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28
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 64 (17 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.
The bidimensionality Theory and Its Algorithmic Applications
 Computer Journal
, 2005
"... This paper surveys the theory of bidimensionality. This theory characterizes a broad range of graph problems (‘bidimensional’) that admit efficient approximate or fixedparameter solutions in a broad range of graphs. These graph classes include planar graphs, map graphs, boundedgenus graphs and gra ..."
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Cited by 52 (3 self)
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This paper surveys the theory of bidimensionality. This theory characterizes a broad range of graph problems (‘bidimensional’) that admit efficient approximate or fixedparameter solutions in a broad range of graphs. These graph classes include planar graphs, map graphs, boundedgenus graphs and graphs excluding any fixed minor. In particular, bidimensionality theory builds on the Graph Minor Theory of Robertson and Seymour by extending the mathematical results and building new algorithmic tools. Here, we summarize the known combinatorial and algorithmic results of bidimensionality theory with the highlevel ideas involved in their proof; we describe the previous work on which the theory is based and/or extends; and we mention several remaining open problems. 1.
Approximation algorithms via contraction decomposition
 Proc. 18th Ann. ACMSIAM Symp. Discrete Algorithms ACMSIAM symposium on Discrete algorithms
, 2007
"... We prove that the edges of every graph of bounded (Euler) genus can be partitioned into any prescribed number k of pieces such that contracting any piece results in a graph of bounded treewidth (where the bound depends on k). This decomposition result parallels an analogous, simpler result for edge ..."
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Cited by 41 (8 self)
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We prove that the edges of every graph of bounded (Euler) genus can be partitioned into any prescribed number k of pieces such that contracting any piece results in a graph of bounded treewidth (where the bound depends on k). This decomposition result parallels an analogous, simpler result for edge deletions instead of contractions, obtained in [Bak94, Epp00, DDO + 04, DHK05], and it generalizes a similar result for “compression ” (a variant of contraction) in planar graphs [Kle05]. Our decomposition result is a powerful tool for obtaining PTASs for contractionclosed problems (whose optimal solution only improves under contraction), a much more general class than minorclosed problems. We prove that any contractionclosed problem satisfying just a few simple conditions has a PTAS in boundedgenus graphs. In particular, our framework yields PTASs for the weighted Traveling Salesman Problem and for minimumweight cedgeconnected submultigraph on boundedgenus graphs, improving and generalizing previous algorithms of [GKP95, AGK + 98, Kle05, Gri00, CGSZ04, BCGZ05]. We also highlight the only main difficulty in extending our results to general Hminorfree graphs.
A Borsuk theorem for antipodal links and a spectral characterization of linklessly embeddable graphs
"... For any undirected graph G, let µ(G) be the graph parameter introduced by Colin de Verdière. In this paper we show that (G) 4 if and only if G is linklessly embeddable (in R 3 ). This forms a spectral characterization of linklessly embeddable graphs, and was conjectured by Robertson, Seymour, and ..."
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Cited by 38 (8 self)
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For any undirected graph G, let µ(G) be the graph parameter introduced by Colin de Verdière. In this paper we show that (G) 4 if and only if G is linklessly embeddable (in R 3 ). This forms a spectral characterization of linklessly embeddable graphs, and was conjectured by Robertson, Seymour, and Thomas. A key ingredient is a Borsuktype theorem on the existence of a pair of antipodal linked (k \Gamma 1) spheres in certain mappings OE : S 2k ! R 2k\Gamma1 . This result might be of interest in its own right. We also derive that (G) 4 for each linklessly embeddable graph G = (V; E), where (G) is the graph paramer introduced by van der Holst, Laurent, and Schrijver. (It is the largest dimension of any subspace L of R V such that for each nonzero x 2 L, the positive support of x induces a nonempty connected subgraph of G.)
Approximation Algorithms for Classes of Graphs Excluding SingleCrossing Graphs as Minors
"... Many problems that are intractable for general graphs allow polynomialtime solutions for structured classes of graphs, such as planar graphs and graphs of bounded treewidth. ..."
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Cited by 27 (18 self)
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Many problems that are intractable for general graphs allow polynomialtime solutions for structured classes of graphs, such as planar graphs and graphs of bounded treewidth.
On the maximum number of cliques in a graph
, 2006
"... Abstract. A clique is a set of pairwise adjacent vertices in a graph. We determine the maximum number of cliques in a graph for various graph classes. For example, we prove that the maximum number of cliques in a planar graph with n vertices is 8(n − 2). 1. ..."
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Cited by 15 (7 self)
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Abstract. A clique is a set of pairwise adjacent vertices in a graph. We determine the maximum number of cliques in a graph for various graph classes. For example, we prove that the maximum number of cliques in a planar graph with n vertices is 8(n − 2). 1.
Planar decompositions and the crossing number of graphs with an excluded minor
 IN GRAPH DRAWING 2006; LECTURE NOTES IN COMPUTER SCIENCE 4372
, 2007
"... Tree decompositions of graphs are of fundamental importance in structural and algorithmic graph theory. Planar decompositions generalise tree decompositions by allowing an arbitrary planar graph to index the decomposition. We prove that every graph that excludes a fixed graph as a minor has a planar ..."
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Cited by 15 (2 self)
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Tree decompositions of graphs are of fundamental importance in structural and algorithmic graph theory. Planar decompositions generalise tree decompositions by allowing an arbitrary planar graph to index the decomposition. We prove that every graph that excludes a fixed graph as a minor has a planar decomposition with bounded width and a linear number of bags. The crossing number of a graph is the minimum number of crossings in a drawing of the graph in the plane. We prove that planar decompositions are intimately related to the crossing number. In particular, a graph with bounded degree has linear crossing number if and only if it has a planar decomposition with bounded width and linear order. It follows from the above result about planar decompositions that every graph with bounded degree and an excluded minor has linear crossing number. Analogous results are proved for the convex and rectilinear crossing numbers. In particular, every graph with bounded degree and bounded treewidth has linear convex crossing number, and every K3,3minorfree graph with bounded degree has linear rectilinear crossing number.
Exponential speedup of fixed parameter algorithms on K_3,3minorfree or K_5minorfree graphs
 in The 13th Anual International Symposium on Algorithms and Computation—ISAAC 2002
, 2002
"... We present a fixed parameter algorithm that constructively solves the kdominating set problem on graphs excluding one of the K5 or K3,3 as a minor in time O(3 6 √ 34k n O(1)). In fact, we present our algorithm for any Hminorfree graph where H is a singlecrossing graph (can be drawn on the plane ..."
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Cited by 15 (7 self)
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We present a fixed parameter algorithm that constructively solves the kdominating set problem on graphs excluding one of the K5 or K3,3 as a minor in time O(3 6 √ 34k n O(1)). In fact, we present our algorithm for any Hminorfree graph where H is a singlecrossing graph (can be drawn on the plane with at most one crossing) and obtain the algorithm for K3,3(K5)minorfree graphs as a special case. As a consequence, we extend our results to several other problems such as vertex cover, edge dominating set, independent set, cliquetransversal set, kernels in digraphs, feedback vertex set and a series of vertex removal problems. Our work generalizes and extends the recent result of exponential speedup in designing fixedparameter algorithms on planar graphs due to Alber et al. to other (nonplanar) classes of graphs.
A polynomial bound for untangling geometric planar graphs
, 2007
"... ABSTRACT. To untangle a geometric graph means to move some of the vertices so that the resulting geometric graph has no crossings. Pach and Tardos [Discrete Comput. Geom., 2002] asked if every nvertex geometric planar graph can be untangled while keeping at least n ɛ vertices fixed. We answer this ..."
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Cited by 14 (1 self)
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ABSTRACT. To untangle a geometric graph means to move some of the vertices so that the resulting geometric graph has no crossings. Pach and Tardos [Discrete Comput. Geom., 2002] asked if every nvertex geometric planar graph can be untangled while keeping at least n ɛ vertices fixed. We answer this question in the affirmative with ɛ = 1/4. The previous best known bound was Ω ( � log n / log log n). We also consider untangling geometric trees. It is known that every nvertex geometric tree can be untangled while keeping at least � n/3 vertices fixed, while the best upper bound was O((n log n) 2/3). We answer a question of Spillner and Wolff
Fast algorithms for hard graph problems: Bidimensionality, minors, and local treewidth
 In Proceedings of the 12th International Symposium on Graph Drawing, volume 3383 of Lecture Notes in Computer Science
, 2004
"... Abstract. This paper surveys the theory of bidimensional graph problems. We summarize the known combinatorial and algorithmic results of this theory, the foundational Graph Minor results on which this theory is based, and the remaining open problems. 1 ..."
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Cited by 13 (3 self)
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Abstract. This paper surveys the theory of bidimensional graph problems. We summarize the known combinatorial and algorithmic results of this theory, the foundational Graph Minor results on which this theory is based, and the remaining open problems. 1