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55
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 46 (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.
List colouring squares of planar graphs
, 2008
"... In 1977, Wegner conjectured that the chromatic number of the square of every planar graph G with maximum degree ∆ ≥ 8 is at most ⌊ 3 2 ∆ ⌋ + 1. We show that it is at most 3 2 ∆ (1 + o(1)), and indeed this is true for the list chromatic number and for more general classes of graphs. ..."
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Cited by 27 (5 self)
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In 1977, Wegner conjectured that the chromatic number of the square of every planar graph G with maximum degree ∆ ≥ 8 is at most ⌊ 3 2 ∆ ⌋ + 1. We show that it is at most 3 2 ∆ (1 + o(1)), and indeed this is true for the list chromatic number and for more general classes of graphs.
Every minorclosed property of sparse graphs is testable
, 2007
"... Suppose G is a graph of bounded degree d, and one needs to remove ɛn of its edges in order to make it planar. We show that in this case the statistics of local neighborhoods around vertices of G is far from the statistics of local neighborhoods around vertices of any planar graph G ′. In fact, a sim ..."
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Cited by 25 (3 self)
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Suppose G is a graph of bounded degree d, and one needs to remove ɛn of its edges in order to make it planar. We show that in this case the statistics of local neighborhoods around vertices of G is far from the statistics of local neighborhoods around vertices of any planar graph G ′. In fact, a similar result is proved for any minorclosed property of bounded degree graphs. As an immediate corollary of the above result we infer that many well studied graph properties, like being planar, outerplanar, seriesparallel, bounded genus, bounded treewidth and several others, are testable with a constant number of queries. None of these properties was previously known to be testable even with o(n) queries. 1
High Girth Graphs Avoiding a Minor are Nearly Bipartite
 J. COMB. THEORY (B
, 1999
"... Let H be a fixed graph. We show that any Hminor free graph of high enough girth has a circularchromatic number arbitrarily close to two. Equivalently, such graphs have homomorphisms into a large odd circuit. In particular, graphs of high girth and of bounded genus or bounded tree width are &quo ..."
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Cited by 13 (0 self)
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Let H be a fixed graph. We show that any Hminor free graph of high enough girth has a circularchromatic number arbitrarily close to two. Equivalently, such graphs have homomorphisms into a large odd circuit. In particular, graphs of high girth and of bounded genus or bounded tree width are "nearly bipartite" in this sense. For example, any planar graph of girth at least 16 admits a homomorphism onto a pentagon. We also obtain tight bounds in a few specific cases of small forbidden minors.
NodeWeighted Steiner Tree and Group Steiner Tree in Planar Graphs
"... Abstract. We improve the approximation ratios for two optimization problems in planar graphs. For nodeweighted Steiner tree, a classical networkoptimization problem, the best achievable approximation ratio in general graphs is Θ(log n), and nothing better was previously known for planar graphs. We ..."
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Cited by 12 (0 self)
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Abstract. We improve the approximation ratios for two optimization problems in planar graphs. For nodeweighted Steiner tree, a classical networkoptimization problem, the best achievable approximation ratio in general graphs is Θ(log n), and nothing better was previously known for planar graphs. We give a constantfactor approximation for planar graphs. Our algorithm generalizes to allow as input any nontrivial minorclosed graph family, and also generalizes to address other optimization problems such as Steiner forest, prizecollecting Steiner tree, and networkformation games. The second problem we address is group Steiner tree: given a graph with edge weights and a collection of groups (subsets of nodes), find a minimumweight connected subgraph that includes at least one node from each group. The best approximation ratio known in general graphs is O(log 3 n), or O(log 2 n) when the host graph is a tree. We obtain an O(log n polyloglog n) approximation algorithm for the special case where the graph is planar embedded and each group is the set of nodes on a face. We obtain the same approximation ratio for the minimumweight tour that must visit each group. 1
On the oddminor variant of Hadwiger’s conjecture
, 2011
"... A Klexpansion consists of l vertexdisjoint trees, every two of which are joined by an edge. We call such an expansion odd if its vertices can be twocoloured so that the edges of the trees are bichromatic but the edges between trees are monochromatic. We show that, for every l, if a graph contains ..."
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Cited by 8 (2 self)
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A Klexpansion consists of l vertexdisjoint trees, every two of which are joined by an edge. We call such an expansion odd if its vertices can be twocoloured so that the edges of the trees are bichromatic but the edges between trees are monochromatic. We show that, for every l, if a graph contains no odd Klexpansion then its chromatic number is O(l √ log l). In doing so, we obtain a characterization of graphs which contain no odd Klexpansion which is of independent interest. We also prove that given a graph and a subset S of its vertex set, either there are k vertexdisjoint odd paths with endpoints in S, or there is a set X of at most 2k − 2 vertices such that every odd path with both ends in S contains a vertex in X. Finally, we discuss the algorithmic implications of these results.
Linear Time Algorithms for Finding a Dominating Set of Fixed Size in Degenerated Graphs
"... Abstract. There is substantial literature dealing with fixed parameter algorithms for the dominating set problem on various families of graphs. In this paper, we give a k O(dk) n time algorithm for finding a dominating set of size at most k in a ddegenerated graph with n vertices. This proves that ..."
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Cited by 8 (0 self)
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Abstract. There is substantial literature dealing with fixed parameter algorithms for the dominating set problem on various families of graphs. In this paper, we give a k O(dk) n time algorithm for finding a dominating set of size at most k in a ddegenerated graph with n vertices. This proves that the dominating set problem is fixedparameter tractable for degenerated graphs. For graphs that do not contain Kh as a topological minor, we give an improved algorithm for the problem with running time (O(h)) hk n. For graphs which are Khminorfree, the running time is further reduced to (O(log h)) hk/2 n. Fixedparameter tractable algorithms that are linear in the number of vertices of the graph were previously known only for planar graphs. For the families of graphs discussed above, the problem of finding an induced cycle of a given length is also addressed. For every fixed H and k, we show that if an Hminorfree graph G with n vertices contains an induced cycle of size k, then such a cycle can be found in O(n) expected time as well as in O(n log n) worstcase time. Some results are stated concerning the (im)possibility of establishing linear time algorithms for the more general family of degenerated graphs. Keywords: Hminorfree graphs, degenerated graphs, dominating set problem, finding an induced cycle, fixedparameter tractable algorithms. 1
Polynomial treewidth forces a large gridlikeminor
, 2008
"... Robertson and Seymour proved that every graph with sufficiently large treewidth contains a large grid minor. However, the best known bound on the treewidth that forces an ℓ × ℓ grid minor is exponential in ℓ. It is unknown whether polynomial treewidth suffices. We prove a result in this direction. ..."
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Cited by 7 (1 self)
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Robertson and Seymour proved that every graph with sufficiently large treewidth contains a large grid minor. However, the best known bound on the treewidth that forces an ℓ × ℓ grid minor is exponential in ℓ. It is unknown whether polynomial treewidth suffices. We prove a result in this direction. A gridlikeminor of order ℓ in a graph G is a set of paths in G whose intersection graph is bipartite and contains a Kℓminor. For example, the rows and columns of the ℓ × ℓ grid are a gridlikeminor of order ℓ + 1. We prove that polynomial treewidth forces a large gridlikeminor. In particular, every graph with treewidth at least cℓ 4 √ log ℓ has a gridlikeminor of order ℓ. As an application of this result, we prove that the cartesian product G □ K2 contains a Kℓminor whenever G has treewidth at least cℓ 4 √ log ℓ.
Dense Minors in Graphs of Large Girth
 Combinatorica
"... this paper is to reduce the upper bound for the required girth to the correct order of magnitude: Theorem 1. For any inte k,e ve graph G of girth g(G) > 6 log k +3and #(G) # 3 has a minor H with #(H) >k. The best lower bound we have found is 8 3 log c, but we note that existing conjectu ..."
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Cited by 7 (0 self)
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this paper is to reduce the upper bound for the required girth to the correct order of magnitude: Theorem 1. For any inte k,e ve graph G of girth g(G) > 6 log k +3and #(G) # 3 has a minor H with #(H) >k. The best lower bound we have found is 8 3 log c, but we note that existing conjectures about cubic graphs of large girth would raise this to about 4 log . Since an average degree of at least cr # log r forces a K r minor [ 5, 8 ], Theorem 1 has the following consequence: Corollary 2.The ee a constant c # R such thate ea graph G of girth g(G) # 6 log r<F1
Relaxing planarity for topological graphs
 Discrete and Computational Geometry, Lecture Notes in Comput. Sci., 2866
, 2003
"... Abstract. According to Euler’s formula, every planar graph with n vertices has at most O(n) edges. How much can we relax the condition of planarity without violating the conclusion? After surveying some classical and recent results of this kind, we prove that every graph of n vertices, which can be ..."
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Cited by 7 (3 self)
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Abstract. According to Euler’s formula, every planar graph with n vertices has at most O(n) edges. How much can we relax the condition of planarity without violating the conclusion? After surveying some classical and recent results of this kind, we prove that every graph of n vertices, which can be drawn in the plane without three pairwise crossing edges, has at most O(n) edges. For straightline drawings, this statement has been established by Agarwal et al., using a more complicated argument, but for the general case previously no bound better than O(n 3/2) was known. 1