Characterizing globally linked pairs in graphs

Characterizing globally link ed pairs in graphs Tib or Jord´ an ∗ Shin-ic hi T aniga w a † Marc h 26, 2026 Abstract A pair { u, v } of vertices is said to b e globally linked in a d -dimensional framework ( G, p ) if there exists no other framew ork ( G, q ) with the same edge lengths, in which the distance b et ween the p oints corresponding to u and v is different from that in ( G, p ). W e sa y that { u, v } is globally link ed in G in R d if { u, v } is globally linked in ev ery generic d -dimensional framew ork ( G, p ). W e giv e a complete com binatorial c haracterization of globally link ed v ertex pairs in graphs in R 2 , solving a conjecture of Jackson, Jord´ an and Szabadk a from 2006 in the affirmative. Our result provides a refinemen t of the characterization of globally rigid graphs in R 2 as well as an efficient algorithm for finding the globally linked pairs in a graph. W e can also deduce that globally link ed pairs in R 2 , globally linked pairs in C 2 , and stress-linked pairs in R 2 are all the same, settling conjectures of Jackson and Owen, and Garamv¨ olgyi, resp ectiv ely . In higher dimensions we determine the globally linked pairs in b o dy-bar graphs in R d , for all d ≥ 1, verifying a conjecture of Connelly , Jord´ an and Whiteley . 1 In tro duction W e briefly in tro duce the basic notions of com binatorial rigidit y theory . Let d ≥ 1 b e an in teger. A (b ar-and-joint) fr amework in R d is a pair ( G, p ), where G = ( V , E ) is a graph and p : V → R d is a function that maps the v ertices of G in to Euclidean space. W e also sa y that ( G, p ) is a r e alization of G in R d . Tw o realizations ( G, p ) and ( G, q ) are e quivalent if the edge lengths coincide in the t wo frameworks, that is, if || p ( u ) − p ( v ) || = || q ( u ) − q ( v ) || for every edge uv ∈ E , where || . || denotes the Euclidean norm in R d . The realizations are c ongruent if || p ( u ) − p ( v ) || = || q ( u ) − q ( v ) || holds for all pairs of v ertices u, v ∈ V . A framew ork ( G, p ) in R d is glob al ly rigid if ev ery equiv alent framew ork ( G, q ) in R d is congruent to ( G, p ). As a lo cal coun terpart, w e define ( G, p ) to b e rigid if there is some ε > 0 suc h that ev ery equiv alent framework ( G, q ) in R d suc h that || p ( v ) − q ( v ) || < ε for all v ∈ V is congruen t to ( G, p ). A framew ork ( G, p ) is generic if the (m ulti)set of co ordinates of p ( v ) , v ∈ V is algebraically indep enden t o ver Q . It is known that for a given dimension d ≥ 1, the rigidit y and global rigidit y of generic realizations of G in R d are determined by G itself, see [1, 3, 11]. W e say that G is rigid in R d (or d -rigid , for short) if ev ery , or equiv alently , if some generic realization of G in R d is rigid. Similarly , w e say that G is glob al ly rigid in R d (or glob al ly d -rigid ) if every , or equiv alently , if some generic realization of G in R d is globally rigid. ∗ Departmen t of Op erations Researc h, EL TE E¨ otv¨ os Lor´ and Universit y , and the HUN-REN-EL TE Egerv´ ary Researc h Group on Com binatorial Optimization, P´ azm´ any P´ eter s´ et´ any 1/C, 1117 Budap est, Hungary . e-mail: tibor.jordan@ttk.elte.hu † Departmen t of Mathematical Informatics, Graduate School of Information Science and T ec hnology , Univ ersit y of T okyo, 7-3-1 Hongo, Bunkyo-ku, 113-8656, T okyo, Japan. e-mail: tanigawa@mist.i.u-tokyo.ac.jp 1 It follo ws from the definitions that globally rigid graphs are rigid. The follo wing stronger necessary conditions of global rigidity are due to Hendric kson [13]. W e say that a graph is r e dundantly rigid in R d if it remains rigid in R d after deleting an y edge. A graph is k -c onne cte d for some p ositive in teger k if it has at least k + 1 v ertices and it remains connected after deleting an y set of less than k v ertices. Theorem 1.1. [13] L et G b e a gr aph on n ≥ d + 2 vertic es for some d ≥ 1 . Supp ose that G is glob al ly rigid in R d . Then G is ( d + 1) -c onne cte d and r e dundantly rigid in R d . F or d = 1 , 2 the conditions of Theorem 1.1 are, in fact, sufficient for global rigidit y . It is w ell- kno wn that a graph is globally rigid in R 1 if and only if it is 2-connected (see e.g. [23, Theorem 63.2.6]). The c haracterization of 2-dimensional global rigidit y is as follo ws. The notion of R 2 - connectivit y (precisely defined in the next section) is cen tral in the theory of global rigidity . Roughly sp eaking, a graph G has this prop ert y if an asso ciated matroid, defined on the edge set of G , cannot b e written as the direct sum of t wo matroids. Theorem 1.2. [14] L et G b e a gr aph on at le ast four vertic es. The fol lowing assertions ar e e quivalent. (a) G is glob al ly rigid in R 2 , (b) G is 3 -c onne cte d and r e dundantly rigid in R 2 , (c) G is 3 -c onne cte d and R 2 -c onne cte d. F or d ≥ 3 the conditions of Theorem 1.1, together, are no longer sufficen t to imply global rigidit y and the combinatorial characterization of globally rigid graphs in these dimensions is a ma jor op en question. In this pap er we shall focus on a refinemen t of global rigidit y , defined as follo ws. Let ( G, p ) b e a framework in R d . F ollowing [17], we define a pair of vertices { u, v } to b e glob al ly linke d in ( G, p ) if for every equiv alen t framework ( G, q ) in R d , w e ha ve || p ( u ) − p ( v ) || = || q ( u ) − q ( v ) || . Th us, ( G, p ) is globally rigid if and only if ev ery pair of vertices is globally linked in ( G, p ). In con trast with global rigidit y , this is not, in general, a generic property: it ma y happen that (in a giv en dimension d ≥ 2) { u, v } is globally link ed in some generic realizations and not globally link ed in others, see [17, Figs. 1 and 2]. W e define the pair { u, v } to be glob al ly linke d in G in R d (or glob al ly d -linke d ) if { u, v } is globally linked in every generic realization of G in R d . One ma y also consider vertex pairs whic h are globally linked in some generic realization of G in R d . These pairs are called we akly glob al ly d -linke d in G . A combinatorial characterization of weakly globally 2-linked pairs can be found in [22]. Global linkedness is well-understoo d for d = 1. F or a graph G and tw o vertices u, v ∈ V ( G ) w e use κ G ( u, v ) to denote the maxim um num b er of pairwise in ternally vertex-disjoin t u - v paths in G . It can be shown that { u, v } is globally 1-link ed in G if and only if uv is an edge of G or κ G ( u, v ) ≥ 2. Finding a characterization of globally 2-link ed pairs (and an efficien t algorithm for testing global 2-linkedness) has been one of the last remaining ma jor op en problems of com binatorial rigidit y theory in R 2 . W e give a complete solution in this pap er (Theorem 1.3 b elo w). Our result giv es an affirmative answer to [17, Conjecture 5.9] and settles sev eral other conjectures of this area. Global rigidit y and globally link ed pairs can b e defined in a similar manner in complex framew orks ( G, p ), where p : V → C d is a complex realization of G . It is kno wn that generic global rigidity in R d and C d are the same, see [12, 19]. Jackson and Owen [19] prov ed that, unlik e in the real setting, global link edness is a generic prop erty in C d . They also c haracterized complex global linkedness in R 2 -connected graphs (where the c haracterization is the same as in 2 the real case, c.f. Theorem 2.2) and conjectured that global link edness in R 2 and C 2 are the same, see [19, Conjecture 5.4]. The follo wing stress-based analogue of globally linked pairs w as introduced by Garamv¨ olgyi [6]. W e refer the reader to [6] for the precise definition. Roughly sp eaking, a pair { u, v } of v ertices of a graph G is d -str ess-linke d in G if for every generic framew ork ( G, p ) in R d and ev ery framew ork ( G, q ) for whic h ev ery equilibrium stress of ( G, p ) is also an equilibrium stress of ( G, q ), w e hav e that every equilibrium stress of ( G + uv , p ) is an equilibrium stress of ( G + uv , q ). In particular, adjacent pairs of vertices are d -stress link ed in G . It was sho wn in [6, Theorems 4.2 and 4.7] that d -stress-link ed pairs are globally d -link ed, and that a graph G is globally d -rigid if and only if { u, v } is d -stress link ed in G for all pairs u, v ∈ V ( G ). F urthermore, a c haracterization of 2-stress linked pairs w as also obtained [6, Theorem 4.15]. A conjecture of the same paper [6, Conjecture 6.1] suggests that d -stress-linked and globally d -linked pairs are the same in R d for all d ≥ 1. Our main result is the follo wing theorem, whic h shows that the three notions mentioned ab o ve are indeed all the same. It solv es eac h of the corresp onding conjectures as w ell as three related conjectures, see [17, Conjectures 5.12, 5.13] and [18, Conjecture 3.13]. Theorem 1.3. L et G = ( V , E ) b e a gr aph and u, v ∈ V . Then the fol lowing ar e e quivalent: (a) { u, v } is glob al ly linke d in G in R 2 , (b) { u, v } is glob al ly linke d in G in C 2 , (c) { u, v } is 2-str ess-linke d in G , (d) either uv ∈ E or ther e is an R 2 -c onne cte d sub gr aph H = ( V ′ , E ′ ) of G with u, v ∈ V ′ and κ H ( u, v ) ≥ 3 . The essential new step, which is the main con tribution of this pap er is the pro of of (a) → (d). The implications (d) → (a) and (d) → (b) w ere prov ed in [17] and [19], resp ectively , while (b) → (a) follo ws from the definitions. F urthermore, the equiv alence of (c) and (d) was sho wn in [6]. W e remark that the natural extension of (d) → (a) fails in d ≥ 3 dimensions, as there exist ( d + 1)-connected and R d -connected graphs whic h are not globally d -rigid, see [21]. How ever, it is still p ossible that the higher dimensional version of the implication (a) → (d) holds in these dimensions, see [8, Conjecture 5.2]. W e use Theorem 1.3 to deduce the characterization of the globally 2-linked clusters in a graph G , whic h are the maximal v ertex sets in whic h all pairs of vertices are globally 2-linked in G , and p oint out a link b etw een these clusters and a conjectured characterization of global 3-rigidit y . The new methods dev elop ed in this pap er can also be used to characterize globally d -link ed pairs for all d ≥ 1 in special families of graphs. W e shall illustrate this b y providing a characterization of the globally d -linked pairs in b o dy-bar graphs. This result confirms a conjecture of Connelly , Jord´ an and Whiteley [4]. The structure of the pap er is as follows. Section 2 contains the basic definitions and results of rigidit y theory w e shall need. Section 3 contains old and new structural results concerning R 2 -connected graphs. Previous results on flexings of frameworks as well as one of our key results, whic h gives a new metho d of using flexings to obtain equiv alent but non-congruen t framew orks, are in Section 4. In Section 5 w e prov e Theorem 1.3 b y showing that prop erty (d) is indeed a necessary condition of global 2-link edness. W e consider the globally 2-link ed clusters and the d -dimensional b o dy-bar graphs in Sections 6 and 7, respectively . Concluding remarks, including a brief discussion of the algorithmic asp ects, are in Section 8. 3 2 Preliminaries The rigidit y matroid of a graph G is a matroid defined on the edge set of G whic h reflects the rigidit y properties of all generic realizations of G . F or a general in tro duction to matroid theory w e refer the reader to [24]. F or a detailed treatment of the 2-dimensional rigidity matroid, see [20]. Let ( G, p ) b e a realization of a graph G = ( V , E ) in R d . The rigidity matrix of the frame- w ork ( G, p ) is the matrix R ( G, p ) of size | E | × d | V | , where, for eac h edge uv ∈ E , in the row corresp onding to uv , the entries in the d columns corresp onding to vertices u and v con tain the d co ordinates of ( p ( u ) − p ( v )) and ( p ( v ) − p ( u )), resp ectiv ely , and the remaining en tries are zeros. The rigidity matrix of ( G, p ) defines the rigidity matr oid of ( G, p ) on the ground set E by linear indep endence of ro ws of the rigidit y matrix. It is kno wn that an y pair of generic framew orks ( G, p ) and ( G, q ) hav e the same rigidity matroid. W e call this the d -dimensional rigidity matr oid R d ( G ) of the graph G , and denote its rank function by r d . F or a subgraph H of G we shall use r d ( H ) to mean r d ( E ( H )). A graph G = ( V , E ) is R d -indep endent if r d ( G ) = | E | and it is an R d -cir cuit if it is not R d -indep enden t but ev ery proper subgraph G ′ of G is R d -indep enden t. An edge e of G is an R d -bridge in G if r d ( G − e ) = r d ( G ) − 1 holds. Equiv alently , e is an R d -bridge in G if it is not contained in an y subgraph of G that is an R d -circuit. A pair { u, v } of vertices is linke d in G in R d (or d -link ed) if r d ( G + uv ) = r d ( G ) holds. By basic matroid theory , this is equiv alent to the existence of an R d -circuit in G + uv con taining the edge uv . The following c haracterization of rigid graphs is due to Gluck. Theorem 2.1. [10] L et G = ( V , E ) b e a gr aph with | V | ≥ d + 1 . Then G is rigid in R d if and only if r d ( G ) = d | V | −  d +1 2  . W e shall need three previous results concerning globally 2-linked pairs. The first one char- acterizes globally 2-link ed pairs in R 2 -connected graphs. Theorem 2.2. [17] L et G = ( V , E ) b e an R 2 -c onne cte d gr aph and x, y ∈ V . Then { x, y } is glob al ly 2-linke d in G if and only if κ G ( x, y ) ≥ 3 . Let H = ( V , E ) b e a graph. The 0-extension op eration adds a new v ertex z to H as w ell as t wo new edges incident with z . The 1-extension op eration deletes an edge xy , and adds a new v ertex z and three new edges inciden t with z , including z x and z y . The following tw o lemmas sho w that these operations preserve the property of being ”not globally 2-link ed”, at least in certain cases. Lemma 2.3. [17] If { u, v } is not glob al ly 2-linke d in H and G is a 0-extension of H , then { u, v } is not glob al ly 2-linke d in G . Theorem 2.4. [18, The or em 3.10] L et H = ( V , E ) b e a 2-rigid gr aph and let G b e obtaine d fr om H by a 1-extension on an R 2 -bridge uw ∈ E . Supp ose that { x, y } is not glob al ly 2-linke d in H for some x, y ∈ V . Then { x, y } is not glob al ly 2-linke d in G . 3 Structural prop erties of R 2 -connected graphs Theorem 1.2 sho ws that global 2-rigidit y and R 2 -connectivit y are closely related. The con- nectivit y prop erties of R 2 ( G ) are also fundamen tal in the (pro of of the) characterization of globally 2-link ed pairs. This section is dev oted to structural results concerning R 2 -connected (sub)graphs. 4 Let M b e a matroid on ground set E . W e can define a relation on the pairs of elemen ts of E b y sa ying that e, f ∈ E are equiv alent if e = f or there is a circuit C of M with { e, f } ⊆ C . This defines an equiv alence relation. The equiv alence classes are the c onne cte d c omp onents of M . Thus the connected components of M form a partition of E . The matroid is c onne cte d if there is only one equiv alence class. A graph G is R d -c onne cte d if R d ( G ) is connected. The subgraphs of G induced by the (edges of the) connected components of R 2 ( G ) are called the R 2 - c omp onents of G . An R 2 -comp onen t H is trivial if | E ( H ) | = 1, or equiv alently , if it corresponds to an R 2 -bridge of G . Otherwise it is non-trivial . Some basic prop erties of the R 2 -comp onen ts are summarized in the next lemma. Lemma 3.1. [14, 20] L et G = ( V , E ) b e a gr aph with R 2 -c omp onents H 1 , H 2 , . . . , H q . Then (a) if H i is non-trivial, then it is a r e dundantly 2-rigid, 2-c onne cte d induc e d sub gr aph of G with | V ( H i ) | ≥ 4 , for 1 ≤ i ≤ t, (b) | V ( H i ) ∩ V ( H j ) | ≤ 1 for 1 ≤ i < j ≤ t , and (c) r 2 ( G ) = P q i =1 r 2 ( H i ) . W e shall also use the following prop erties. Lemma 3.2. L et G = ( V , E ) b e a gr aph with R 2 -c omp onents H 1 , H 2 , . . . , H q , and let u, v ∈ V b e a non-adjac ent vertex p air with u, v ∈ V ( H 1 ) . Then the R 2 -c omp onents of G + uv ar e H 1 + uv , H 2 , . . . , H q . In p articular, the vertex sets of the R 2 -c omp onents of G and G + uv ar e the same. Pr o of. Since H 1 is 2-rigid b y Lemma 3.1(a), H 1 + uv is an R 2 -connected subgraph of G + uv . It suffices to sho w that there is no R 2 -circuit C in G + uv with uv ∈ E ( C ) and E ( C ) ∩ E ( H i )  = ∅ for some 2 ≤ i ≤ q . Supp ose, for a con tradiction, that such an R 2 -circuit exists and let f ∈ E ( C ) ∩ E ( H i ). Let C ′ b e an R 2 -circuit in H 1 + uv with uv ∈ E ( C ′ ). Then the strong circuit axiom implies that there exists an R 2 -circuit C ′′ in G with f ∈ E ( C ′′ ) ⊆ ( E ( C ) ∪ E ( C ′ )) − uv . Since E ( C ′′ ) ∩ E ( H 1 )  = ∅ , it contradicts the assumption that H 1 and H i are different R 2 - comp onen ts of G . Let H = ( V , E ) be an R 2 -connected graph with | V | ≥ 4. By Lemma 3.1(a) H is 2-connected. Let a, b ∈ V . W e say that the pair { a, b } is a 2-sep ar ator of H if H − { a, b } is disconnected. W e sa y that t wo 2-separators { a, b } and { a ′ , b ′ } of H are cr ossing , if a and b are in differen t comp onen ts of H − { a ′ , b ′ } . The next lemma is a corollary of [14, Lemma 3.6]. Lemma 3.3. [14] Supp ose that H is an R 2 -c onne cte d gr aph. Then ther e ar e no cr ossing 2- sep ar ators in H . Let { a, b } b e a 2-separator of H and let X b e the union of the v ertex sets of some, but not all comp onen ts of H − { a, b } . W e say that the graphs H 1 = H [ X ∪ { a, b } ] + ab and H 2 = ( H − X ) + ab are obtain ed from H by cle aving H along the 2-sep ar ator { a, b } . Note that if ab ∈ E , then H 1 , H 2 con tain only one cop y of ab . Lemma 3.4. [14, L emma 3.4] Supp ose that H 1 , H 2 ar e obtaine d fr om H by cle aving along a 2-sep ar ator. If H is R 2 -c onne cte d, then H 1 , H 2 ar e also R 2 -c onne cte d. The augmente d gr aph ˆ H is obtained from H b y adding an edge ab for all 2-separators { a, b } of H with ab / ∈ E . A maximal 3-connected subgraph of ˆ H is called a 3-blo ck . It was sho wn in [14, Section 3] that each 3-block is R 2 -connected, every edge e = ab of ˆ H belongs to at least one 3-block, and if e belongs to tw o or more 3-blocks, then { a, b } is a 2-separator. The 5 3-blo c ks can be obtained from ˆ H b y recursiv ely cleaving the graph along 2-separators. Note that r 2 ( H ) = r 2 ( ˆ H ) and the 2-separators of H and ˆ H are the same b y Lemma 3.1(a) and Lemma 3.3, resp ectively . Let J 1 , J 2 , . . . , J t b e the 3-blo cks of H . F or eac h 2-separator { a, b } of H let h H ( ab ) denote the num b er of 3-blo c ks J i , 1 ≤ i ≤ t , with { a, b } ⊂ V ( J i ), and let k ( H ) = P ( h H ( ab ) − 1), where the summation is ov er all 2-separators { a, b } of H . W e say that an ordering ( X 1 , X 2 , . . . , X p ) of p subsets of V is m -shel lable , for some integer m ≥ 0, if | ( ∪ j − 1 i =1 X i ) ∩ X j | ≤ m for all 2 ≤ j ≤ p . Lemma 3.5. L et H = ( V , E ) b e an R 2 -c onne cte d gr aph with 3-blo cks J 1 , J 2 , . . . , J t . Then (a) r 2 ( H ) = P t i =1 r 2 ( J i ) − k ( H ) , (b) t = k ( H ) + 1 , (c) for every e ∈ E ( H ) the vertex sets V ( J i ) , 1 ≤ i ≤ t , have a 2-shel lable or dering such that e is induc e d by the first set of the or dering. Pr o of. The pro of is b y induction on | V | . F or | V | = 4 we ha v e H = K 4 and the lemma trivially holds. Supp ose | V | ≥ 5. If H has no 2-separators, or equiv alently , if H is 3-connected, then t = 1, k ( H ) = 0, and the lemma is straigh tforward. Let us assume that H is not 3-connected and let X ⊂ V b e a minimal subset of v ertices satisfying | N H ( X ) | = 2 and V − X − N H ( X )  = ∅ . Let N H ( X ) = { a, b } . By Lemma 3.3 w e ha ve N H ( X ) = N ¯ H ( X ). Let H 1 , H 2 b e the graphs obtained from ¯ H by clea ving along { a, b } , suc h that V ( H 1 ) = X ∪ { a, b } . Note that for every designated edge w e can c ho ose X so that e ∈ E ( H 2 ) holds. By Lemma 3.3 and the c hoice of X , H 1 is a 3-block of H (sa y , H 1 = J 1 ), and the 3-blocks of H 2 are J 2 , J 3 , . . . , J t . F urthermore, b oth H 1 and H 2 are R 2 -connected by Lemma 3.4. By induction, r 2 ( H 2 ) = P t i =2 r 2 ( J i ) − k ( H 2 ) , and t − 1 = k ( H 2 ) + 1. Let B 1 , B 2 b e R 2 -bases of H 1 , H 2 , resp ectively , with ab ∈ E ( B 1 ) ∩ E ( B 2 ). Then B 1 ∪ B 2 is an R 2 -base of H by Lemma 3.4. Hence r 2 ( H ) = r 2 ( H 1 ) + r 2 ( H 2 ) − 1. Now w e can deduce that (a) and (b) hold for H b y using that k ( H ) = k ( H 2 ) + 1. Moreo ver, the v ertex sets of J 2 , J 3 , ..., J t ha ve a 2-shellable ordering suc h that e is induced b y the first set of the ordering, by induction. By adding V ( J 1 ) to the end of this ordering w e obtain the ordering as required b y (c). A 3-blo ck of some non-trivial R 2 -comp onen t of a graph G is said to b e an R 2 -blo ck of G . 4 Equiv alent realizations and flexings In this section we pro ve a key result, whose pro of is based on con tinuous motions of frameworks. Let G = ( V , E ) be a graph and let ( G, p ) b e a d -dimensional framework. A flexing of the framew ork ( G, p ) is a contin uous function ϕ : [0 , 1] → R d | V | suc h that (i) ϕ (0) = p , (ii) ( G, ϕ ( t )) is equiv alent to ( G, p ) for all t ∈ [0 , 1], and (iii) ( G, ϕ ( t )) is not congruent to ( G, p ) for all t ∈ (0 , 1]. The framework ( G, p ) is said to b e flexible if it has a flexing, and rigid otherwise. Let us fix an ordering of the edges of G and define the rigidity map f G : R d | V | → R | E | of G b y f G ( p ) = ( . . . , || p ( u ) − p ( v ) || 2 , . . . ) , where uv ∈ E , and p ( w ) ∈ R d for w ∈ V . F or a smo oth map f : R n → R m , let k = max { rank d f ( x ) : x ∈ R n } , the maxim um of the rank of the Jacobian of f . W e sa y that x ∈ R n is a r e gular p oint of f if rank d f ( x ) = k . The image f ( p ) is a r e gular value if each p oint in f − 1 ( f ( p )) is a regular p oin t. Note that the Jacobian 6 d f G ( p ) of the rigidit y map at some point p ∈ R d | V | is giv en b y 2 R ( G, p ), where R ( G, p ) is the rigidit y matrix of ( G, p ). The next lemma is a w ell-known ”rigidity predictor”. Lemma 4.1. [25, Pr op osition 5.1] L et ( G, p ) b e a d -dimensional fr amework. Supp ose that p is a r e gular p oint of the rigidity map f G and the fr amework do es not lie in a hyp erplane of R d . Then ( G, p ) is rigid if and only if rank R ( G, p ) = d | V | −  d +1 2  and ( G, p ) is flexible if and only if rank R ( G, p ) < d | V | −  d +1 2  . W e shall need the follo wing statement concerning flexings that change the distance betw een a designated v ertex pair. Lemma 4.2. L et ( G, p ) b e a generic r e alization of G = ( V , E ) in R d and supp ose that { x, y } is not d -linke d in G for some x, y ∈ V . L et ( G, q ) b e an e quivalent r e alization with a d -dimensional affine sp an. Then ( G, q ) has a flexing ϕ : [0 , 1] → R d | V | for which || ϕ ( t )( x ) − ϕ ( t )( y ) ||  = || p ( x ) − p ( y ) || for al l t ∈ (0 , 1] . Pr o of. Let H be a graph on vertex set V . Since p is generic and f H is a p olynomial map b et ween manifolds, basic results of algebraic geometry imply that f H ( p ) is a regular v alue, see, e.g., [11, Prop osition 2.32] or [2, Theore m 9.6.2]. In particular, q is a regular point of f G and { x, y } is not link ed in ( G, q ). Th us rank R ( G, q ) < d | V | −  d +1 2  and rank R ( G + xy , q ) = rank R ( G, q ) + 1. Moreov er, since ( G, q ) has a d -dimensional affine span, G has a supergraph ¯ G for which rank R ( ¯ G, p ) = d | V | −  d +1 2  − 1 and rank R ( ¯ G + xy , p ) = d | V | −  d +1 2  . Hence ( G, q ) has the desired flexing by Lemma 4.1. In a graph G = ( V , E ) the degree of a v ertex v ∈ V (resp. the set of neighbours of v ) is denoted by deg G ( v ) (resp. N G ( v )). Hence w e ha v e deg G ( v ) = | N G ( v ) | if G con tains no parallel edges. The next theorem is the main result of this section. Theorem 4.3. L et G = ( V , E ) b e a d -rigid gr aph, let xy ∈ E b e an R d -bridge in G with deg G ( y ) ≥ d + 2 , and supp ose that the vertic es in N G ( y ) − { x } ar e p airwise glob al ly d -linke d in G − y . L et { u, v } b e a p air of vertic es with y / ∈ { u, v } , such that { u, v } is not glob al ly d -linke d in G − y . Then { u, v } is not glob al ly d -linke d in G . Pr o of. Let H = G − y and let ( H , p ) b e a generic d -dimensional realization of H in whic h { u, v } is not globally link ed. Then there exists an equiv alent realization ( H , q ) for whic h | p ( u ) − p ( v ) |  = | q ( u ) − q ( v ) | . Since the vertices in N G ( y ) − { x } are pairwise globally d -linked in H , w e ma y supp ose, by applying an isometry , if necessary , that p ( v ) = q ( v ) for all v ∈ N G ( y ) − { x } . Claim 4.4. Ther e exists a vertex w ∈ N G ( y ) − { x } for which { w , x } is not d -linke d in H . Pr o of. Supp ose that { w , x } is d -linked in H for all w ∈ N G ( y ) − { x } . The fact that the v ertices in N G ( y ) − { x } are pairwise globally d -link ed in H implies that they are also pairwise d -link ed in H . Th us the v ertex set N G ( y ) forms a d -rigid cluster of size at least d + 1 in H . It follo ws that the edge xy is induced by a d -rigid cluster in G − xy . Hence xy is d -link ed in G − xy , con tradicting the assumption that xy is an R d -bridge of G . The facts that deg G ( y ) ≥ d + 2 and the subframew ork of ( H , q ) on vertex set N G ( y ) − { x } is congruen t to that of ( H, p ) sho ws that ( H, q ) do es not lie in a hyperplane of R d . Th us w e can apply Lemma 4.2 to ( H , p ) and ( H , q ) to deduce that ( H, q ) has a flexing ϕ : [0 , 1] → R d | V | suc h that for some w ∈ N G ( y ) − { x } w e hav e || ϕ ( t )( x ) − ϕ ( t )( w ) ||  = || q ( x ) − q ( w ) || for all t ∈ (0 , 1]. 7 W e ma y assume, by con tinuit y , and scaling the flexing, that || p ( u ) − p ( v ) ||  = || ϕ ( t )( u ) − ϕ ( t )( v ) || for all t ∈ [0 , 1]. Since the vertices in N G ( y ) are pairwise d -link ed in H , and q is a regular p oin t, they are pairwise link ed in ( H , q ). Hence w e ma y assume that ϕ ( t )( w ) = q ( w ) for all w ∈ N G ( y ) − { x } and t ∈ [0 , 1]. It follows that the flexing c hanges the p osition of x . W e shall compare ϕ ( t )( x ) to a ”fixed” p oin t p ( x ) ∈ R d , the position of x in ( H , p ). F or eac h t ∈ [0 , 1] the points a ∈ R d with || a − p ( x ) || = || a − ϕ ( t )( x ) || lie on a h yp erplane. As ϕ ( t )( x ) mo v es during the flexing, the union of these hyperplanes con tains an op en ball B ⊂ R d . Let us choose a p oin t a ∗ ∈ B for which the multiset of the co ordinates of p ( v ), v ∈ V − { y } , together with the co ordinates of a ∗ , are algebraically indep endent o ver the rationals. Let t ′ ∈ [0 , 1] suc h that | a − p ( x ) | = | a − ϕ ( t ′ )( x ) | . With these points in hand we are ready to define tw o frameworks that v erify the statemen t of the theorem. Let ( G, p 1 ) b e obtained from ( H , p ) b y adding vertex y and putting p 1 ( y ) = a ∗ . Let ( G, p 2 ) b e obtained from ( H , ϕ ( t ′ )) b y adding v ertex y and putting p 2 ( y ) = a ∗ . It follows from the c hoice of a ∗ that ( G, p 1 ) is generic. Since the subframew orks of ( G, p 1 ) and ( G, p 2 ) on v ertex set N G ( y ) − { x } are iden tical, the c hoice of a ∗ implies that ( G, p 1 ) and ( G, p 2 ) are equiv alent. As w e ha v e || p ( u ) − p ( v ) ||  = || ϕ ( t ′ )( u ) − ϕ ( t ′ )( v ) || , w e obtain that { u, v } is not globally d -link ed in G , as required. 5 Globally linked pairs in R 2 In this section w e complete the pro of of Theorem 1.3. As w e discussed in the In tro duction, the theorem follows from the next statement. Theorem 5.1. L et G = ( V , E ) b e a gr aph and let u, v ∈ V b e a p air of non-adjac ent vertic es. If { u, v } is glob al ly 2-linke d in G , then ther e is an R 2 -c omp onent H of G with u, v ∈ V ( H ) and κ H ( u, v ) ≥ 3 . Pr o of. Let { u, v } b e a non-adjacent vertex pair in G . W e shall prov e, by induction on | V | , that if there is no R 2 -comp onen t H with u, v ∈ V ( H ) and κ H ( u, v ) ≥ 3 , (1) in G , then { u, v } is not globally 2-link ed in G . The cases | V | ≤ 3 are trivial, so we ma y assume that | V | ≥ 4. First we sho w that w e can add new edges to G so that the resulting graph is 2-rigid, each of its R 2 -blo c ks is a complete subgraph, u and v remain non-adjacen t, and (1) is preserv ed. T o pro ve this first observe that, since K | V | min us an edge is 2-rigid for | V | ≥ 4, there exists a set B of new edges, uv / ∈ B , such that G + B is 2-rigid and each edge of B is an R 2 -bridge in G + B . Th us the addition of B mak es the graph 2-rigid and preserv es (1). Next observ e that adding an edge that connects a pair of non-adjacent v ertices of an R 2 -blo c k do es not c hange the vertex sets of the R 2 -blo c ks and also preserves (1). Hence w e can make all the R 2 -blo c ks complete subgraphs. Since it suffices to sho w that { u, v } is not globally 2-linked in a sup ergraph of G , in the rest of the pro of we ma y assume that G is 2-rigid and each R 2 -blo c k is a complete subgraph of G . W e sa y that a v ertex y ∈ V is r e ducible if either y belongs to a unique R 2 -blo c k of G and y is inciden t with at most one R 2 -bridge, or y belongs to no R 2 -blo c k of G and y is inciden t with at most three R 2 -bridges of G . Claim 5.2. G has at le ast thr e e r e ducible vertic es. 8 Pr o of. Let H 1 , H 2 , ..., H q b e the non-trivial R 2 -comp onen ts and let J 1 , J 2 , ..., J t b e the R 2 -blo c ks of G . Let n i = | V ( J i ) | for 1 ≤ i ≤ t . W e hav e n i ≥ 4 for all 1 ≤ i ≤ t . By Lemma 3.5 we hav e r 2 ( H i ) = X J i ⊆ H i (2 n i − 3) − k ( H i ) , (2) for 1 ≤ i ≤ q . Let G ′ b e the subgraph of G induced by the non-trivial R 2 -comp onen ts and let F ⊆ E b e the set of R 2 -bridges in G . F or eac h R 2 -blo c k J i let X i b e the set, and x i b e the n umber of v ertices in J i that belong to no other R 2 -blo c k, let Y i = V ( J i ) − X i , and y i = | Y i | . T hen w e ha ve n i = x i + y i , 1 ≤ i ≤ t . Let X = ∪ t i =1 X i , Y = ∪ t i =1 Y i , and let Z = V − ( X ∪ Y ) b e the set of vertices that b elong to no non-trivial R 2 -comp onen t in G . Note that P t i =1 y i ≥ 2 | Y | , since eac h v ertex of Y contributes to the left hand side b y at least tw o. W e shall pro ve the claim by a coun ting argumen t, fo cusing on the v ertex set X ∪ Z in the subgraph G F , where G F = ( V , F ). F or eac h x ∈ X let c ( x ) = max { 2 − d G F ( x ) , 0 } and for each z ∈ Z let c ( z ) = max { 4 − d G F ( z ) , 0 } . Since G is 2-rigid with | V | ≥ 4, we ha ve c ( v ) ≤ 2 for all v ∈ X ∪ Z . By coun ting degrees in G F w e obtain | F | ≥ P t i =1 2 x i + 4 | Z | − P v ∈ X ∪ Z c ( v ) 2 = t X i =1 x i + 2 | Z | − P v ∈ X ∪ Z c ( v ) 2 . (3) Let us define C = P v ∈ X ∪ Z c ( v ) 2 for simplicity . Then 2 | V | − 3 = r 2 ( G ) (b y the 2-rigidity of G ) = q X i =1 r 2 ( H i ) + | F | (b y Lemma 3.1( c )) ≥ t X i =1 (2 n i − 3) − q X i =1 k ( H i ) + t X i =1 x i + 2 | Z | − C (b y (2) and (3)) = 2 t X i =1 x i + t X i =1 y i + t X i =1 ( x i + y i ) − 3 t − q X i =1 k ( H i ) + 2 | Z | − C (by n i = x i + y i ) ≥ 2 | V | + 4 t − 3 t − q X i =1 k ( H i ) − C (b y x i + y i ≥ 4 and t X i =1 2 x i + t X i =1 y i + 2 | Z | ≥ 2 | V | ) = 2 | V | + q − C (by t + q X i =1 k ( H i ) = q by Lemma 3.5(b)). As q ≥ 0, w e m ust ha ve C ≥ 3. Since c ( v ) ≤ 2 for each v ∈ X ∪ Z , this implies that there exist at least three v ertices v in X ∪ Z with c ( v ) ≥ 1. The claim follows, as these v ertices are all reducible. By Claim 5.2 there exists a reducible v ertex y ∈ V with y / ∈ { u, v } . First supp ose that y b elongs to a unique R 2 -blo c k J of G and v is incident with no R 2 -bridges. Then G [ N G ( y )] is a complete graph on at least three v ertices. F urthermore, G − y satisfies (1). W e can no w deduce b y induction, that { u, v } is not globally 2-linked in G − y . Since the neighbour set of y in G is complete, this implies that { u, v } is not globally 2-link ed in G . 9 Next suppose that y belongs to a unique R 2 -blo c k J of G and y is inciden t with one R 2 - bridge, sa y xy . Then G [ N G ( y ) − { x } ] is a complete graph on at least three vertices. Thus Lemma 4.3 implies that { u, v } is not globally 2-linked in G . Finally , suppose that y b elongs to no R 2 -blo c k of G and y is incident with at most three R 2 -bridges of G . Since G is 2-rigid, w e hav e 2 ≤ d G ( v ) ≤ 3. If deg G ( y ) = 2, then Lemma 2.3 implies that { u, v } is not globally 2-link ed in G , so w e ma y assume that deg G ( v ) = 3. Claim 5.3. L et N G ( y ) = { x 1 , x 2 , x 3 } . Then { x i , x j } is not 2-linke d in G − y for some 1 ≤ i < j ≤ 3 . Pr o of. Supp ose that eac h pair of vertices in N G ( y ) is 2-linked in G − y . Then the w ell-known fact that the 0-extension op eration preserves 2-rigidity implies that G − y x i is 2-rigid for 1 ≤ i ≤ 3. This contradicts the assumption that the edges inciden t with y are R 2 -bridges in G . By Claim 5.3 w e may supp ose, by relab elling the neighbours of y , if necessary , that { x 1 , x 2 } is not 2-linked in G − y . Then G ′ = G − y + x 1 x 2 is 2-rigid, and x 1 x 2 is an R 2 -bridge in G ′ . Since G can b e obtained from G ′ b y a 1-extension on edge x 1 x 2 , Lemma 2.4 implies that { u, v } is not globally 2-linked in G . This completes the pro of of the theorem. Theorem 5.1 implies the follo wing statemen t, which is a w eaker version of the 2- dimensional case of Theorem 1.1. Corollary 5.4. L et G = ( V , E ) b e a gr aph with | V | ≥ 4 and supp ose that every generic r e aliza- tion of G in R 2 is glob al ly rigid. Then G is r e dundantly 2-rigid. Pr o of. W e may assume that G is 2-rigid. W e shall pro ve that G has no R 2 -bridges. F or a con tradiction supp ose that G − xy is not 2-rigid for some xy ∈ E . Since | V | ≥ 4, there exists a pair { u, v } (  = { x, y } ) such that u and v do not b elong to the same 2-rigid subgraph of G − xy . The fact that R 2 -connected graphs are 2-rigid implies that there is no R 2 -connected subgraph of G − xy whic h con tains both u and v . As xy is an R 2 -bridge in G , the same holds in G , to o. By Theorem 5.1 we obtain that { u, v } is not globally 2-link ed in G , whic h means that there exists a generic realization ( G, p ) of G in whic h { u, v } is not globally 2-link ed. Th us ( G, p ) is not globally rigid in R 2 , a con tradiction. By using that global rigidit y is a generic prop ert y , w e obtain the original statemen t of Theorem 1.1 in the d = 2 case. 6 The globally link ed clusters in R 2 The 2-dimensional glob al ly linke d closur e , denoted b y glc 2 ( G ), is the graph obtained from G b y adding an edge uv for all pairs { u, v } of non-adjacen t globally 2-link ed v ertices of G . The glob al ly 2-linke d clusters of G are the v ertex sets of the maximal complete subgraphs in glc 2 ( G ). Theorem 1.3 and Lemma 3.2 giv e the following c haracterization. Lemma 6.1. L et G = ( V , E ) b e a gr aph. Then the glob al ly 2-linke d clusters of G of size at le ast four ar e pr e cisely the vertex sets of the R 2 -blo cks of G . F urthermor e, an e dge e ∈ E (glc 2 ( G )) is not induc e d by a glob al ly 2-linke d cluster of size at le ast four if and only if e is an R 2 -bridge of G . 10 The main result of this section, Theorem 6.3 b elo w, is motiv ated by a conjectured character- ization of global 3-rigidit y , see [5, Conjecture 4.9]. The truth of this conjecture w ould imply that if a graph G (whic h is not a cop y of K 5 , 5 ) is not globally 3-rigid, then this fact can b e certified b y a set F ⊆ E ( G ) and a family of subsets of V ( G ) of size at least five whic h forms a 4-shellable ”non-trivial tight co ver” of G − F , see [5]. The theorem implies that there is a similar certificate of b eing not globally 2-rigid, and it can b e obtained from the globally 2-link ed clusters of the graph. It is conceiv able that a similar phenomenon holds in R 3 . W e shall need one more lemma on R 2 -comp onen ts in the pro of. F or a graph G = ( V , E ) and its subgraph H w e call ( N G ( V − V ( H )) ∩ V ( H ) the vertic es of attachment of H . Lemma 6.2. L et G = ( V , E ) b e a gr aph with at le ast thr e e R 2 -c omp onents. Then G has at le ast thr e e R 2 -c omp onents with at most two vertic es of attachment. Pr o of. The pro of is b y induction on | V | . F or | V | = 3 we ha ve G = K 3 , in which case the statemen t is clear. Supp ose | V | ≥ 4. If G is disconnected, or has a cut-v ertex, then the lemma follo ws easily b y induction, using Lemma 3.1. So w e may assume that G is 2-connected. Let H 1 , H 2 , . . . , H q b e the R 2 -comp onen ts of G , let n i = | V ( H i ) | , and let y i b e the num b er of attac hment vertices of H i . Let x i = n i − y i . Since G is 2-connected, w e ha ve y i ≥ 2 for all 1 ≤ i ≤ q . F or a con tradiction supp ose that n i ≥ y i ≥ 3 for all but at most t wo R 2 -comp onen ts of G . Since eac h H i is 2-rigid, we can use Lemma 3.1(c) to deduce that 2 | V | − 3 ≥ r 2 ( G ) = P q i =1 r 2 ( H i ) = P q i =1 (2 n i − 3) = 2 P q i =1 n i − 3 q = = (2 P q i =1 x i + P q i =1 y i ) + P q i =1 y i − 3 q ≥ 2 | V | + 3 q − 2 − 3 q ≥ 2 | V | − 2 , a contradiction. Let G = ( V , E ) b e a graph and let X b e a family of subsets of V . F or a pair u, v ∈ V let h X ( uv ) denote the num b er of sets X ∈ X with u, v ∈ X . Let H ( X ) = {{ u, v } : h X ( uv ) ≥ 2 } . Theorem 6.3. L et G = ( V , E ) b e a gr aph, let C = { C 1 , C 2 , ..., C s } b e the glob al ly 2-linke d clusters of G of size at le ast four, and let F ⊆ E b e the set of e dges of G not induc e d by the memb ers of C . Then r 2 ( G ) = | F | + s X i =1 (2 | C i | − 3) − X { u,v }∈ H ( C ) ( h C ( uv ) − 1) . (4) F urthermor e, C has a 3-shel lable or dering. Pr o of. W e may assume that G = glc 2 ( G ). By Lemma 6.1 F is the set of R 2 -bridges of G . Let H 1 , H 2 , ..., H q b e the non-trivial R 2 -comp onen ts of G . By Lemma 3.1(b) H ( C ) consists of the v ertex pairs of the 2-separators { a, b } of the R 2 -comp onen ts, and h C ( ab ) = h H i ( ab ), where a, b ∈ V ( H i ) and h H i ( ab ) denotes the num b er of R 2 -blo c ks in H i that contain b oth a and b (as defined in Section 3). W e can now use Lemma 3.1(c), Lemma 3.5(a), and the fact that eac h R 2 -blo c k is 2-rigid to deduce that r 2 ( G ) = | F | + q X i =1 r 2 ( H i ) = | F | + q X i =1 ( X { r 2 ( J ) : J is an R 2 -blo c k of H i } ) − k ( H i )) = = | F | + s X i =1 (2 | C i | − 3) − X u,v ∈ H ( C ) ( h C ( uv ) − 1) , whic h pro ves the first part of the statemen t. 11 Figure 1: The globally 2-linked clusters in this graph are the vertex sets of the six copies of K 4 . No ordering of these sets is 2-shellable. W e prov e the second part by induction on q . F or q = 1 w e are done b y Lemma 3.5(c). Supp ose that q ≥ 2. By Lemma 6.2, applied to G − F , there is a non-trivial R 2 -comp onen t, sa y H q , of G with at most t wo vertices of attachmen t. Let G ′ b e obtained from G deleting the non-attac hment v ertices and the edges of H q . The non-trivial R 2 -comp onen ts of G ′ are H 1 , H 2 , ..., H q − 1 . By induction, the globally 2-link ed clusters of G ′ ha ve a 3-shellable ordering. W e can extend this to a 3-shellable ordering of C b y adding a 2-shellable ordering of the vertex sets of the R 2 -blo c ks of H q in such a w ay that if there exists an attac hment v ertex x ∈ V ( H q ), then w e choose suc h an ordering in whic h the first R 2 -blo c k contains an edge of H q inciden t with x . Such an ordering exists by Lemma 3.5(c). Since H q has at most tw o v ertices of attachmen t, its first R 2 -blo c k has at most t wo vertices in common with the preceeding sets. F urthermore, the c hoice of x and the 2-shellability within H q imply that the extended ordering is 3-shellable. In Theorem 6.3 we cannot replace 3-shellable b y 2-shellable, see Figure 1. 7 Globally linked pairs in d -dimensional b o dy-bar graphs Let H = ( V , E ) b e a lo opless multigraph. The b o dy–b ar gr aph induc e d by H , denoted by G H , is the graph obtained from H b y replacing each v ertex w ∈ V by a complete graph B w (the ‘b o dy’ of w ) on deg H ( w ) vertices and replacing each edge w z by an edge (a ‘bar’) b et ween B w and B z in such a wa y that the bars are pairwise disjoint. The d -rigidit y and the global d -rigidity of b o dy-bar graphs ha v e b een c haracterized in terms of the ”tree-connectivity” of the underlying m ultigraph H = ( V , E ). F or a partition P of V let e H ( P ) denote the n um b er of edges of H that connect distinct parts of P . W e say that H is k -tr e e-c onne cte d , for some in teger k ≥ 1, if e H ( P ) ≥ k ( t − 1) (5) for all partitions P of V in to t ≥ 1 parts. W e call H highly k -tr e e-c onne cte d if (5) holds with strict inequality whenever t ≥ 2. Note that a single vertex, K 1 , is highly k -tree connected for all k ≥ 1. The following theorem is due to T ay . Theorem 7.1. [26] L et H = ( V , E ) b e a multigr aph with | V | ≥ 2 and | E | ≥ 2 and let G H b e the b o dy–b ar gr aph induc e d by H . L et d ≥ 1 b e an inte ger. Then G H is d -rigid if and only if H is  d +1 2  -tr e e-c onne cte d. Globally d -rigidity for b o dy-bar graphs turns out to b e the same as redundant d -rigidit y , in the following sense. 12 Theorem 7.2. [4] L et H = ( V , E ) b e a multigr aph with | V | ≥ 2 and | E | ≥ 2 and let G H b e the b o dy–b ar gr aph induc e d by H . L et d ≥ 1 b e an inte ger. Then G H is glob al ly d -rigid if and only if H is highly  d +1 2  -tr e e-c onne cte d. It w as conjectured in [4] that a non-adjacent pair { u, v } is globally d -link ed in G H if and only if there is a globally d -rigid subgraph of G H that contains b oth u and v . In the proof of this conjecture w e need some further notions and structural results. Let k ≥ 1 b e an in teger. A maximal k -tree-connected subgraph of a m ultigraph H = ( V , E ) is called a k -sup erbrick of H . It w as shown in [16] that the vertex sets of the sup erbric ks of H form a partition of V . Let M k ( H ) be the matroid union of k copies of the cycle matroid of H . The k -sup erbricks corresp ond to the non-trivial connected comp onen ts of this matroid: Lemma 7.3. [16, L emma 2.11] L et H = ( V , E ) b e a multigr aph and let k ≥ 1 b e an inte ger. L et F ⊆ E b e the set of bridges in M k ( H ) . Then the k -sup erbricks of G ar e the c onn e cte d c omp onents of the gr aph ( V , E − F ) . W e are ready to prov e the main result of this section and confirm the ab ov e mentioned conjecture of Connelly , Jord´ an, and Whiteley [4, Section 6.2] on globally d -linked pairs, whic h has b een unsolved even for d = 2. F or a multigraph H = ( V , E ) and X ⊆ V let B X = ∪ w ∈ X V ( B w ) b e the union of the v ertex sets of the corresp onding b o dies in G H . Theorem 7.4. L et H = ( V , E ) b e a multigr aph with | V | ≥ 2 and | E | ≥ 2 and let G H b e the b o dy- b ar gr aph induc e d by H . L et u, v ∈ V ( G H ) b e a non-adjac ent p air and let d ≥ 1 . Then { u, v } is glob al ly d -linke d in G H if and only if ther e exists a  d +1 2  -sup erbrick S of H with u, v ∈ B V ( S ) . Pr o of. Let S b e a  d +1 2  -sup erbric k of H with u, v ∈ B V ( S ) . Since u and v are non-adjacen t, we ha ve | V ( S ) | ≥ 2. Theorem 7.2 implies that G S , which is a b o dy-bar subgraph of G H , is globally d -rigid. This subgraph ma y prop erly in tersect some bo dies in G H . Ho w ever, b y using that S has minim um degree at least  d +1 2  + 1 ≥ d + 1, it follo ws that these intersections ha ve cardinality at least d + 1. Th us G H [ B V ( S ) ] is also globally d -rigid. This pro v es sufficiency , and shows that the  d +1 2  -sup erbric ks of H induce a partition of G H in to globally d -rigid subgraphs. Let us consider necessit y . Our goal is to show that if the v ertices of a non-adjacen t pair { u, v } of G H b elong to different mem b ers of this partition, then they are not globally d -linked. By adding edges to H without in tro ducing new  d +1 2  -sup erbric ks (and hence adding new v ertices and edges to G H ) we may assume that H is  d +1 2  -tree-connected, and hence G H is d -rigid b y Theorem 7.1. Let C 1 , C 2 , . . . , C q b e the  d +1 2  -sup erbric ks of H , and let B i = B V ( C i ) for 1 ≤ i ≤ q . Then B = { B 1 , B 2 , ..., B q } is a partition of V ( G H ), and by our assumption, u and v b elong to differen t mem b ers of B . Since G H is d -rigid and uv / ∈ E ( G H ), G H + uv con tains an R d -circuit J with uv ∈ E ( J ). Since J is 2-edge-connected, there is an edge xy ∈ E ( J ), different from uv , suc h that x and y b elong to different mem b ers of B . By symmetry w e may assume that y / ∈ { u, v } . Hence, by the construction of the b o dy-bar graph, H has a unique edge f ∈ E corresp onding to xy , and f / ∈ E ( C i ) for 1 ≤ i ≤ q . Therefore, f is a bridge in M ( d +1 2 ) ( H ) by Lemma 7.3, hence xy is an R d -bridge in G H b y Theorem 7.1. It follo ws that G H − xy is not d -rigid. F urthermore, the existence of the R d -circuit J with xy , uv ∈ E ( J ) implies that G H − xy + uv is d -rigid. By using that G H − xy is not d -rigid, w e obtain that { u, v } is not d -linked in G H − xy . Th us { u, v } is not globally d -link ed in G H − y . The b o dy-bar structure and the d -rigidit y of G H , together with Theorem 7.1, imply that deg G H ( y ) ≥ d + 2 and the v ertices in N G H ( y ) − { x } induce a complete s ubgraph in G H . So 13 the conditions of Theorem 4.3 are satisfied, and w e obtain that { u, v } is not globally d -linked in G H . This completes the pro of. W e ha ve the following corollary , whic h gives affirmativ e answ ers to t wo conjectures from [6] and [8], respectively , men tioned in the Introduction, in the sp ecial case of b o dy-bar graphs. Corollary 7.5. L et H = ( V , E ) b e a multigr aph with | V | ≥ 2 and | E | ≥ 2 , and let G H b e the b o dy-b ar gr aph induc e d by H . L et u, v ∈ V ( G H ) and let d ≥ 1 . Then (a) X ⊆ V ( G H ) is a glob al ly d -linke d cluster of G H if and only if X = B V ( S ) for some  d +1 2  - sup erbrick S of H , or X is the end-vertex p air of an e dge of G H not induc e d by such a cluster, (b) { u, v } is glob al ly d -linke d in G H if and only if either uv ∈ E ( G H ) or ther e is an R d -c onne cte d sub gr aph G ′ of G H with κ G ′ ( u, v ) ≥ d + 1 , and (c) { u, v } is d -str ess-linke d in G H if and only if { u, v } is glob al ly d -linke d in G H . Pr o of. (a) follo ws directly from Theorem 7.4. (b) Let us consider a non-adjacent globally d -link ed pair { u, v } . It b elongs to a non-trivial globally d -link ed cluster of G H . By (a) this cluster induces a globally d -rigid subgraph G ′ of G H . Hence G ′ is ( d + 1)-connected. By a result of [7] G ′ is R d -connected. Th us (b) holds. (c) Necessit y was prov ed in [6, Theorem 4.2]. Supp ose that { u, v } is a non-adjacen t globally d -link ed pair in G H . It follo ws from (the proof of ) (b) that there is a globally d -rigid subgraph G ′ of G H with u, v ∈ V ( G ′ ). The result in [6, Prop osition 4.3] implies that a graph is globally d -rigid if and only if each pair of its v ertices is d -stress-link ed. Thus { u, v } is d -stress-link ed in G ′ . By [6, Lemma 4.9] it is also d -stress-link ed in G H . Corollary 7.5(a) implies that the non-trivial globally d -linked clusters of G H are determined b y an appropriate partition of the vertex set of H . W e conjecture that this prop erty holds in b o dy-hinge graphs, too. The global d -rigidit y of these graphs, whic h can b e describ ed as collec- tions of rigid b o dies in whic h some pairs of b o dies share d − 1 v ertices, has been characterized in [21]. Given a multigraph H and k ≥ 1, the graph k H is obtained from H by replacing every edge e b y k copies of e . Conjecture 7.6. L et G H b e the b o dy-hinge gr aph induc e d by multigr aph H and let d ≥ 3 . Then a p air { u, v } is glob al ly d -linke d in G H if and only if ther e is a  d +1 2  -sup erbrick S of (  d +1 2  − 1) H which c ontains the vertic es of the b o dies of u and v . 8 Concluding remarks 8.1 Algorithms The com binatorial c haracterizations giv en in Theorems 1.3 and 7.4 imply that the globally 2- link ed pairs in a graph G and the globally d -link ed pairs in a b o dy-bar graph G H can be found in p olynomial time. These corollaries follo w from the fact that the R 2 -comp onen ts, the pairs { u, v } of v ertices with κ G ( u, v ) ≥ 3, and the k -superbricks of a m ultigraph can be found efficien tly , see [16, 20]. Another algorithmic implication is concerned with the family of d -joined graphs, defined in [8]. A graph G = ( V , E ) is said to b e d -joine d , for some d ≥ 1, if G is d -rigid, and for all u, v ∈ V the pair { u, v } is globally d -link ed in G if and only if uv ∈ E or κ G ( u, v ) ≥ d + 1. F or example, R 2 -connected graphs are 2-joined b y Theorem 2.2. It was pointed out in [8] that d -joined graphs ha ve v arious in teresting properties, but it remained an op en problem to dev elop an algorithm for testing whether a graph is d -joined, for d ≥ 2. Theorem 1.3 gives rise to suc h an algorithm 14 for d = 2. It also provides an affirmativ e answer to the (2-dimensional version of ) [8, Conjecture 5.7], which has further algorithmic consequences, see [8]. 8.2 Uniquely lo calizable v ertices The theory of globally rigid graphs and globally linked pairs has several applications, for example, in lo calization problems of wireless sensor netw orks, see, e.g., [15]. The following definition in [17] was motiv ated b y this context. Let us assume d = 2 and let ( G, p ) b e a generic framework with a designated set P ⊆ V ( G ) of vertices. W e say that a vertex v ∈ V ( G ) is uniquely lo c alizable with respect to P if whenev er ( G, q ) is equiv alen t to ( G, p ) and p ( b ) = q ( b ) for all v ertices b ∈ P , then we also hav e p ( v ) = q ( v ). W e call a vertex v uniquely lo c alizable in graph G with resp ect to P ⊆ V ( G ) if v is uniquely lo calizable with resp ect to P in all generic framew orks ( G, p ). Let G + K ( P ) denote the graph obtained from G by adding all edges bb ′ for which bb ′ / ∈ E ( G ) and b, b ′ ∈ P . Theorem 1.3 implies the following characterization of uniquely lo calizable vertices, confirming [17, Conjecture 6.3]. Theorem 8.1. L et G = ( V , E ) b e a gr aph, P ⊆ V , v ∈ V − P . 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