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... Minors In Graphs Of Bounded TreeWidth
 J. Combin. Theory Ser. B
, 2000
"... It is shown that for any positive integers k and w there exists a ..."
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Cited by 10 (3 self)
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It is shown that for any positive integers k and w there exists a
Graph Minors and Graphs on Surfaces
, 2001
"... Graph minors and the theory of graphs embedded in surfaces are ..."
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Cited by 8 (3 self)
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Graph minors and the theory of graphs embedded in surfaces are
Structure and enumeration of twoconnected graphs with prescribed threeconnected components
, 2009
"... ..."
The ArcWidth of a Graph
, 2001
"... The arcrepresentation of a graph is a mapping from the set of vertices to the arcs of a circle such that adjacent vertices are mapped to intersecting arcs. The width ofsucharepresentationisthemaximumnumberofarcshavingapointincommon. The arcwidth(aw) of a graph is the minimum width of its arcrepre ..."
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The arcrepresentation of a graph is a mapping from the set of vertices to the arcs of a circle such that adjacent vertices are mapped to intersecting arcs. The width ofsucharepresentationisthemaximumnumberofarcshavingapointincommon. The arcwidth(aw) of a graph is the minimum width of its arcrepresentations. We show how arcwidth is related to pathwidth and vortexwidth. We prove that aw(K s,s )=s. 1
The number of graphs not containing K3,3 as a minor
"... We derive precise asymptotic estimates for the number of labelled graphs not containing K3,3 as a minor, and also for those which are edge maximal. Additionally, we establish limit laws for parameters in random K3,3minorfree graphs, like the number of edges. To establish these results, we translat ..."
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We derive precise asymptotic estimates for the number of labelled graphs not containing K3,3 as a minor, and also for those which are edge maximal. Additionally, we establish limit laws for parameters in random K3,3minorfree graphs, like the number of edges. To establish these results, we translate a decomposition for the corresponding graphs into equations for generating functions and use singularity analysis. We also find a precise estimate for the number of graphs not containing the graph K3,3 plus an edge as a minor. 1
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
and
"... It is shown that for any positive integers k and w there exists a constant N = N(k, w) such that every 7connected graph of treewidth less than w and of order at least N contains K3,k as a minor. Similar result is proved for Ka,k minors where a is an arbitrary fixed integer and the required connect ..."
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It is shown that for any positive integers k and w there exists a constant N = N(k, w) such that every 7connected graph of treewidth less than w and of order at least N contains K3,k as a minor. Similar result is proved for Ka,k minors where a is an arbitrary fixed integer and the required connectivity depends only on a. These are the first results of this type where fixed connectivity forces arbitrarily large (nontrivial) minors.