A ′ C ∩ H ) ∩ ( A ′ E \ C ∩ H ))= rank( A ′ C ) + rank( A ′ E \ C ) − rank( A ′ C ∩ A ′ E \ C ) −
1= rank( A ′ ) − , implying (30). This completes the proof of the lemma.For F ⊆ E , let χ u ( F ) = ( u ∈ V ( F )0 otherwise , where u is a vertex shown in Lemma 4.18. Then, Lemma 4.19 implies, for almost all bar-configurations r ′ on R − u and its generic extension r ,rank( A ( r )) = min { P i (rank( A E i ( r ′ )) − χ u ( E i )) : a partition { E , . . . , E k } of E } . (42)22et R ′ = R − u . The induction hypothesis on | R | impliesrank( A E i ( r ′ )) = min { P j ( D | V ( E i,j ) | − D − | R ′ ( E i,j ) | ) : a partition { E i, , . . . , E i,k ′ } of E i } (43)for each E i ⊆ E . Since χ u ( E i ) ≤ P j χ u ( E i,j ) for any E i ⊆ E and any partition { E i, , . . . , E i,k ′ } of E i , (42) and (43) implyrank( A ( r )) ≥ min { P i ( D | V ( E i ) | − D − | R ′ ( E i ) | − χ u ( E i )) : a partition { E , . . . , E k } of E } . (44)Note that, for any F ⊆ E , we have | R ( F ) | = | R ′ ( F ) | + χ u ( F ). Thus, (44) implies “ ≥ ” directionof (19) for case B = ∅ . This completes the proof of Theorem 4.12. An identified body-hinge framework (simply called a body-hinge framework) is a structure con-sisting of rigid bodies connected by hinges (that is, ( d − G, h ),where • G = ( B, H ; E ) is a bipartite graph with vertex classes B and H , representing bodies andhinges, respectively; • h : H → Gr ( d − , W ) is a hinge-configuration.Note that each v ∈ B and v ∈ H correspond to a body and a hinge, respectively, and e ∈ E indicates their incidence.A motion of ( G, h ) is defined as a mapping m : B → V d − W such that m ( u ) − m ( v ) iscontained in h ( w ) for any neighbors u, v ∈ B of w ∈ H . A motion m is called trivial if m ( v )’sare equal for all v ∈ B . ( G, h ) is said to be infinitesimally rigid if every motion is trivial.For a bipartite graph G = ( B, H ; E ), the graph obtained from G by duplicating each edgeby ( D −
1) parallel copies is denoted by ( D − ◦ G , and ( D − ◦ E denotes the edge set of( D − ◦ G . Tay showed a combinatorial characterization of identified body-hinge frameworksby converting to rod-bar frameworks. Below, we give a more natural proof. Corollary 5.1 (Tay [31]) . Let G = ( B, H ; E ) be a bipartite graph. Then, there exists a hinge-configuration h such that ( G, h ) is infinitesimally rigid if and only if ( D − ◦ G contains anedge subset I ⊆ ( D − ◦ E satisfying the following counting conditions: • | I | = D | B | + ( D − | H | − D ; • | F | ≤ D | B ( F ) | + ( D − | H ( F ) | − D for each nonempty F ⊆ I .Proof. Let ( G, h ) be an identified body-hinge framework. For an edge e = uv ∈ E with u ∈ H and v ∈ B , we can regard h ( u ) as a rod (generically) linked by ( D −
1) bars with the bodyassociated with v (see Figure 4). Hence, the identified body-hinge framework ( G, h ) is equalto the body-rod-bar framework ( G ′ , q , r ), where G ′ is the graph with V ( G ′ ) = B ∪ H and E ( G ′ ) = ( D − ◦ E , r = h , and q is a generic bar-configuration. Since D ( | B ( F ) ∪ H ( F ) | − − | H ( F ) | = D | B ( F ) | + ( D − | H ( F ) | − D for any F ⊆ ( D − ◦ E , the statement followsfrom Theorem 4.9. 23igure 4: Conversion of the body-hinge model to the body-rod-bar model.The proof can be extended to frameworks consisting of bodies, rods, bars, and hinges withoutdifficulty.Katoh and Tanigawa [16] showed that, if each hinge is allowed to connect only two bodies,then each body can be realized as a rigid panel (i.e., a hyperplane). Namely, a panel-hingeframework, which consists of rigid panels connected by hinges, is generically characterized bythe counting condition of Corollary 5.1. A natural question is whether we can drop the restrictionor not. Problem 1.
Let G = ( B, H ; E ) be a bipartite graph satisfying the counting condition of Corol-lary 5.1. Is there a hinge-configuration h such that ( G, h ) is an infinitesimally rigid panel-hingeframework? Indeed, this problem was already discussed in, e.g., [31, 34, 40] and is unsolved even for2-dimensional case. In [40], Whiteley presented a partial solution for 2-dimensional case.In the context of combinatorial rigidity, three types of characterizations are typically con-sidered; Maxwell/Laman-type counting conditions, Henneberg-type graph constructions, andtree-decompositions. In particular, tree-decompositions often provide very short proofs for com-binatorial characterizations. See, e.g., [14, 33, 39]. It is hence natural to ask a tree-decompositionfor identified body-hinge frameworks, which leads to Corollary 5.1.
Problem 2.
Let G = ( B, H ; E ) be a bipartite graph. Suppose there is an edge set I ⊆ ( D − ◦ E such that | I | = D | B | +( D − | H |− D and | F | ≤ D | B ( F ) | +( D − | H ( F ) |− D for each nonempty F ⊆ I . Then, does ( D − ◦ G contain D edge-disjoint trees such that each vertex of B is spannedby all of them and each vertex of H is spanned by exactly D − trees among them. The problem may be false since the problem of deciding whether a hypergraph contains k edge-disjoint spanning connected subgraphs is NP-hard even for k = 2 [5].As for computational issue, O ( | V | ) time algorithms are known for computing the rankof the counting (poly)matroids appeared in this paper (see, e.g., [2, 8, 12, 19] for more detail).Developing a sub-quadratic algorithm is indeed a challenging problem. As a direct application of Dilworth truncation, we shall briefly discuss direction-rigidity of bar-joint frameworks.Recall that a d -dimensional bar-joint framework is a pair ( G, p ), where G = ( V, E ) is a graphand p : V → R d . Each vertex represents a joint and each edge represents a bar which usuallyconstraints the distance between two endpoints. As a variant of length-constraint, direction-constraint (and the mixture of length and direction constraints) has been considered in theliterature (see, e.g., [15, 29, 41]). In [41], Whiteley showed a combinatorial characterization of24irection-rigidity as a corollary of a combinatorial characterization of reconstructivity of picturesappeared in scene analysis (see, e.g.,[40–42]). In this section we provide a direct proof of thischaracterization.For a d -dimensional bar-joint framework ( G, p ), an infinitesimal motion m : V → R d of( G, p ) under direction-constraint is an assignment of m ( v ) ∈ R d to each v ∈ V such that m ( u ) − m ( v ) is parallel to p ( u ) − p ( v ) for any uv ∈ E , i.e., m ( u ) − m ( v ) = t ( p ( u ) − p ( v )) forsome t ∈ R . Of course, the direction-constraint for each uv ∈ E can be written as( m ( u ) − m ( v )) · α = 0 for any α ∈ R d with ( p ( u ) − p ( v )) · α = 0 . (45)It is easy to observe that the space of infinitesimal motions of ( G, p ) has dimension at least d + 1; a linear combination of parallel transformations to d directions and the dilation centeredat the origin (see, e.g., [41, Section 8] for more detail). We say that ( G, p ) is direction-rigid ifthe dimension of the motion space is exactly d + 1.In this section, we shall use V u to denote a d -dimensional vector space associated with u (which was D -dimensional in the preceding sections), and let V V denote the direct product of V u for all u ∈ V . Hence, V V is d | V | -dimensional in this case. For each uv ∈ E , let us define a( d − P ( V V ) by A uv ( p ) = { [0 , · · · , , u α , , · · · , , v − α , , · · · ,
0] : α ∈ R d , ( p ( u ) − p ( v )) · α = 0 } , (46)and let A ( p ) = { A e ( p ) : e ∈ E } . Then, it is easy to see that direction-rigidity is characterizedby the polymatroid PM ( A ( p )) in the sense that ( G, p ) is direction-rigid if and only if therank of PM ( A ( p )) is equal to d | V | − ( d + 1). The following theorem provides a combinatorialcharacterization of this polymatroid. Theorem 6.1.
Let f ′ : 2 E → Z be an integer-valued monotone submodular function defined by f ′ ( F ) = d | V ( F ) | − ( d + 1) ( F ⊆ E ) . (47) Then, for almost all joint-configurations p : V → R d , PM ( A ( p )) is equal to the polymatroid PM f ′ ( G ) = ( E, ˆ f ′ ) induced by f ′ .Proof. We prove rank( A F ( p )) = ˆ f ′ ( F ) for any nonempty F ⊆ E (see (1) for the definition ofˆ f ). The idea is exactly the same as the alternative proof of Laman’s theorem by Lov´asz andYemini[21].Recall that g = | V ( · ) | − d copies of the graphic matroid is the matroid inducedby dg as well as the generic matroid associated with the family A = { A e : e ∈ E } of flats A uv = { [0 , · · · , , u α , , · · · , , v − α , , · · · ,
0] : α ∈ R d } . In other words, PM ( A ) = ( E, c dg ).Denote V = { v , v , . . . , v n } . For p : V → R d , we define a hyperplane H of P ( V V ) by H = { [ x v , x v , . . . , x v n ] : x v ∈ V v = R d , P v ∈ V p ( v ) · x v = 0 } . Then, observe A e ( p ) = A e ∩ H for any e ∈ E . Therefore, if we take p so that the set ofcoordinates of p is algebraically independent over Q , we found that PM ( A ( p )) is obtained from25 M ( A ) by Dilworth truncation. By Theorem 2.2, we obtain, for any F ⊆ E ,rank( A F ( p )) = min { P i (rank( A F i ) −
1) : a partition { F , . . . , F k } of F } = min { P i ( c dg ( F i ) −
1) : a partition { F , . . . , F k } of F } = min { P i ((min { P j dg ( F i,j ) : a partition of F i } ) −
1) : a partition of F } = min { P i ( dg ( F i ) −
1) : a partition { F , . . . , F k } of F } = min { P i f ′ ( F i ) : a partition { F , . . . , F k } of F } = ˆ f ′ ( F ) , where we used f ′ ( F ) = dg ( F ) −
1. This completes the proof.Let ( d − ◦ G be the graph obtained from G by replacing each edge by ( d −
1) copies, andlet ( d − ◦ E be the edge set. Notice f ′ ( e ) = d − e ∈ E . Hence, applying the sameargument given in Lemma 4.1, it is not difficult to see that the rank of PM f ′ ( G ) = ( E, ˆ f ′ )is equal to the rank of M f ′ (( d − ◦ G ), that is, the matroid on ( d − ◦ E induced by f ′ .Thus, Theorem 6.1 implies a combinatorial characterization of direction-rigidity of bar-jointframeworks proved by Whiteley [41]. Corollary 6.2 (Whiteley[41]) . For almost all joint-configurations p : V → R d , ( G, p ) isdirection-rigid if and only if ( d − ◦ G contains an edge subset I ⊆ ( d − ◦ E satisfyingthe following counting conditions: • | I | = d | V | − ( d + 1) ; • | F | ≤ d | V ( F ) | − ( d + 1) for any nonempty F ⊆ E . Servatius and Whiteley [29] further proved a combinatorial characterization of generic rigidityof two-dimensional bar-joint frameworks having both length and direction constraints. It canbe observed that the representation of the associated rigidity matrix can be obtained from therepresentation of the union of two copies of the graphic matroid by restricting some of rows toa generic hyperplane H and the others to a hyperplane (determined by H ). It is still unclearwhy Theorem 2.2 can be extended in this situation. Acknowledgments
The work was supported by Grant-in-Aid for JSPS Research Fellowships for Young Scientists.
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A Description of Bar-constraints
Here we give a note on how to obtain bar-constraints (10). This note also appears in [17,Appendix].We can coordinatize the exterior product R d ∧ R d as follows: For a = ( a , a , . . . , a d ) ∈ R d and b = ( b , b , . . . , b d ) ∈ R d , a ∧ b = (1 , (cid:12)(cid:12)(cid:12)(cid:12) a a b b (cid:12)(cid:12)(cid:12)(cid:12) , (1 , − (cid:12)(cid:12)(cid:12)(cid:12) a a b b (cid:12)(cid:12)(cid:12)(cid:12) , · · · , ( i,j ) ( − i + j +1 (cid:12)(cid:12)(cid:12)(cid:12) a i a j b i b j (cid:12)(cid:12)(cid:12)(cid:12) , · · · , ( d − ,d ) (cid:12)(cid:12)(cid:12)(cid:12) a d − a d b d − b d (cid:12)(cid:12)(cid:12)(cid:12) ! ∈ R ( d ) . (48)Suppose we are given rigid bodies B and B in R d , which can be identified with a pair( p i , M i ) of a point p i ∈ R d and an orthogonal matrix M i ∈ SO ( d ) for each i = 1 ,
2. Namely,each ( p i , M i ) is a local Cartesian coordinate system for each body. We consider a situation,where the bodies B and B are connected by a bar. We denote the endpoints of the bars by p + M q and p + M q , where q i is the coordinate of each endpoint (joint) in the coordinatesystem of each body.The constraint by the bar can be written by h p + M q − p − M q , p + M q − p − M q i = ℓ (49)for some ℓ ∈ R . If we take the differentiation with variables p i and M i , we get h p + M q − p − M q , ˙ p + ˙ M q − ˙ p − ˙ M q i = 0 (50)We may simply assume p i = 0 and M i = I d . Then by setting h = q − q and ˙ M i = A i with askew-symmetric matrix A i , h h, ˙ p + A q − ˙ p − A q i = 0 . (51)29lso we denote a skew-symmetric matrix A by A = − w , · · · · · · · · · · · · ( − d +1 w ,d w , − i + j w i,j ...... 0 ...... ( − i + j +1 w i,j . . . ...... 0 w d − ,d ( − d w ,d · · · · · · · · · · · · − w d − ,d (52)and let w = (cid:0) w , w , · · · w d − ,d (cid:1) ∈ R ( d ). Then, for any h ∈ R d and q ∈ R d , we have h h, Aq i = h q ∧ h, w i . (53)Therefore, we can simply describe the infinitesimal bar-constraint (51) by h q − q , ˙ p − ˙ p i + h q ∧ q , w − w i = 0 , (54)where w ∈ R ( d ) and w ∈ R ( d ) denote the (cid:0) d (cid:1) -dimensional vectors corresponding to A and A ,respectively.We call a pair s i = ( w i , p i ) ∈ R ( d ) × R d a screw motion , which can be identified with a vectorin V d − R d +1 . Using the homogeneous coordinate of q i in P d , (54) is written as h ( q , ∧ ( q , , s − s i = 0 , (55)where [( q , ∧ ( q ,,