~ 2.J2,;N, Such That W(A - H), W(H - A) ~ Jw(V(G))

JOURNAL OF THE AMERICAN MATHEMATICAL SOCIETY Volume 3, Number 4, October 1990

A SEPARATOR THEOREM FOR NONPLANAR GRAPHS

NOGA ALON, PAUL SEYMOUR, AND ROBIN THOMAS

1. INTRODUCTION A separation of a graph G is a pair (A, H) of subsets of V(G) with AuH = V (G) , such that no edge of G joins a vertex in A - H to a vertex in H - A . Its order is IA n HI. A well-known theorem of Lipton and Tarjan [2] asserts the following. (R+ denotes the set of nonnegative real numbers. If w: V (G) --+ R+ is a function and X ~ V(G) , we denote 2::)w(v): v E X) by w(X).)

(1.1) Let G be a planar graph with n vertices, and let w: V(G) --+ R+ be a function. Then there is a separation (A, H) of G of order ~ 2.J2,;n, such that w(A - H), w(H - A) ~ jw(V(G)). Our object is to prove an extension of (1.1) for nonplanar graphs with a fixed excluded "minor." A graph H is a minor of a graph G if H can be obtained from a subgraph of G by contracting edges. By an H-minor of G we mean a minor of G isomorphic to H. Thus, the Kuratowski-Wagner theorem asserts that planar graphs are those without Ks- or K3 , 3-minors. We prove the following:

( 1.2) Let h ~ 1 be an integer, let G be a graph with n vertices and with no Kh-minor, and let w: V(G) -+ R+ be a function. Then there is a separation (A, H) ofG of order ~ h3/ 2n l/2 such that w(A-H), w(H-A) ~ jw(V(G)). Our thanks to N, Linial, who pointed out several years ago to the second author that a result like (1.2) was probably true. We think that the expression h3/2nl/2 in (1.2) is not the best possible, and that O(hnl/2) is the correctanswer, but have not been able to decide this. If true, this would generalize a result of Gilbert, Hutchinson, and Tarjan [1] that every graph with n vertices and genus g has a "separator" of order ~ O(gl/2nl/2) , because Kh has genus ~ Q(h2). Every 3-regular expander with n vertices is a graph with no Kh -minor for h = cn l/2 , and with no separator of size dn, for appropriately chosen positive constants c and d; and hence the estimate O( h n 1/2) would be the best possible.

Received by the editors December 11, 1989 and, in revised form, March 1990; the contents of this paper were presented at the 22nd STOC conference, Baltimore. Maryland, 1990. 1980 Mathematics Subject Classification (1985 Revision). Primary 05C40. The first author's research was carried out under a consulting agreement with Bellcore.

801

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We observe also that since Kh contains an H-minor for every simple graph H with h vertices, (1.2) is equivalent to the following.

(1.3) Let h ~ 1 be an integer, let G be a graph with n vertices and with no H -minor, where H is an arbitrary simple graph with h vertices, and let w: V( G) ~ R+ be a function. Then there is a separation (A, B) of G of order ~ h3/ 2n l / 2 such that w(A - B), weB - A) ~ jw(V(G)).

If G is a graph and X ~ V (G) , an X -flap is the vertex set of some compo- nent of G\X (the graph obtained from G by deleting X). Let w: V(G) ~ R+ be a function. If X ~ V(G) is such that w(F) ~ jw(V(G)) for every X-flap F then it is easy to find a separation (A, B) with A n B = X such that w(A-B), w(B-A) ~ jw(V(G)). (If w(F) ~ iw(V(G)) for some X-flap F, take the separation (F u X, V(G) - F). If not, let the X-flaps be F1 , ••• , Fk and choose j with 1 ~ j ::5 k maximal such that L:I::5i::5j w(F) ~ jw(V(G)); and take the separation (H u X, V(G) - H), where H = U1::5i::5j Fi .) Thus, (1.2) is implied by the following: (1.4) Let h ~ 1 be an integer, let G be a graph with n vertices and with no Kh-minor, and let w: V(G) ~ R+ be a function. Then there exists X ~ V(G) with IXI ~ h3/ 2n l / 2 such that w(F) ~ !w(V(G)) for every X-flap F. Lipton and Tarjan [2] gave an algorithm to find a separation (A, B) as in (1.1) in linear time. We have not been able to do as well, but we shall show the following:

(1.5) There is an algorithm with running time O(h l / 2 n l / 2m), which takes as input an integer h ~ 1, a graph G (where n = W(G)I and m = W(G)I + IE(G)I), and a/unction w: V(G) ~ R+ . It outputs either (a) a Kh-minor of G, or (b) a subset X ~ V(G) with IXI ~ h3/ 2n l / 2 such that w(F) ~ !w(V(G)) for every X -flap F. Several algorithmic applications of this result appear in [4]. By a haven 0/ order k in G we mean a function P which assigns to each subset X ~ V(G) with IXI ~ k an X-flap P(X) , in such a way that if X ~ Y and IYI ~ k then P(Y) ~ P(X). Now if (1.4) is false, then for each X ~ V(G) with IXI ~ h3/ 2n'/2 there is a unique X-flap, say P(X), with w(P(X)) > !w(V(G)); and P thus defined is evidently a haven of order h3/ 2n l / 2 • Thus (1.4) is implied by the following:

(1.6) Let h ~ 1 be an integer and let G be a graph with n vertices with a haven %rder h3/ 2n l / 2 • Then G has a Kh-minor. While (1.6) is more compact and more general than (1.4), it seems difficult to formulate a corresponding generalization of (1.5). We shall content ourselves, therefore, with proving (1.5) and (1.6) separately.

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Let us mention an application of (1.6). A tree-decomposition of a graph G is a pair (T, W), where T is a tree and W = (~: t E V(T)) is a family of subsets of V (G) , such that (i) U(~: t E V(T)) = V(G), and for every e E £(G) there exists t E V (T) such that ~ contains both ends of e; (ii) if tl , t2, t3 E V(T) and t2 lies on the path between tl and t3 then ~n~s;~. I 3 2 The tree-width of G is the minimum k such that there is a tree-decomposition (T, W) of T satisfying I~ I ~ k + 1 for all t E V (T) . The following is proved in [3]: (1.7) If k ;::: 0 is an integer then G has a haven of order ;::: k if and only if the tree-width of G is at least k. From (1.6) and (1.7) we deduce

(1.8) Let h ~ 1 be an integer and let G be a graph with n vertices and with tree-width at least h 3/2n l/2. Then G has a Kh-minor.

2. FINDING SMALL CONNECTING TREES We shall need the following lemma: (2.1) Let G be a graph with n vertices, let AI' ... ' Ak S; V(G), and let r E R+ with r;::: I. Then either (i) there is a tree T in G with W(T)I ~ r such that V(T) n Ai =F 0 for i = I, ... , k, or (ii) there exists Z ~ V(G) with IZI :5 (k - l)njr, such that no Z-jlap intersects all of Al ' ... , Ak . Proof. We may assume that k ;::: 2. Let d, ... ,Gk- I be isomorphic copies of G, mutually disjoint. For each v E V(G) and 1 :5 i :5 k - I, let Vi be the corresponding vertex of d . Let J be the graph obtained from d u· .. U Gk - I by adding, for 2 :5 i :5 k - 1 and all v E Ai ' an edge joining Vi - I and Vi . Let X = {Vi: v E AI} and Y = {V k- I: v E Ak}. For each u E V(J), let d(u) be the number of vertices in the shortest path of J between X and u (or 00 if there is no such path). There are two cases:

Case 1. d(u):5 r for some u E Y. Let P be a path of J between X and Y with ~ r vertices. Let S = {v E V(G): Vi E V(P) for some i, 1:5 i:5 k - I}. Then lSI :5 W(P)I ~ r, the subgraph of G induced on S is connected, and IS nAil =F 0 for 1 :5 i :5 k . Thus (i) holds.

Case 2. d (u) > r for all u E Y .

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Let t be the least integer with t ~ r. For 1 ~ j ~ t, let Zj = {u E V(J): d(u) = j}. Since IV(J)I = (k - l)n and ZI"'" Zt are mutually disjoint, one of them, say Zj' has cardinality ~ (k - 1)njt ~ (k - 1)njr. Now every path of J between X and Y has a vertex in Zj' because d(u) ~ j for all UE Y. Let Z = {v E V(G): Vi E Zj for some i, 1 ~ i ~ k - I}. Then IZI ~ IZ) ~ (k - l)njr, and we claim that Z satisfies (ii). Suppose that F is a Z-flap of G which intersects all of AI' ... , Ak . Let ai E F n Ai (1 ~ i ~ k), and for 1 ~ i ~ k - 1 let Pi be a path of G with V(Pi ) ~ F and with ends ai' ai+1 • Let pi be the path of d corresponding to Pi' Then V (P I) u ... u V (pk-I) includes the vertex set of a path of J between X and Y , and yet is disjoint from Zj' a contradiction. Thus, there is no such F , and so (ii) holds. D We observe that the proof of (2.1) is easily converted to an algorithm with running time O(km) , which, with input G, r, and AI"" ,Ak as in (2.1) (where m = W(G)I + IE(G)I), computes either a tree T as in (i) or a set Z as in (ii). It would be desirable to replace the expression (k - 1) n j r in (ii) by some f(k)njr, where f(k) = o(k), because there would be a corresponding improve- ment in the expression h3/ 2n l / 2 of (1.2). We do not know if this is possible, but we suspect not. Indeed, let G be the "cube" with vertex set

{(XI' ... , xd): XI' x 2' ... , xd E {O, I}}, where (XI"" ,xd) and (x;, ... , x~) are adjacent if Ixl-x;I+" '+lxd-x~1 = 1 . For 1 ~ i ~ d , let Ai = {(XI' ... , Xd): Xi = A}, Ad+i = {(XI' ... ,Xd): Xi = I}, and let k = 2d . Then certainly every tree in G which meets all of AI ' ... , Ak has at least d + 1 vertices, and yet we suspect that, for any Z ~ V(G) with IZI < 2d- 1 , some Z-flap intersects all of AI' ... , Ak . If so, this would show that (2.1) is best possible up to a constant factor.

3. PROOF OF THE THEOREM First we prove (1.6), and then adapt the proof to yield an algorithm for (1.5). Let G be a graph. By a covey in G we mean a set ~ of (nonnull) trees of G, mutually vertex-disjoint, such that for all distinct CI , C2 E ~ there is an edge of G with one end in V ( C1) and the other in V ( C2 ). Thus, if G has a covey of cardinality h then it has a Kh-minor.

Proofof(1.6). Let P be a haven in G of order h3/ 2n l / 2 • Choose X ~ V(G) and a covey ~ with I~I ~ h such that (i) X ~ U(V(C): C E~), (ii) IX n V(C)I ~ h l / 2n l / 2 for each C E ~,

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(iii) V(C) n P(X) = 0 for each C E Cjf, and (iv) subject to (i), (ii), and (iii), I~I + IXI + 3IP(X)1 is minimum. (This is certainly possible; setting ~ = X = 0 satisfies (i), (ii), and (iii).) Let Cjf = {CI ' ... , Ck }. We suppose for a contradiction that k < h. For 1 $ i $ k, let Ai be the set of all v E P(X) adjacent in G to a vertex of Ci • Let G' be the restriction of G to P(X). By (2.1) applied to G' with r = h 1/2 n 1/2 , one of the following cases holds:

Case l. There is a tree T of G' with W(T)I $ h l/ 2n l / 2 ,such that V(T)nAi =f o for 1 $ i $ k. Let Cjf' = Cjf u {T} and X' = Xu V(T); then Cjf' is a covey and for each C E Cjf' ,

V(C) n P(X') ~ V(C) n (P(X) - V(T)) = 0. This contradicts (iv).

Case 2. There exists Z ~ P(X) with IZI $ (k - 1)IP(X)I/hl / 2n l / 2 $ h l/ 2n l/ 2 such that no Z-flap of G' intersects all of AI' ... , Ak • Let Y = Xu Z . Since k $ h - 1, it follows that IYI $ h3/ 2n l / 2 , and so P(Y) exists and P(Y) ~ P(X). Since P(Y) isa Z-flapof G' there exists i with 1 $ i $ k such that p(Y)nAi = 0. Extend Ci to a maximaltree C; of G disjoint from P(Y) and from each Cj (j i= i). Let Z' = V(C;) n Z , let X' = Z' u (X - V(C)) , and let W = V(C;) u (V(G) - P(X)). We claim that P(X') n W = 0. For suppose not. Since P(Y) ~ P(X') , there is a path of G between Wand P(Y) contained within P(X') and hence disjoint from X' . Since W n P(Y) = 0 , there are two consecutive vertices u, v of this path with u E Wand v E V (G) - W ~ P(X). Since u, v are adjacent it follows that u E Xu P(X), and so

U E (X U P(X)) n (W - X') ~ V(C).

Since v ¢ W it follows from the maximality of C; that v E P(Y). Since U ¢ P(Y) we deduce that U E Y, and so

U E Y n (V(C;) - X') ~ V(CJ

But then v E Ai' which is impossible since Ai n P(Y) = 0. This proves our claim that p(X')nW = 0. Hence, P(X') ~ P(X). Let Cjf' = (Cjf-{CJ)u{C;}; then Cjf' is a covey. We observe that (i) X' ~ U(V(C): C E Cjf'); for Z' ~ V(C;) , (ii) IX' n V(C)I $ h l / 2n l / 2 for each C E Cjf'; for if C i= C; then X' n V(C) = X n V(C), and X' n V(C;) = z' , and (iii) V(C) n P(X') = 0 for each C = Cjf'; for P(X') n W = 0, as we have seen.

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By (iv), I~'I + IX'I + 3IP(X')1 ~ I~I + IXI + 3IP(X)I· But I~'I = I~I and X'uP(X') ~ (XuP(X))-(XnV(C;)) , and so XnV(C;) = 0. Then ~ - {C;}, X satisfy (i), (ii), and (iii), contrary to (iv). In both cases, therefore, we have obtained a contradiction. Thus our as- sumption that k < h was incorrect, and so k = h and G has a Kh -minor, as required. 0

Now let us convert the proof of (1.6) to an algorithm for (1.5). The main difference will be that we shall keep the sets X n V (C) as large as possible, to improve the running time. Let h, G, w be the input, and let r = Lh l / 2n l / 2 J. Set Xo = ~ = 0 and Bo = V(G) , and begin the first iteration. In general, at the beginning of the tth iteration, we have a subset XI_I ~ V(G) , a covey ~_I

with I~_II $ h, and a subset BI _ I ~ V (G) which is a union of XI_I-flaps, such that (i) XI_I ~ U(V(C): C E ~_I), (ii) lXI_I n V(C)I = r for each C E ~_I ' (iii) V(C) n BI _ I = 0 for each C E ~_I ' (iv) w(F) $ !w(V(G)) for each XI_I-flap F with F ~ BI_I . (I) Let I~-d = k. If k = h we have found a Kh-minor, we output (a) and stop. Otherwise we go to step (2). (2) We compute all the XI_I-flaps included in BI_I . If w(F) $ !w(V(G)) for every such XI_I-flap F, we output (b) (with X = XI_I) and stop. Other- wise, let F be the unique Xl_I-flap with w(F) > !w(V(G)). If IFI < h l / 2n l / 2 we output (b) (with X = XI_I U F) and stop. Otherwise we go to step (3). (3) Let ~_I = {CI ' ... , Ck }. For I $ i $ k, let A; be the set of all v E F with a neighbor in V(C;). If A; = 0 for some i, we set XI = XI_I - V(C;), ~ = ~_I - {C;}, BI = F, and return to step (1) for the next iteration. Otherwise we go to step (4). (4) Let G' be the restriction of G to F. We apply (2.1) to G' and AI ' ... , Ak . We obtain either: (i) a tree T of G' with W(T)I $ r such that VeT) n A; =j:. 0 for each i, or (ii) a subset Z ~ V(G') with IZI $ (k - l)n/r $ r such that no Z-flap of G' intersects all of AI' ... , Ak . In the first case we go to step (5), and in the second to step (6). (5) Given T as in (4)(i), we enlarge T to a tree T' of G' with W(T)I = r (this is possible since IFI ~ r). We set XI = XI_I u V(T), ~ = ~_I U {T}, BI = F - V (1") , and return to step (I) for the next iteration. (6) Given Z as in (4)(ii), let Y = XI_I U Z. If weD) $ !w(V(D)) for every Y -flap D of G, we output (b) (with X = Y) and stop. Otherwise, let D be the unique Y-flap with weD) > !w(V(G)). Now D is a Z-flap of G' ,

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and so we may choose i with 1 ~ i ~ k such that D n Ai = 0. Extend Ci to a maximal tree C; of G disjoint from D and from each Cj (j i= i). Since Ai n P(Y) = 0, it follows that Ai ~ V(C;). Let

X' = (V(C;) n Z) U (Xt _ 1 - V(CJ);

then, as in the proof of (1.6), it follows that

Fin (V(c)uU(V(C): 1 ~j~k, ji=i)) =0

for any X'-flap F' of G with W(F') > !w(V(G)). Extend C; to a maximal tree T of G disjoint from each Cj (j i= i) and with IV(T) n (Z U D)I ~ r. Since 0 i= Ai ~ V(C;) and IFI ~ r, it follows that IV(T) n FI ~ r, and so we may choose Z' with IZ/I = r such that

v (T) n (Z u D) ~ Z' ~ V (T) n F.

We set Xt = (Xt_1 - V(Ci )) U ZI, ~ = (~_I - {CJ) u {T}, Bt = D - ZI, and return to (1) for the next iteration. (We observe that any Xt-flap F' of G with W(F') > !w(V(G)) is a subset of an X'-flap with the same property, since X' ~ X t ' and hence is disjoint from each member of ~ and is a subset of Bt .) This completes the description of the algorithm (apart from the simple data structures used, which we omit) and the proof of correctness. To analyze the running time, we observe that the most time-consuming part of each iteration is step (4), which takes time ~ O(hm). Since IXol + 21Bol = 2n and in each iteration the quantity IXtl + 21Btl is reduced by at least r, there are at most 2n/r ~ O(h-I/2nl/2) iterations, and so the total running time is at most O(h l / 2n l / 2m) , as claimed. It may be that by using more sophisticated, dynamic data structures, the algorithm can be implemented more efficiently, but at the moment we do not see how to do so.

REFERENCES

1. J. R. Gilbert, J. P. Hutchinson, and R. E. Tarjan, A separation theorem for graphs o/bounded genus, J. Algorithms 5 (1984), 391-407. 2. R. J. Lipton and R. E. Tarjan, A separator theorem/or planar graphs, SIAM J. Appl. Math. 36 (1979),177-189. 3. P. D. Seymour and R. Thomas, Graph searching, and a minimax theorem for tree-width, J. Combin. Theory, Ser. B (to appear). 4. N. Alon, P. D. Seymour, and R. Thomas, A separator theorem for graphs with an excluded minor and its applications (Proc. 22nd STOC, Baltimore, Maryland, 1990), ACM Press, 293-299.

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ABSTRACT. Let G be an n-vertex graph with no minor isomorphic to an h- vertex complete graph. We prove that the vertices of G can be partitioned into three sets A, B , C such that no edge joins a vertex in A with a vertex in B, neither A nor B contains more than 2nj3 vertices, and C contains no more than h3/2n l /2 vertices. This extends a theorem of Lipton and Tarjan for planar graphs. We exhibit an algorithm which finds such a partition (A, B, C) in time O(h l / 2n l / 2m), where m = JV(G)I + IE(G)I.

DEPARTMENT OF MATHEMATICS, SACKLER FACULTY OF EXACT SCIENCE, TEL AVIv UNIVERSITY, RAMAT AVIV, TEL AVIv 69978, ISRAEL BELLCORE, 445 SOUTH STREET, MORRISTOWN, NEW JERSEY 07960

DIMACS CENTER, HILL CENTER, BUSCH CAMPUS, RUTGERS UNIVERSITY, NEW BRUNSWICK, NEW JERSEY 08903 ScHOOL OF MATHEMATICS, GEORGIA INSTITUTE OF TECHNOLOGY, ATLANTA, GEORGIA 30332

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