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43
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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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.
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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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 ...
Two Graphs Without Planar Covers
, 2001
"... In this note we prove that two speci c graphs do not have nite planar covers. The graphs are K 7 C 4 and K 4;5 4K 2 . This research is related to Negami's 121 Conjecture which states \A graph G has a nite planar cover if and only if it embeds in the projective plane". In particular ..."
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Cited by 6 (0 self)
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In this note we prove that two speci c graphs do not have nite planar covers. The graphs are K 7 C 4 and K 4;5 4K 2 . This research is related to Negami's 121 Conjecture which states \A graph G has a nite planar cover if and only if it embeds in the projective plane". In particular,
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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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.
On possible counterexamples to Negami's planar cover conjecture
 J. Graph Theory
, 1999
"... A simple graph H is a cover of a graph G if there exists a mapping ϕ from H onto G such that ϕ maps the neighbors of every vertex v in H bijectively to the neighbors of ϕ(v) in G. Negami conjectured in 1986 that a connected graph has a finite planar cover if and only if it embeds in the projective p ..."
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Cited by 5 (3 self)
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A simple graph H is a cover of a graph G if there exists a mapping ϕ from H onto G such that ϕ maps the neighbors of every vertex v in H bijectively to the neighbors of ϕ(v) in G. Negami conjectured in 1986 that a connected graph has a finite planar cover if and only if it embeds in the projective plane. The conjecture is still open. It follows from the results of Archdeacon, Fellows, Negami, and the first author that the conjecture holds as long as the graph K1,2,2,2 has no finite planar cover. However, those results seem to say little about counterexamples if the conjecture was not true. We show that there are, up to obvious constructions, at most 16 possible counterexamples to Negami’s conjecture. Moreover, we exhibit a finite list of sets of graphs such that the set of excluded minors for the property of having finite planar cover is one of the sets in our list.
Obstructions for Embedding Cubic Graphs on the Spindle Surface
 Journal of Combinatorial Theory. Series B
"... The spindle surface S is the pinched surface formed by identifying two points on the sphere. In this paper we examine cubic graphs that minimally do not embed on the spindle surface. We give the complete list of 21 cubic graphs that form the topological obstruction set in the cubic order for gra ..."
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The spindle surface S is the pinched surface formed by identifying two points on the sphere. In this paper we examine cubic graphs that minimally do not embed on the spindle surface. We give the complete list of 21 cubic graphs that form the topological obstruction set in the cubic order for graphs that embed on S.
Simpler Projective Plane Embedding
, 2000
"... A projective plane is equivalent to a disk with antipodal points identified. A graph is projective planar if it can be drawn on the projective plane with no crossing edges. A linear time algorithm for projective planar embedding has been described by Mohar. We provide a new approach that takes O(n ..."
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A projective plane is equivalent to a disk with antipodal points identified. A graph is projective planar if it can be drawn on the projective plane with no crossing edges. A linear time algorithm for projective planar embedding has been described by Mohar. We provide a new approach that takes O(n 2 ) time but is much easier to implement. We programmed a variant of this algorithm and used it to computationally verify the known list of all the projective plane obstructions. Key words: graph algorithms, surface embedding, graph embedding, projective plane, forbidden minor, obstruction 1 Background A graph G consists of a set V of vertices and a set E of edges, each of which is associated with an unordered pair of vertices from V . Throughout this paper, n denotes the number of vertices of a graph, and m is the number of edges. A graph is embeddable on a surface M if it can be drawn on M without crossing edges. Archdeacon's survey [2] provides an excellent introduction to topologica...