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Subexponential Parameterized Algorithms on Graphs of Bounded Genus and HMinorFree Graphs
, 2003
"... We introduce a new framework for designing fixedparameter algorithms with subexponential running time2 . Our results apply to a broad family of graph problems, called bidimensional problems, which includes many domination and covering problems such as vertex cover, feedback vertex set, minimum m ..."
Abstract

Cited by 41 (13 self)
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We introduce a new framework for designing fixedparameter algorithms with subexponential running time2 . Our results apply to a broad family of graph problems, called bidimensional problems, which includes many domination and covering problems such as vertex cover, feedback vertex set, minimum maximal matching, dominating set, edge dominating set, cliquetransversal set, and many others restricted to bounded genus graphs. Furthermore, it is fairly straightforward to prove that a problem is bidimensional. In particular, our framework includes as special cases all previously known problems to have such subexponential algorithms. Previously, these algorithms applied to planar graphs, singlecrossingminorfree graphs, and/or map graphs; we extend these results to apply to boundedgenus graphs as well. In a parallel development of combinatorial results, we establish an upper bound on the treewidth (or branchwidth) of a boundedgenus graph that excludes some planar graph H as a minor. This bound depends linearly on the size (H) of the excluded graph H and the genus g(G) of the graph G, and applies and extends the graphminors work of Robertson and Seymour. Building on these results...
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
"... ..."
Circumferences of graphs with no K3,tminors
"... It has recently been shown that every 3connected planar graph G contains a cycle of length at least G  log 3 2, where G  denotes the number of vertices of G. A planar graph contains no K3,3minor. Thomas conjectured that there exists a function β(t)> 0 for t ≥ 3 such that, for any integer t ≥ 3 ..."
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Cited by 1 (1 self)
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It has recently been shown that every 3connected planar graph G contains a cycle of length at least G  log 3 2, where G  denotes the number of vertices of G. A planar graph contains no K3,3minor. Thomas conjectured that there exists a function β(t)> 0 for t ≥ 3 such that, for any integer t ≥ 3 and for any 3connected graph G with no K3,tminor, G contains a cycle of length at least G  β(t). In this paper this conjecture is established with β(t) = log 8t t+1 2. We also show that if t ≥ 3 is an integer and G is a 2connected graph containing no K2,tminor, then there exists a cycle in G of length at least G/t t−1.
The Bidimensionality Theory and Its . . .
, 2005
"... Our newly developing theory of bidimensional graph problems provides general techniques for designing efficient fixedparameter algorithms and approximation algorithms for NPhard graph problems in broad classes of graphs. This theory applies to graph problems that are bidimensional in the sense that ..."
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Our newly developing theory of bidimensional graph problems provides general techniques for designing efficient fixedparameter algorithms and approximation algorithms for NPhard graph problems in broad classes of graphs. This theory applies to graph problems that are bidimensional in the sense that (1) the solution value for the k × k grid graph (and similar graphs) grows with k, typically as Ω(k 2), and (2) the solution value goes down when contracting edges and optionally when deleting edges. Examples of such problems include feedback vertex set, vertex cover, minimum maximal matching, face cover, a series of vertexremoval parameters, dominating set, edge dominating set, rdominating set, connected dominating set, connected edge dominating set, connected rdominating set, and unweighted TSP tour (a walk in the graph visiting all vertices). Bidimensional problems have many
and
, 2005
"... We prove that every sufficiently large 6connected graph of bounded treewidth either has a K6 minor, or has a vertex whose deletion makes the graph planar. This is a step toward proving that the same conclusion holds for all sufficiently large 6connected graphs. Jørgensen conjectured that it holds ..."
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We prove that every sufficiently large 6connected graph of bounded treewidth either has a K6 minor, or has a vertex whose deletion makes the graph planar. This is a step toward proving that the same conclusion holds for all sufficiently large 6connected graphs. Jørgensen conjectured that it holds for all 6connected graphs.
Linear Connectivity Forces . . . Bipartite Minors
, 2004
"... Let a be an integer. It is proved that for any s and k, there exists a constant N = N(s, k, a) such that every 31 2 (a+1)connected graph with at least N vertices either contains a subdivision of Ka,sk or a minor isomorphic to s disjoint copies of Ka,k. In fact, we prove that connectivity 3a + 2 and ..."
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Let a be an integer. It is proved that for any s and k, there exists a constant N = N(s, k, a) such that every 31 2 (a+1)connected graph with at least N vertices either contains a subdivision of Ka,sk or a minor isomorphic to s disjoint copies of Ka,k. In fact, we prove that connectivity 3a + 2 and minimum degree at least 31 2 (a + 1) − 3 are enough. The condition “a subdivision of Ka,sk ” is necessary since G could be a complete bipartite graph K31 where m could be arbitrarily 2 (a+1),m, large. The requirement on N(s, k, a) vertices is necessary since there exist graphs without Kaminor whose connectivity is Θ(a √ log a). When s = 1 and k = a, this implies that every 31 2 (a+1)connected graph with at least N(a) vertices has a Kaminor. This is the first result where a linear lower bound on the connectivity in terms of a forces a Kaminor. This was also conjectured in [68, 47, 69, 39]. Our result generalizes a recent result of Böhme and Kostochka [4] and resolves a conjecture of FonDerFlaass [16]. Our result together with a recent result in [25] also implies that there exists an absolute constant c such that there are only finitely many ckcontractioncritical graphs without Kk as a minor and there are only finitely many ckconnected ckcolorcritical graphs without Kkminors. These results are related to the wellknown conjecture of Hadwiger [17]. Our result was also motivated by the wellknown result of Erdős and Pósa [15]. Suppose that G is 31