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43
A linear time algorithm for embedding graphs in an arbitrary surface
 SIAM J. Discrete Math
, 1999
"... Ljubljana, February 2, 2009A simpler linear time algorithm for embedding graphs into an arbitrary surface and the genus of graphs of bounded treewidth ..."
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Cited by 56 (10 self)
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Ljubljana, February 2, 2009A simpler linear time algorithm for embedding graphs into an arbitrary surface and the genus of graphs of bounded treewidth
ThreeDimensional Orthogonal Graph Drawing
, 2000
"... vi Declaration . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . vii Acknowledgements . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . viii List of Figures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ix List of Tables . . . . . . . . . . . . ..."
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Cited by 27 (10 self)
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vi Declaration . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . vii Acknowledgements . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . viii List of Figures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ix List of Tables . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xiii List of Algorithms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xiv I Orthogonal Graph Drawing 1 1
The minor crossing number
 SIAM J. Discrete Math
"... The minor crossing number of a graph G is defined as the minimum crossing number of all graphs that contain G as a minor. Basic properties of this new invariant are presented. Topological structure of graphs with bounded minor crossing number is determined and a new strong version of a lower bound b ..."
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Cited by 18 (4 self)
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The minor crossing number of a graph G is defined as the minimum crossing number of all graphs that contain G as a minor. Basic properties of this new invariant are presented. Topological structure of graphs with bounded minor crossing number is determined and a new strong version of a lower bound based on the genus is derived. An inequality of Moreno and Salazar [15] between crossing numbers of a graph and its minors is generalized.
Branch and Tree Decomposition Techniques for Discrete Optimization
, 2005
"... This chapter gives a general overview of two emerging techniques for discrete optimization that have footholds in mathematics, computer science, and operations research: branch decompositions and tree decompositions. Branch decompositions and tree decompositions along with their respective connecti ..."
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Cited by 16 (3 self)
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This chapter gives a general overview of two emerging techniques for discrete optimization that have footholds in mathematics, computer science, and operations research: branch decompositions and tree decompositions. Branch decompositions and tree decompositions along with their respective connectivity invariants, branchwidth and treewidth, were first introduced to aid in proving the Graph Minors Theorem, a wellknown conjecture (Wagner’s conjecture) in graph theory. The algorithmic importance of branch decompositions and tree decompositions for solving NPhard problems modelled on graphs was first realized by computer scientists in relation to formulating graph problems in monadic second order logic. The dynamic programming techniques utilizing branch decompositions and tree decompositions, called branch decomposition and tree decomposition based algorithms, fall into a class of algorithms known as fixedparameter tractable algorithms and have been shown to be effective in a practical setting for NPhard problems such as minimum domination, the travelling salesman problem, general minor containment, and frequency assignment problems.
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.
Recent Excluded Minor Theorems for Graphs
 IN SURVEYS IN COMBINATORICS, 1999 267 201222. THE ELECTRONIC JOURNAL OF COMBINATORICS 8 (2001), #R34 8
, 1999
"... A graph is a minor of another if the first can be obtained from a subgraph of the second by contracting edges. An excluded minor theorem describes the structure of graphs with no minor isomorphic to a prescribed set of graphs. Splitter theorems are tools for proving excluded minor theorems. We disc ..."
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Cited by 9 (0 self)
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A graph is a minor of another if the first can be obtained from a subgraph of the second by contracting edges. An excluded minor theorem describes the structure of graphs with no minor isomorphic to a prescribed set of graphs. Splitter theorems are tools for proving excluded minor theorems. We discuss splitter theorems for internally 4connected graphs and for cyclically 5connected cubic graphs, the graph minor theorem of Robertson and Seymour, linkless embeddings of graphs in 3space, Hadwiger’s conjecture on tcolorability of graphs with no Kt+1 minor, Tutte’s edge 3coloring conjecture on edge 3colorability of 2connected cubic graphs with no Petersen minor, and Pfaffian orientations of bipartite graphs. The latter are related to the even directed circuit problem, a problem of Pólya about permanents, the 2colorability of hypergraphs, and signnonsingular matrices.
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
KuratowskiType Theorems for Average Genus
 J. Combinatorial Theory B
, 1992
"... Graphs of small average genus are characterized. In particular, a Kuratowskitype theorem is obtained: except for finitely many graphs, a cutedgefree graph has average genus less than or equal to 1 if and only if it is a necklace. We provide a complete list of those exceptions. A Kuratowskitype th ..."
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Cited by 6 (3 self)
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Graphs of small average genus are characterized. In particular, a Kuratowskitype theorem is obtained: except for finitely many graphs, a cutedgefree graph has average genus less than or equal to 1 if and only if it is a necklace. We provide a complete list of those exceptions. A Kuratowskitype theorem for graphs of maximum genus 1 is also given. Some of the methods used in obtaining these results involve variations of a classical result of Whitney. April 27, 1992 1 Supported by Engineering Excellence Award from Texas A&M University, and by the National Science Foundation under Grant CCR9110824. 2 Supported by ONR Contract N00014850768. CUCS01992 1 Introduction By the average genus of a graph G, we mean the average value of the genus of the imbedding surface, taken over all orientable imbeddings of G. This value is evidently a rational number, and it is clearly an invariant of the homeomorphism type of a graph. The average genus of individual graphs is in the GrossFurst ...
Lightness of digraphs in surfaces and directed game chromatic number
"... The lightness of a digraph is the minimum arc value, where the value of an arc is the maximum of the indegrees of its terminal vertices. We determine upper bounds for the lightness of simple digraphs with minimum indegree at least 1 (resp., graphs with minimum degree at least 2) and a given girth ..."
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Cited by 6 (5 self)
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The lightness of a digraph is the minimum arc value, where the value of an arc is the maximum of the indegrees of its terminal vertices. We determine upper bounds for the lightness of simple digraphs with minimum indegree at least 1 (resp., graphs with minimum degree at least 2) and a given girth k, and without 4cycles, which can be embedded in a surface S. (Graphs are considered as digraphs each arc having a parallel arc of opposite direction.) In case k ≥ 5, these bounds are tight for surfaces of nonnegative Euler characteristics. This generalizes results of He et al. [11] concerning the lightness of planar graphs. From these bounds we obtain directly new bounds for the game coloring number, and thus for the game chromatic number of (di)graphs with girth k and without 4cycles embeddable in S. The game chromatic resp. game coloring number were introduced by Bodlaender [3] resp. Zhu [22] for graphs. We generalize these notions to arbitrary digraphs. We prove that the game coloring number of a directed simple forest is at most 3.