A Separator Theorem for Graphs with an Excluded Minor and its Applications

Noga Alon IBM Almaden Research Center, San Jose, CA 95120 ,USA and Sackler Faculty of Exact Sciences, Tel Aviv University, Tel Aviv, Israel Paul Seymour BellCore, 445 South St., Morristown, NJ 07960, USA Robin Thomas School of Mathematics, Georgia Institute of Technology, Atlanta, GA 30332, USA Abstract corresponds to G in time O(n3/2). We also describe a combinatorial application of our result which re- Let G be an n-vertex graph with nonnegative weights lates the tree-width of a graph to the maximum size whose sum is 1 assigned to its vertices, and with no of a Kh-minor in it. minor isomorphic to a given h-vertex graph H. We prove that there is a set X of no more than h3/2n1/2 vertices of G whose deletion creates a graph in 1 Introduction which the total weight of every connected compo- nent is at most 1/2. This extends significantly a A separation of a graph G is a pair (A, B) of subsets well-known theorem of Lipton and Tarjan for pla- of V (G) with A B = V (G), such that no edge of G joins a vertex∪ in A B to a vertex in B nar graphs. We exhibit an algorithm which finds, − − given an n-vertex graph G with weights as above A. Its order is A B . A well-known theorem of | ∩ | + and an h-vertex graph H, either such a set X or a Lipton and Tarjan [7] asserts the following. ( R minor of G isomorphic to H. The algorithm runs denotes the set of non-negative real numbers. If + in time O(h1/2n1/2m), where m is the number of w : V (G) R is a function and X V (G), we denote (→w(v): v X) by w(X).) ⊆ edges of G plus the number of its vertices. Our P ∈ results supply extensions of the many known ap- Theorem 1.1 (Lipton-Tarjan [7]) Let G be a pla- plications of the Lipton-Tarjan separator theorem nar graph with n vertices, and let w : V (G) R+ from the class of planar graphs (or that of graphs be a function. Then there is a separation (A,→ B) of with bounded genus) to any class of graphs with G of order 2√2n1/2, such that an excluded minor. For example, it follows that for w(A B), w≤(B A) 2 w(V (G)). any fixed graph H , given a graph G with n vertices − − ≤ 3 and with no H-minor one can approximate the size Here we prove an extension of this theorem to non- of the maximum independent set of G up to a rela- planar graphs with a fixed excluded “minor”. A tive error of 1/√log n in polynomial time, find that graph H is a minor of a graph G if H can be ob- size exactly and find the chromatic number of G in tained from a subgraph of G by contracting edges. time 2O(√n) and solve any sparse system of n linear By an H-minor of G we mean a minor of G isomor- equations in n unknowns whose sparsity structure phic to H. Thus, the Kuratowski-Wagner Theorem (see, e.g., [1] ) asserts that planar graphs are those without K5 or K3,3 minors. We prove the following result. Theorem 1.2 Let h 1 be an integer, let H be a simple graph with h vertices,≥ and let G be a graph with n vertices, no H-minor and with a weight func- tion w : V (G) R+ . Then there is a separa- → 3/2 1/2 0 tion (A, B) of G of order h n , such that w(A B), w(B A) 2 w(V≤(G)). − − ≤ 3 Note that since the complete graph Kh contains known , though, how to find such an embedding every simple graph on h vertices it suffices to prove in time which is polynomial in both the genus and this theorem for the case H = Kh. Hence we deal, the size of the graph, and in fact this is impossible from now on, only with this case. if P = NP , since the problem of determining the We suspect that the estimate h3/2n1/2 in Theorem genus6 of a graph is, as proved by C. Thomassen [11], 1.2 can be replaced by O(hn1/2). Since the genus NP-complete. Moreover, even for bounded genus, 2 of Kh is Θ(h ) this would extend, if true, a result of the best known algorithm for finding such an em- Gilbert, Hutchinson and Tarjan [3] who proved that bedding is much slower than our separator algo- any graph on n vertices with genus g has a separator rithm). Although our algorithm is not as efficient, of order O(g1/2n1/2). Although an O(hn1/2) result it is much more general. As a very special case it would be an asymptotically better result than the gives an algorithm for finding small separators in one stated above, an advantage of the formulation graphs with genus g, which is polynomial in both given in Theorem 1.2 is that there are no hidden n and g, even when we are not given an embedding constants in the expression for the size of the sepa- of the graph in its genus surface. The question of rator given by it. finding such an algorithm was raised in [3]. If G is a graph and X V (G), an X-flap is By a haven of order k in G we mean a function β the vertex set of some component⊆ of G X (=the which assigns to each subset X V (G) with X graph obtained from G by deleting X.)\ Let w : k an X-flap β(X), in such a way⊆ that if X Y| and| ≤ V (G) R+ be a function. If X V (G) is such Y k then β(Y ) β(X). Clearly, if Proposition⊆ → 2 ⊆ | | ≤ ⊆ that w(F ) 3 w(V (G)) for every X-flap F then it 1.3 is false for G, then for each X V (G) with is easy to find≤ a separation (A, B) with A B = X X h3/2n1/2 there is a unique X-flap,⊆ say β(X), such that w(A B), w(B A) 2 w(V (G)).∩ Thus, with| | ≤w(β(X)) > 1 w(V (G)); and β thus defined − − ≤ 3 2 Theorem 1.2 is implied by the following. is evidently a haven of order h3/2n1/2. Therefore, Proposition 1.3 is implied by the following more Proposition 1.3 Let h 1 be an integer, let G ≥ general and more compact result. be a graph with n vertices and with no Kh-minor and let w : V (G) R+ be a weight function. Then Theorem 1.5 Let h 1 be an integer, and let G → there exists X V (G) with X h3/2n1/2 such be a graph with n vertices≥ with a haven of order 1 ⊆ | | ≤ that w(F ) w(V (G)) for every X-flap F . h3/2n1/2. Then G has a K -minor. ≤ 2 h Our proof is algorithmic and in fact we can show: Lipton and Tarjan [8] , [9] and Lipton, Rose and Tarjan [6] gave many applications of the planar sep- Theorem 1.4 There is an algorithm with running arator theorem (and noted that most of them would time O(h1/2n1/2m), which takes as input an integer generalize to any family of graphs with small sep- h 1, a graph G (where n = V (G) and m = arators.) Indeed our results supply simple gener- V ≥(G) + E(G) ), and a function|w : V|(G) R+. alizations of all these applications. In particular it It| outputs| | either| → follows that for any fixed graph H , given a graph (a) a K -minor of G, or h G with n vertices and with no H-minor one can ap- (b) a subset X V (G) with X h3/2n1/2 such proximate the size of the maximum independent set that w(F ) 1 w⊆(V (G)) for every| |X ≤-flap F . ≤ 2 of G up to a relative error of 1/√log n in polynomial O(√n) We note that (unlike some other recent polynomial- time. In time 2 one can find that size exactly time algorithms involving graph minors) this algo- and find the chromatic number of G. Also, any rithm is practical and easy to implement, and there sparse system of n linear equations in n unknowns are no large constants hidden in the O notation whose sparsity structure corresponds to G can be 3/2 above. solved in time O(n ). It can also be shown that Note that our algorithm is not as efficient as the graphs with an excluded minor are easy to pebble, linear time one given by Lipton and Tarjan for the can be imbedded with a small average proximity in planar case. In fact, even for the case of bounded binary trees, and cannot be the underlying graphs (orientable) genus, the algorithm of [3] for finding of efficient Boolean circuits computing certain func- a small separator runs in linear time, given an em- tions. bedding of the graph in its genus surface. (It is not The rest of this paper is organized as follows. In Y has a vertex in Zj, for d(u) j for all u Y . i ≥ ∈ the next two sections we present the proofs of the Let Z = v V (G): v Zj for some i , 1 { ∈ ∈ ≤ main results; Theorem 1.4 and Theorem 1.5. Our i k 1 . Then Z Zj (k 1)n/r, and we method, unlike the ones used for proving the sep- claim≤ − that} Z satisfies| | ≤(ii). | For| ≤ suppose− that F is a arator theorems for planar graphs and for graphs Z-flap of G which intersects all of A1,...,Ak. Let of bounded genus, does not involve any topologi- ai F Ai (1 i k), and for 1 i k 1 let ∈ ∩ ≤ ≤ ≤ ≤ − cal considerations concerning the embedding of the Pi be a path of G with V (Pi) F and with ends i ⊆ i graphs, and is purely combinatorial. A simple but ai, ai+1. Let P be the path of G corresponding 1 k 1 useful tool is a lemma concerning connecting trees to Pi. Then V (P ) ... V (P − ) includes the presented in Section 2. In Section 4 we discuss the vertex set of a path of∪J between∪ X and Y , and yet applications of our results; outline the extensions is disjoint from Zj, a contradiction. Thus, there is of the applications given in [9] and [6] to graphs no such F , and so (ii) holds. 2 with excluded minors and mention another combi- We observe that the last proof is easily converted natorial application that relates the tree-width of to an algorithm with running time O(km), which, a graph and the existence of minors of complete with input G, r and A1,...,Ak as in the lemma graphs in it. (where m = V (G) + E(G) ), computes either a tree T as in (i)| or a| set|Z as in| (ii). 2 Finding small connecting trees 3 The algorithm and the proof We shall need the following lemma. First, we present the proof of Theorem 1.5, and Lemma 2.1 Let G be a graph with n vertices, let then adapt the proof to yield an algorithm for The- A ,...,A be k subsets of V (G) and let r 1 be a 1 k orem 1.4. Let G be a graph. By a covey in G real number. Then either ≥ we mean a set of (non-null) trees of G, mutually (i) there is a tree T in G with V (T ) r such that vertex-disjoint,C such that for all distinct C ,C V (T ) A = for i = 1, . . . , k|or | ≤ 1 2 i there is an edge of G with one end in V (C ) and∈ C (ii) there∩ exists6 ∅ Z V (G) with Z (k 1)n/r, 1 ⊆ | | ≤ − the other in V (C2). Thus, if G has a covey of car- such that no Z-flap intersects all of A1,...,Ak. dinality h then it has a Kh-minor. 1 k 1 Proof. We may assume that k 2. Let G ,...,G − Proof of Theorem 1.5 Let β be a haven in G of be isomorphic copies of G, mutually≥ disjoint. For order h3/2n1/2. Choose X V (G) and a covey ⊆ C each v V (G) and 1 i k 1, let vi be the corre- with h such that |C| ≤ sponding∈ vertex of Gi≤. Let≤ J −be the graph obtained (i) X (V (C): C ) ⊆ ∪ ∈ C from G1 ... Gk 1 by adding, for 2 i k 1 (ii) X V (C) h1/2n1/2 for each C − | ∩ | ≤ ∈ C and all v∪ A∪, an edge joining vi 1 and≤ ≤vi.− Let (iii) V (C) β(X) = for each C i − ∩ ∅ ∈ C X = v1 :∈v A , and Y = vk 1 : v A . For (iv) subject to (i), (ii) and (iii), + 2( β(X) + 1 − k |C| | | each u{ V (J∈), let} d(u) be the{ number∈ of vertices} X β(X) ) is minimum. | ∪ | in the shortest∈ path of J between X and u (or (This is certainly possible, for = X = satisfy ∞ C ∅ if there is no such path). There are two cases: (i), (ii) and (iii).) Let = C1,...,Ck . We sup- C { } Case 1: d(u) r for some u Y . pose for a contradiction that k < h. For 1 i k, ≤ ∈ ≤ ≤ Let P be a path of J between X and Y with r ver- let Ai be the set of all v β(X) adjacent in G to a ≤ ∈ tices. Let S = v V (G): vi V (P ) for some i, 1 vertex of Ci. Let G0 be the restriction of G to β(X). { ∈ ∈ ≤ 1/2 1/2 i k 1 . Then S V (P ) r, the subgraph By Lemma 2.1 applied to G0 with r = h n , one ≤ − } | | ≤ | | ≤ of G induced on S is connected, and S Ai = for of the following cases holds: ∩ 6 ∅ 1 i k. Thus (i) holds. Case 1: there is a tree T of G0 with V (T ) ≤ ≤ 1/2 1/2 | | ≤ Case 2: d(u) > r for all u Y . h n , such that V (T ) Ai = for 1 i k. ∩ 6 ∅ ≤ ≤ Let t be the least integer with∈ t r. For 1 j t, Case 2: there exists Z β(X) with Z (k ≥ ≤ ≤ 1/2 1/2 1/⊆2 1/2 | | ≤ − let Zj = u V (J): d(u) = j . Since V (J) = 1) β(X) /h n h n such that no Z-flap { ∈ } | | | | ≤ (k 1)n and Z1,...,Zt are mutually disjoint, one of G0 intersects all of A1,...,Ak. − of them, say Zj, has cardinality (k 1)n/t In the first case we replace by 0 = T ≤ − ≤ C C C ∪ { } (k 1)n/r. Now every path of J between X and and X by X0 = X V (T ) . It can be easily checked − ∪ that this supplies a contradiction to (iv). Now let us convert the proof above to an al- The second case is somewhat more complicated. gorithm, as promised in Theorem 1.4. Let h, G, w In this case we can enlarge one of the members of be the input. Set X0 = 0 = and B0 = V (G), to include some vertices of Z together with severalC and begin the first iteration.C ∅ In general, at the additional properly chosen vertices. This is done beginning of the tth iteration, we have a subset as follows. Define Y = X Z. Since k h 1 Xt 1 V (G), a covey t 1 with t 1 h, and a 3/2 1∪/2 ≤ − − ⊆ C − |C − | ≤ it follows that Y h n , and so β(Y ) exists subset Bt 1 V (G) which is a union of Xt 1-flaps, | | ≤ − ⊆ − and β(Y ) β(X). Since β(Y ) is a Z-flap of G0 such that ⊆ there exists an index i with 1 i k such that (i) Xt 1 (V (C): C t 1) ≤ ≤ − ⊆ ∪ 1∈/2 C 1−/2 β(Y ) Ai = . Extend Ci to a maximal tree Ci0 (ii) Xt 1 V (C) = [h n ] for each C t 1 ∩ ∅ | − ∩ | ∈ C − of G disjoint from β(Y ) and from each Cj,(j = i). (iii) V (C) Bt 1 = for each C t 1 6 ∩ 1 − ∅ ∈ C − Let Z0 = V (Ci0) Z, let X0 = Z0 (X V (Ci)), (iv) w(F ) w(V (G)) for each Xt 1-flap F which ∩ ∪ − ≤ 2 − and let W = V (Ci0) (V (G) β(X)). is not a subset of Bt 1. ∪ − − We claim that β(X0) W = . For suppose ∩ ∅ 1. Let t 1 = k. If k = h we have found a Kh- not. Since β(Y ) β(X0), there is a path of G |C − | between W and β(⊆Y ) contained within β(X ) and minor; we output (a) and stop. Otherwise we 0 go to step (2). hence disjoint from X0. Since W β(Y ) = , there are two consecutive vertices u,v ∩of this path∅ with 2. Compute the Xt 1-flaps of the induced sub- u W and v V (G) W β(X). Since u, v are − 1 graph of G on Bt 1. If w(F ) w(V (G)) for adjacent∈ it follows∈ that−u ⊆X β(X), and so − ≤ 2 ∈ ∪ each such flap F we output (b) (with X = Xt 1 ) and stop. Otherwise we let F be u (X β(X)) (W X0) V (Ci0). − 1 ∈ ∪ ∩ − ⊆ the unique Xt 1-flap with w(F ) > 2 w(V (G)) and go to step− (3). Since v W it follows from the maximality of C0 6∈ i that v β(Y ). Since u β(Y ) we deduce that 1/2 1/2 ∈ 6∈ 3. Let t 1 = C1,...,Ck . If F [h n ] u Y , and so C − { } | | ≤ ∈ we output (b) , with X = Xt 1 F , and stop. − ∪ Otherwise we let G0 be the induced subgraph u Y (V (Ci0) X0) V (Ci). ∈ ∩ − ⊆ of G on F . For 1 i k, let Ai V (G0) ≤ ≤ ⊆ But then v Ai, which is impossible, since Ai be the set of all v V (G0) with a neighbour ∈ ∩ ∈ β(Y ) = . This proves our claim that β(X0) W = in V (Ci). If Ai = for some i, we set Xt = ∅ ∩ ∅ . Hence β(X0) β(X). Let 0 = ( Ci ) Ci0 ; Xt 1 V (Ci), t = t 1 Ci and Bt = F , ∅ ⊆ C C−{ } ∪{ } − − C C − − { } then 0 is a covey. We observe that and return to step (1) for the next iteration. C (i) X0 (V (C): C 0); for Z0 V (Ci0) Otherwise, we go to step (4). ⊆ ∪ 1∈/2 C 1/2 ⊆ (ii) X0 V (C) h n for each C 0; for if | ∩ | ≤ ∈ C 4. We apply Lemma 2.1 to G0 and A1,...,Ak, C = C0 then X0 V (C) = X V (C), and X0 6 i ∩ ∩ ∩ taking r = h1/2n1/2. We obtain either: V (Ci0) = Z0 1/2 1/2 (i) a tree T of G0 with V (T ) h n such (iii) V (C) β(X0) = for each C 0; for β(X0) | | ≤ ∩ ∅ ∈ C ∩ that V (T ) Ai = for each i, or W = , as we have seen. ∩ 6 ∅ ∅ (ii) a subset Z V (G0) such that no Z-flap By (iv), ⊆ of G0 intersects all of A1,...,Ak. 0 + 2( β(X0) + X0 β(X0) ) In the first case we go to step (5), and in the |C | | | | ∪ | ≥ second to step (6). + 2( β(X) + X β(X) ). |C| | | | ∪ | 5. Given T as in (4)(i), extend it to a tree T 0 of But 0 = , β(X0) β(X) and X0 β(X0) 1/2 1/2 |C | |C| ⊆ ∪ ⊆ size [h n ] in G0 and then set Xt = Xt 1 (X β(X)) (X V (Ci)), and so X V (Ci) = . − ∪ ∪ − ∩ ∩ ∅ V (T 0), Bt = F V (T 0), and t = t 1 T 0 , Then Ci ,X satisfy (i),(ii) and (iii), contrary and return to step− (1) for theC nextC − iteration.∪{ } to (iv).C − { } In both cases, therefore, we have obtained a con- 6. Given Z as in (4)(ii), let Y = Xt 1 Z. If tradiction. Thus our assumption that k < h was w(L) 1 w(V (G)) for every Y -flap− ∪L, we ≤ 2 incorrect, and so k = h and G has a Kh-minor, as output (b) (with X = Y ) and stop. Other- required. 2 wise, there is a unique Y -flap L with w(L) > 1 w(V (G)), and it satisfies L Bt 1. Choose extension of the one for the planar case. In all the 2 ⊆ − i with 1 i k such that L Ai = . propositions in this section, when G is a graph we ≤ ≤ ∩ ∅ Extend Ci to a maximal tree Ci0 of G with let m denote the sum of the number of its vertices the property that V (Ci0) is disjoint from L and the number of its edges. and from each V (Cj)(j = i). Put Z0 = By repeatedly applying the separating algorithm 1/2 6 1/2 Z V (Ci0). If Z0 < [h n ] extend it to of Theorem 1.5 to a graph one can obtain the fol- a set∩ of that size by adding to it sufficiently lowing immediate generalization of a result of [9]. many vertices from F such that the graph Proposition 4.1 Let G be an n-vertex graph with induced on V (Ci0) Z0 would still be con- ∪ no Kh-minor, and with nonnegative weights whose nected and replace C0 by a spanning tree on i total sum is 1 assigned to its vertices. Then, for any this union. Let Xt = (Xt 1 V (Ci)) Z0, − − ∪ 0 <  1 there is a set of at most O(h3/2n1/2/1/2) and t = ( t 1 Ci ) Ci0 , and let Bt be ≤ C C − − { } ∪ { } vertices of G whose removal leaves G with no con- the Xt-flap of maximum weight among those contained in F . We return to step (1) for the nected component whose total weight exceeds . Such 1/2 1/2 next iteration. a set can be found in time O(h n m), (indepen- dent of  ). This completes the description of the algorithm. Its correctness can be argued as in the proof of The- This proposition can be used to obtain a polynomial orem 1.5. We have omitted details of the (obvious) time algorithm for approximating the size of the data structures used, but if they are chosen appro- maximum independent set of a graph with an ex- priately, then each of the steps (1), (2), (3), (5) and cluded minor, (and for approximating several simi- (6) takes time O(m) in each iteration in which it lar quantities, whose exact determination is known is called, and step (4) takes time O(hm) in each to be NP-complete). We simply break the graph into pieces of small size (say, size 1 log n ), find the iteration it is called. Notice that in each iteration 2 maximum independent set in each piece by exhaus- the quantity Bt + Bt Xt decreases by at least [h1/2n1/2] (and| this| | quantity∪ | never increases dur- tive search, and combine the results to obtain the ing the algorithm). It follows that each of the steps desired approximation. To estimate the relative er- 1/2 1/2 ror here one can apply the result of Kostochka [4] (1)-(6) is performed O(h− n ) times. The al- gorithm thus has running time O(h1/2n1/2m), as and Thomason [12] and a simple greedy-argument required. This completes the proof of Theorem 1.4. to obtain a lower bound for the size of the maxi- 2 mum independent set in an n-vertex graph with no It may be that by using more sophisticated, dy- Kh-minor. This gives: namic data structures, a more efficient implementa- Proposition 4.2 There is an algorithm that ap- tion of the algorithm can be found. At the moment proximates, given an n-vertex graph G with no Kh- this remains an open problem. minor , the size of a maximum independent set in h5/2(log h)1/2 it with a relative error of O( (log n)1/2 ) in time 4 Applications O(h1/2n1/2m).

Efficient algorithms for finding small separators in Separator theorems are useful in designing effi- graphs are useful in the layout of circuits in a model cient divide-and-conquer algorithms. An example of VLSI (see, e.g., [5]). Thus our results can be given in [9] is that of nonserial dynamic program- applied for finding efficiently embeddings of graphs ming (see, e.g., [13]). with excluded minors. Proposition 4.3 Let f(x , . . . , x ) be a function All the applications of the Lipton-Tarjan planar 1 n of n variables, where each variable takes values in a separator theorem given in [9] and in [6] carry over, finite set S of s elements, and suppose f is a sum of by our result, to any class of graphs with an ex- functions f , where each f is a function of some of cluded minor. Since most of the proofs are straight- i i the variables x . Let G be the graph whose vertices forward extensions of those given in the above pa- j are the variables x in which x is adjacent to x pers we merely state each result, and only add a j j k iff they both appear in a common term f . Suppose few words about its proof when it is not an obvious i , further, that G has no Kh-minor. Then one can find the maximum of f in the domain xj S for to the fixed-excluded-minor case; although its run- 3/2 1/2 ∈ all j in time sO(h n ). ning time matches that of the best general algo- rithm it seems simpler to implement. Note that the graph G is not planar if a single term Finally we mention a combinatorial application fi is a function of more than 4 variables, whereas of our results. A tree-decomposition of a graph G the extension to graphs with an excluded minor al- is a pair (T,W ), where T is a tree and W = (Wt : lows much more complicated functions fi. t V (T )) is a family of subsets of V (G), such that ∈ It is not too difficult to see that the last proposition (Wt : t V (T )) = V (G), for every e E(G) ∪ ∈ ∈ implies that one can find the maximum size of an there exists t V (T ) such that Wt contains both ∈ independent set in an n-vertex graph G with no Kh- ends of e, and if t1, t2, t3 V (T ) and t2 lies on the O(h3/2n1/2) ∈ minor in time 2 and check if such a graph path between t1 and t3 then Wt1 Wt3 Wt2 . 3/2 1/2 ∩ ⊆ is s-colorable in time sO(h n ). For fixed h this The tree-width of G is the minimum k such that shows that the maximum size of an independent there is a tree-decomposition (T,W ) of T satisfying 2 set in a graph with no K -minor and with log n Wt k + 1 for all t V (T ). Combining Theorem h | | ≤ ∈ vertices can be found in time nO(1). This can be 1.5 with one of the results of [10] we can prove the applied to modify the proof of Proposition 4.2 and following. obtain a polynomial-time approximation algorithm Proposition 4.5 Let h 1 be an integer, and let for the size of the maximum independent set in an G be a graph with n vertices≥ and with tree-width at n-vertex graph with a fixed excluded minor with 3/2 1/2 1 least h n . Then G has a Kh-minor. relative error of O( log n ). Pebbling is a one person game that arises in Acknowledgement We would like to thank N. the study of time-space trade-offs (see, e.g., [2]). Linial for fruitful discussions that led us to the An immediate extension of a result from [9] gives; problem considered in this work. Proposition 4.4 Any n-vertex acyclic digraph with no Kh-minor and with maximum in-degree k can be References pebbled using O(h3/2n1/2 + k log n) pebbles. [1] J. A. Bondy and U. S. R. 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