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A Separator Theorem for Graphs with an Excluded Minor and its Applications Paul Seymour Noga Alon Robin Thomas Lecturer : Daniel Motil

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19.01.2009 A Separator Theorem for Graphs with an Excluded Minor and its Applications Introduction Let H be a h-vertex graph Let G be a n-vertex graph with nonnegative weights whose sum is 1, and with no minor isomorphic H We prove that there is a set X with less than h 3/2 n 1/2 vertices whose deletion creates a graph in which the total weight of every connected component is at most 1/2

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19.01.2009 A Separator Theorem for Graphs with an Excluded Minor and its Applications Main result An algorithm which finds, given an n-vertex graph G with weights as above and an h-vertex graph H, either such a set X or a minor of G isomorphic to H The algorithm runs in time O(h 1/2 n 1/2 m), where m is the number of edges of G plus the number of its vertices

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19.01.2009 A Separator Theorem for Graphs with an Excluded Minor and its Applications Motivation Extensions of the well-known theorem of Lipton and Tarjan for planar graphs Lipton-Tarjan separator theorem has many known applications, which can be extended now from the class of planar graphs to any class of graphs with an excluded minor

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19.01.2009 A Separator Theorem for Graphs with an Excluded Minor and its Applications Separation of graph A separation of a graph G=(V,E) is a pair (A,B) such as – A,B ⊆ V – A ∪ B = V – No edges between A – B and B – A The order of a separation is |A ∩ B|

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19.01.2009 A Separator Theorem for Graphs with an Excluded Minor and its Applications Theorem 1.1 (Lipton-Tarjan) Let G = (V,E) be a planar graph with n vertices, and let w : V → R + be a weight function. Then there is a separation (A,B) of G of order ≤ 2√2∙n 1/2, such that w(A - B), w(B - A) ≤ ⅔w(V )

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19.01.2009 A Separator Theorem for Graphs with an Excluded Minor and its Applications Minor of a graph A graph H is called a minor of a graph G if it can be obtained from a subgraph of G by a series of edge contractions edge contractions – removing the edge and combining its two endpoints

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19.01.2009 A Separator Theorem for Graphs with an Excluded Minor and its Applications Lipton-Tarjan Theorem extension Kuratowski-Wagner Theorem asserts that planar graphs are those without K 5 or K 3,3 minors We now can extend Lipton-Tarjan theorem to graphs with an excluded minor

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19.01.2009 A Separator Theorem for Graphs with an Excluded Minor and its Applications Theorem 1.2 Let H be a simple graph with h ≥ 1 vertices, let G = (V, E) be a graph with n vertices, no H-minor and with a weight function w : V → R + Then there is a separation (A,B) of G of order ≤ h 3/2 n 1/2, such that w(A - B), w(B - A) ≤ ⅔w(V )

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19.01.2009 A Separator Theorem for Graphs with an Excluded Minor and its Applications Theorem 1.2 – Some notes Since K h contains every simple graph with h vertices it suffices to prove this theorem for the case H = K h We suspect that the estimate h 3/2 n 1/2 in this theorem can be replaced by O(h ∙ n 1/2 )

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19.01.2009 A Separator Theorem for Graphs with an Excluded Minor and its Applications Theorem 1.2 – X-flap For a graph G = (V,E) and X ⊆ V, an X-flap is the vertex set of some connected component of G \ X If X ⊆ V is such that w(F) ≤ ⅔w(V ) for every X-flap F then it is easy to find a separation (A,B) with A ∩ B = X such that w(A - B), w(B - A) ≤ ⅔w(V ) Thus, Theorem 1.2 is implied by the following

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19.01.2009 A Separator Theorem for Graphs with an Excluded Minor and its Applications Proposition 1.3 Let G be a graph with n vertices and with no K h -minor and let w : V → R + be a weight function. Then there exists X ⊆ V with |X| ≤ h 3/2 n 1/2 such that w(F) ≤ ½w(V ) for every X-flap F

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19.01.2009 A Separator Theorem for Graphs with an Excluded Minor and its Applications Theorem 1.4 There is an algorithm which takes as input an integer h ≥ 1, a graph G = (V,E), and a function w : V → R + and output either 1. a K h -minor of G 2. A subset X ⊆ V with |X| ≤ h 3/2 n 1/2 such that w(F) ≤ ½w(V ) for every X-flap F

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19.01.2009 A Separator Theorem for Graphs with an Excluded Minor and its Applications Algorithm running time The algorithm running time is O(h 1/2 n 1/2 m), where n = V and m = V + E Unlike some other recent polynomial time algorithms involving graph minors this algorithm has no large constants hidden in the O notation above On the other hand it is not as efficient as the linear time one given by Lipton and Tarjan for the planar case.

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19.01.2009 A Separator Theorem for Graphs with an Excluded Minor and its Applications Haven of order A haven of order k in a graph G = (V,E) is a function β which assigns to each subset X ⊆ V with |X| ≤ k an X-flap β(X), in such a way that if X ⊆ Y then β(Y) ⊆ β(X) V X Y β(X) β(Y)

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19.01.2009 A Separator Theorem for Graphs with an Excluded Minor and its Applications Proposition 1.3 - Haven of order In Proposition 1.3 we claim that for graph G with no K h -minor and a weight function: there exists X ⊆ V with |X| ≤ h 3/2 n 1/2 such that for every X-flap F, w(F) ≤ ½w(V ) So, if Proposition 1.3 is false then for each X ⊆ V with |X| ≤ h 3/2 n 1/2 there exists X-flap F such that w(F) > ½w(V)

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19.01.2009 A Separator Theorem for Graphs with an Excluded Minor and its Applications Proposition 1.3 - Haven of order We now can define for each X ⊆ V with |X| ≤ h 3/2 n 1/2, β(X) to be that X-flap w(β(X)) > ½w(V) Clearly, β(Y) ⊆ β(X), because other connected component of G \ Y has weight less than ½w(V) So, β defined a haven of order h 3/2 n 1/2 X Y β(X)

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19.01.2009 A Separator Theorem for Graphs with an Excluded Minor and its Applications Theorem 1.5 Therefore, Proposition 1.3 is implied by the following more general and more compact result: Theorem 1.5 : Let h ≥ 1 be an integer, and let G be a graph with n vertices with a haven of order h 3/2 n 1/2. Then G has a K h -minor

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19.01.2009 A Separator Theorem for Graphs with an Excluded Minor and its Applications Lemma 2.1 Let G = (V,E) be a graph with n vertices, let A 1,…, A k be k subsets of V, let r ≥ 1 be real number. Then either: 1. there is a tree T in G with |V(T)| ≤ r such that V(T) ∩ A i ≠ ∅ for i = 1,…,k 2. there exists Z ⊆ V with |Z| ≤ (k - 1)n/r, such that no Z-flap intersects all of A 1,…, A k

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19.01.2009 A Separator Theorem for Graphs with an Excluded Minor and its Applications Proof – Some notations G 1,…,G k-1 - isomorphic copies of G For v ∈ V, v i is the corresponding vertex of G i J - the graph obtained from G 1 ∪ … ∪ G k-1 by adding an edge joining v i-1 and v i for all v ∈ A i G i-1 G i G i+1 Ai Ai+1

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19.01.2009 A Separator Theorem for Graphs with an Excluded Minor and its Applications Proof – More notations X = { v 1 : v ∈ A 1 } and Y = { v k-1 : v ∈ A k } G 1 G k-1 Finally, for each u ∈ V(J), let d(u) be the number of vertices in the shortest path between X and u (or ∞ if there is no such path) There are two cases: A 1 =X A k =Y

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19.01.2009 A Separator Theorem for Graphs with an Excluded Minor and its Applications Lemma 2.1 – Case 1 Case 1: d(u) ≤ r for some u ∈ Y Let P be a path in J between X and Y with less than r vertices Let S = { v ∈ V(G) : v i ∈ V(P) } Clearly, |S| ≤ |V(P)| ≤ r, the sub graph of G induced on S is a tree and S ∩ A i ≠ ∅ for i = 1,…,k

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19.01.2009 A Separator Theorem for Graphs with an Excluded Minor and its Applications Lemma 2.1 – Case 2 Case 2: d(u) > r for all u ∈ Y Let t be the least integer with t ≥ r For 1 ≤ j ≤ t, let Z j = { u ∈ V(J) : d(u) = j } Since |V(J)| = (k - 1)n and Z 1,.., Z t are mutually disjoint, one of them, say Z j, has cardinality ≤ (k - 1)n/t ≤ (k - 1)n/r

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19.01.2009 A Separator Theorem for Graphs with an Excluded Minor and its Applications Lemma 2.1 – Case 2 Clearly every path in J between X and Y has a vertex in Z j since d(u) ≥ j for all u ∈ Y Let Z = { v ∈ V(G) : v i ∈ Z j } |Z| ≤ |Z j | ≤ (k - 1)n/r Let us now show that Z satisfies the second option of the lemma

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19.01.2009 A Separator Theorem for Graphs with an Excluded Minor and its Applications Lemma 2.1 – Case 2 Suppose that F is a Z-flap intersects all of A 1,…,A k Let a i ∈ F ∩ Ai (1 ≤ i ≤ k), and P i be a path of G with V(P i ) ⊆ F and with ends a i,a i+1 Let P i be the path of G i corresponding to P i Then V(P 1 ) ∪ … ∪ V(P k-1 ) includes the vertex set of a path in J between X and Y, and yet is disjoint from Z j, a contradiction

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19.01.2009 A Separator Theorem for Graphs with an Excluded Minor and its Applications Applications Lipton and Tarjan and Rose gave many applications of the planar separator theorem (and noted that most of them would generalize to any family of graphs with small separators) Indeed our results supply simple generalizations of all these applications.

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19.01.2009 A Separator Theorem for Graphs with an Excluded Minor and its Applications Applications In particular it follows that for any fixed graph H, given a graph G with n vertices and with no H-minor one can approximate the size of the maximum independent set of G up to a relative error of 1/√(logn) in polynomial time. In time 2 O(√n) one can nd that size exactly and find the chromatic number of G

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19.01.2009 A Separator Theorem for Graphs with an Excluded Minor and its Applications Applications In general, All the applications of the Lipton- Tarjan planar separator theorem carry over, by our result, to any class of graphs with an excluded minor Let us see some of them

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19.01.2009 A Separator Theorem for Graphs with an Excluded Minor and its Applications Proposition 4.1 Let G be an n-vertex graph with no K h -minor, and with nonnegative weights whose total sum is 1 assigned to its vertices. Then, for any 0 < ε ≤ 1 there is a set of at most O(h 3/2 n 1/2 / ε 1/2 ) vertices of G whose removal leaves G with no connected component whose total weight exceeds ε Such a set can be found in time O(h 1/2 n 1/2 m)

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19.01.2009 A Separator Theorem for Graphs with an Excluded Minor and its Applications Proposition 4.1 This proposition can be used to obtain a polynomial time algorithm for approximating the size of the maximum independent set of a graph with an excluded minor

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19.01.2009 A Separator Theorem for Graphs with an Excluded Minor and its Applications Proposition 4.2 There is an algorithm that approximates, given an n-vertex graph G with no K h -minor, the size of a maximum independent set in it with a relative error of O(h 5/2 (logh) 1/2 / (logn) 1/2 ) in time O(h 1/2 n 1/2 m)

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