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Algorithmic Graph Minor Theory: Decomposition, Approximation, and Coloring
 In 46th Annual IEEE Symposium on Foundations of Computer Science
, 2005
"... At the core of the seminal Graph Minor Theory of Robertson and Seymour is a powerful structural theorem capturing the structure of graphs excluding a fixed minor. This result is used throughout graph theory and graph algorithms, but is existential. We develop a polynomialtime algorithm using topolog ..."
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Cited by 47 (12 self)
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At the core of the seminal Graph Minor Theory of Robertson and Seymour is a powerful structural theorem capturing the structure of graphs excluding a fixed minor. This result is used throughout graph theory and graph algorithms, but is existential. We develop a polynomialtime algorithm using topological graph theory to decompose a graph into the structure guaranteed by the theorem: a cliquesum of pieces almostembeddable into boundedgenus surfaces. This result has many applications. In particular, we show applications to developing many approximation algorithms, including a 2approximation to graph coloring, constantfactor approximations to treewidth and the largest grid minor, combinatorial polylogarithmicapproximation to halfintegral multicommodity flow, subexponential fixedparameter algorithms, and PTASs for many minimization and maximization problems, on graphs excluding a fixed minor. 1.
Some recent progress and applications in graph minor theory, Graphs Combin
"... In the core of the seminal Graph Minor Theory of Robertson and Seymour lies a powerful theorem capturing the “rough ” structure of graphs excluding a fixed minor. This result was used to prove Wagner’s Conjecture that finite graphs are wellquasiordered under the graph minor relation. Recently, a n ..."
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Cited by 10 (5 self)
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In the core of the seminal Graph Minor Theory of Robertson and Seymour lies a powerful theorem capturing the “rough ” structure of graphs excluding a fixed minor. This result was used to prove Wagner’s Conjecture that finite graphs are wellquasiordered under the graph minor relation. Recently, a number of beautiful results that use this structural result have appeared. Some of these along with some other recent advances on graph minors are surveyed.
Algorithmic graph minor theory: Improved grid minor bounds and wagner’s contraction
 Proceedings of the Third International Conference on Distributed Computing and Internet Technology
, 2006
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A nearly linear time algorithm for the half integral disjoint paths packing
 ACMSIAM SYMPOSIUM ON DISCRETE ALGORITHMS (SODA’08)
, 2008
"... We consider the following problem, which is called the half integral parity disjoint paths packing problem. Input: A graph G, k pair of vertices (s1, t1), (s2, t2),..., (sk, tk) in G (which are sometimes called terminals), and a parity li for each i with 1 ≤ i ≤ k, where li = 0 or 1. Output: Paths P ..."
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Cited by 4 (0 self)
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We consider the following problem, which is called the half integral parity disjoint paths packing problem. Input: A graph G, k pair of vertices (s1, t1), (s2, t2),..., (sk, tk) in G (which are sometimes called terminals), and a parity li for each i with 1 ≤ i ≤ k, where li = 0 or 1. Output: Paths P1,..., Pk in G such that Pi joins si and ti for i = 1, 2,..., k and parity of length of the path Pi is li, i.e, if li = 0, then length of Pi is even, and if li = 1, then length of Pi is odd for i = 1, 2,..., k. In addition, each vertex is on at most two of these paths. We present an O(mα(m, n) log n) algorithm for fixed k, where n, m are the number of vertices and the number of edges, respectively, and the function α(m, n) is the inverse of the Ackermann function (see by Tarjan [43]). This is the first polynomial time algorithm for this problem, and generalizes polynomial time algorithms by Kleinberg [23] and Kawarabayashi and Reed [20], respectively, for the half integral disjoint paths packing problem, i.e., without the parity requirement. As with the RobertsonSeymour algorithm to solve the k disjoint paths problem, in each iteration, we would like to either use a huge clique minor as a ”crossbar”, or exploit the structure of graphs in which we cannot find such a minor. Here, however, we must maintain the parity of the paths and can only use an ”odd clique minor”. We must also describe the structure of those graphs in which we cannot find such a minor and discuss how to exploit it. We also have algorithms running in O(m (1+ε)) time for any ε> 0 for this problem, if k is up to o(log log log n) for general graphs, up to o(log log n) for
An (almost) Linear Time Algorithm For Odd Cycles Transversal
, 2009
"... We consider the following problem, which is called the odd cycles transversal problem. Input: A graph G and an integer k. Output: A vertex set X ∈ V (G) with X  ≤ k such that G − X is bipartite. We present an O(mα(m, n)) time algorithm for this problem for any fixed k, where n, m are the number o ..."
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Cited by 3 (0 self)
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We consider the following problem, which is called the odd cycles transversal problem. Input: A graph G and an integer k. Output: A vertex set X ∈ V (G) with X  ≤ k such that G − X is bipartite. We present an O(mα(m, n)) time algorithm for this problem for any fixed k, where n, m are the number of vertices and the number of edges, respectively, and the function α(m, n) is the inverse of the Ackermann function (see by Tarjan [38]). This improves the time complexity of the algorithm by Reed, Smith and Vetta [29] who gave an O(nm) time algorithm for this problem. Our algorithm also implies the edge version of the problem, i.e, there is an edge set X ′ ∈ E(G) such that G − X ′ is bipartite. Using this algorithm and the recent result in [16], we give an O(mα(m, n) + n log n) algorithm for the following problem for any fixed k: Input: A graph G and an integer k. Output: Determine whether or not there is a halfintegral k disjoint odd cycles packing, i.e, k odd cycles C1,..., Ck in G such that each vertex is on at most two of these odd cycles. This improves the time complexity of the algorithm by Reed, Smith and Vetta [29] who gave an O(n 3) time algorithm for this problem. We also give a much simpler and much shorter proof for the following result by Reed [28]. The ErdősPósa property holds for the halfintegral disjoint odd cycles packing problem. I.e. either G has a halfintegral k
Some Remarks on the Odd Hadwiger's Conjecture
, 2005
"... We say that H has an odd complete minor of order at least l if there are l vertex disjoint trees in H such that every two of them are joined by an edge, and in addition, all the vertices of trees are twocolored in such a way that the edges within the trees are bichromatic, but the edges between tre ..."
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Cited by 3 (3 self)
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We say that H has an odd complete minor of order at least l if there are l vertex disjoint trees in H such that every two of them are joined by an edge, and in addition, all the vertices of trees are twocolored in such a way that the edges within the trees are bichromatic, but the edges between trees are monochromatic. Gerards and Seymour conjectured that if a graph has no odd complete minor of order l, then it is (l − 1)colorable. This is substantially stronger than the wellknown conjecture of Hadwiger. Recently, Geelen et al. proved that there exists a constant c such that any graph with no odd Kkminor is ck √ log kcolorable. However, it is not known if there exists an absolute constant c such that any graph with no odd Kkminor is ckcolorable.
The Extremal Function for K9 Minors
, 2005
"... We prove that every (simple) graph on n ≥ 9 vertices and at least 7n − 27 edges either has a K9 minor, or is isomorphic to K2,2,2,3,3, or is isomorphic to a graph obtained from disjoint copies of K1,2,2,2,2,2 by identifying cliques of size six. The proof of one of our lemmas is computerassisted. 1 ..."
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Cited by 2 (0 self)
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We prove that every (simple) graph on n ≥ 9 vertices and at least 7n − 27 edges either has a K9 minor, or is isomorphic to K2,2,2,3,3, or is isomorphic to a graph obtained from disjoint copies of K1,2,2,2,2,2 by identifying cliques of size six. The proof of one of our lemmas is computerassisted. 1