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

This paper introduces a separator theorem for graphs with an excluded minor and its applications. The main result is an algorithm that finds either a subset of vertices whose deletion creates a graph with desired properties or a minor isomorphic to a given graph. The algorithm runs in polynomial time.

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

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

  2. 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 h3/2n1/2 vertices whose deletion creates a graph in which the total weight of every connected component is at most 1/2 A Separator Theorem for Graphs with an Excluded Minor and its Applications

  3. 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(h1/2n1/2m), where m is the number of edges of G plus the number of its vertices A Separator Theorem for Graphs with an Excluded Minor and its Applications

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

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

  6. 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∙n1/2, such that w(A - B), w(B - A) ≤ ⅔w(V ) A Separator Theorem for Graphs with an Excluded Minor and its Applications

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

  8. Lipton-Tarjan Theorem extension • Kuratowski-Wagner Theorem asserts that planar graphs are those without K5 or K3,3 minors • We now can extend Lipton-Tarjan theorem to graphs with an excluded minor A Separator Theorem for Graphs with an Excluded Minor and its Applications

  9. 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 ≤ h3/2n1/2 , such that w(A - B), w(B - A) ≤ ⅔w(V ) A Separator Theorem for Graphs with an Excluded Minor and its Applications

  10. Theorem 1.2 – Some notes • Since Kh contains every simple graph with h vertices it suffices to prove this theorem for the case H = Kh • We suspect that the estimate h3/2n1/2 in this theorem can be replaced by O(h∙n1/2 ) A Separator Theorem for Graphs with an Excluded Minor and its Applications

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

  12. Proposition 1.3 Let G be a graph with n vertices and with no Kh-minor and let w : V → R+ be a weight function. Then there exists X ⊆ V with |X| ≤ h3/2n1/2 such that w(F) ≤ ½w(V ) for every X-flap F A Separator Theorem for Graphs with an Excluded Minor and its Applications

  13. 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 • a Kh -minor of G • A subset X ⊆ V with |X| ≤ h3/2n1/2 such that w(F) ≤ ½w(V ) for every X-flap F A Separator Theorem for Graphs with an Excluded Minor and its Applications

  14. Algorithm running time • The algorithm running time is O(h1/2n1/2m), 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. A Separator Theorem for Graphs with an Excluded Minor and its Applications

  15. 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 Y X β(X) β(Y) A Separator Theorem for Graphs with an Excluded Minor and its Applications

  16. Proposition 1.3 - Haven of order • In Proposition 1.3 we claim that for graph G with no Kh-minor and a weight function: there exists X ⊆ V with |X| ≤ h3/2n1/2 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| ≤ h3/2n1/2 there exists X-flap F such that w(F) > ½w(V) A Separator Theorem for Graphs with an Excluded Minor and its Applications

  17. Proposition 1.3 - Haven of order • We now can define for each X ⊆ V with |X| ≤ h3/2n1/2, 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 h3/2n1/2 Y β(X) X A Separator Theorem for Graphs with an Excluded Minor and its Applications

  18. 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 h3/2n1/2. Then G has a Kh-minor A Separator Theorem for Graphs with an Excluded Minor and its Applications

  19. Lemma 2.1 Let G = (V,E) be a graph with n vertices, let A1,…, Ak be k subsets of V, let r ≥ 1 be real number. Then either: • there is a tree T in G with |V(T)| ≤ r such that V(T) ∩ Ai ≠ ∅ for i = 1,…,k • there exists Z ⊆ V with |Z| ≤ (k - 1)n/r, such that no Z-flap intersects all of A1,…, Ak A Separator Theorem for Graphs with an Excluded Minor and its Applications

  20. Proof – Some notations • G1,…,Gk-1 - isomorphic copies of G • For v ∈ V, viis the corresponding vertex of Gi • J- the graph obtained from G1∪… ∪ Gk-1 by adding an edge joining vi-1 and vi for all v ∈ Ai Gi-1 Gi Gi+1 Ai+1 Ai+1 Ai Ai A Separator Theorem for Graphs with an Excluded Minor and its Applications

  21. Proof – More notations • X = { v1 : v ∈ A1 } and Y = { vk-1 : v ∈ Ak } • G1Gk-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: Ak=Y A1=X A Separator Theorem for Graphs with an Excluded Minor and its Applications

  22. 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) : vi∈ V(P) } • Clearly, |S| ≤ |V(P)| ≤ r, the sub graph of G induced on S is a tree and S ∩ Ai ≠ ∅ for i = 1,…,k A Separator Theorem for Graphs with an Excluded Minor and its Applications

  23. 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 Zj = { u ∈ V(J) : d(u) = j } • Since |V(J)| = (k - 1)n and Z1,.., Zt are mutually disjoint, one of them, say Zj, has cardinality ≤ (k - 1)n/t ≤ (k - 1)n/r A Separator Theorem for Graphs with an Excluded Minor and its Applications

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

  25. Lemma 2.1 – Case 2 • Suppose that F is a Z-flap intersects all of A1,…,Ak • Let ai∈ F ∩ Ai (1 ≤ i ≤ k), and Pi be a path of G with V(Pi) ⊆ F and with ends ai,ai+1 • Let Pi be the path of Gi corresponding to Pi • Then V(P1) ∪ … ∪ V(Pk-1) includes the vertex set of a path in J between X and Y , and yet is disjoint from Zj , a contradiction A Separator Theorem for Graphs with an Excluded Minor and its Applications

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

  27. 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 2O(√n) one can nd that size exactly and find the chromatic number of G A Separator Theorem for Graphs with an Excluded Minor and its Applications

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

  29. Proposition 4.1 Let G be an n-vertex graph with no Kh-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(h3/2n1/2 / ε1/2) 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(h1/2n1/2m) A Separator Theorem for Graphs with an Excluded Minor and its Applications

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

  31. Proposition 4.2 There is an algorithm that approximates, given an n-vertex graph G with no Kh-minor, the size of a maximum independent set in it with a relative error of O(h5/2(logh)1/2/ (logn)1/2) in time O(h1/2n1/2m) A Separator Theorem for Graphs with an Excluded Minor and its Applications

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