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Topic: Topological graph theory

Related Topics

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Max flow min cut theorem

Mereotopology

	PlanetMath: graph theory
		Graph theory is the branch of mathematics that concerns itself with graphs.
		The only remainder of the topological past is the topological graph theory, a branch of graph theory that primarily deals with drawing of graphs on surfaces.
		Now, a (finite) graph is usually thought of as a subset of pairs of elements of a finite set (called vertices), or more generally as a family of arbitrary sets in the case of hypergraphs.
	planetmath.org /encyclopedia/GraphTheory.html (506 words)

	PlanetMath: planar graph
		A planar graph is a graph which can be drawn on a plane (a flat 2-d surface) or on a sphere, with no edges crossing.
		Every graph drawn on a sphere can be drawn on a plane (puncture the sphere in the interior of any one of the countries) and vice versa.
		planar graphs and embeddings by archibal on 2004-03-30 23:54:21
	planetmath.org /encyclopedia/Planar.html (540 words)

	Wikinfo \| Graph theory
		Definitions of graphs vary in style and substance, according to the level of abstraction that is approriate to a particular approach or application.
		Graphs with weights can be used to represent many different concepts; for example if the graph represents a road network, the weights could represent the length of each road.The only information a weighted graph provides as such is (a) the vertices, (b) the edges and (c) the weights.
		Graph theory is also used to study molecules in chemistry and physics.
	www.wikinfo.org /wiki.php?title=Graph_theory (2257 words)

	Topological graph theory - Wikipedia, the free encyclopedia
		In mathematics topological graph theory is a branch of graph theory.
		It studies the embedding of graphs in surfaces, and graphs as topological spaces.
		Embedding a graph in a surface means that we want to draw the graph on a surface, a sphere for example, without two edges intersecting.
	en.wikipedia.org /wiki/Topological_graph_theory (425 words)

	Graph theory - Wikipedia, the free encyclopedia
		In mathematics and computer science, graph theory is the study of graphs, mathematical structures used to model pairwise relations between objects from a certain collection.
		Enumerative graph theory then rose from the results of Cayley and the fundamental results published by Pólya between 1935 and 1937 and the generalization of these by De Bruijn in 1959.
		The introduction of probabilistic methods in graph theory, specially in the study of Erdös and Rényi of the asymptotic probability of graph connexity is at the origin of yet another branch, known as random graph theory.
	en.wikipedia.org /wiki/Graph_theory (1871 words)

	Graph theory (Site not responding. Last check: 2007-10-09)
		Graph theory is the branch of mathematics that examines the properties of graphs.
		Informally a graph is a set of called vertices (or nodes) connected by links called edges (or arcs).
		Every graph gives rise to a matroid but in general the graph cannot recovered from its matroid so matroids are truly generalizations of graphs.
	www.freeglossary.com /Graph_theory (1059 words)

	Combinatorial Hypermaps vs Topic Maps
		Topological graph theory mainly developed around 1968, from the resolution by Ringel and Youngs [29] of Heawood's conjecture on map coloring [21].
		R. Tamassia and I.G. Tollis, Tessalation representation of planar graphs, Proc.
		He is co-author of Public Implementation of a Graph Algorithm Library and Editor (Pigale) (http://www.ehess.fr/centres/cams/person/pom/pigale.html), contributors of the GPL Pliant programming environment (http://pliant.cx/) and author of more than 10 papers in the field of Topological Graph Theory.
	www.gca.org /papers/xmleurope2001/papers/html/s23-1.html (1973 words)

	A Survey of Embedding Problems in Topological Graph Theory by Joerg Sawollek
		The main purpose of topological graph theory is to consider geometric realizations of a graph and its embeddings in 3-space or its embeddings or immersions in the plane or surfaces of higher genus.
		A topological graph is a 1-dimensional cell complex where the 0-cells correspond to the vertices and the 1-cells correspond to the edges of an underlying abstract graph.
		In [9] the intrinsically chiral 3-connected graphs are characterized by the existence of certain automorphisms of the graph group.
	at.yorku.ca /t/a/i/c/38.htm (2029 words)

	05C: Graph theory
		A graph is a set V of vertices and a set E of edges -- pairs of elements of V. This simple definition makes Graph Theory the appropriate language for discussing (binary) relations on sets, which is clearly a broad topic.
		A graph may be viewed as a one-dimensional CW-complex and hence studied with tools from Algebraic Topology, in particular, questions of planarity (and genus).
		Determining the genus of a graph is NP-complete.
	www.math.niu.edu /~rusin/known-math/index/05CXX.html (1204 words)

	Topological Graph Theory (Site not responding. Last check: 2007-10-09)
		Clear, comprehensive introduction emphasizes graph imbedding but also covers thoroughly the connections between topological graph theory and other areas of mathematics.
		Authors explore the role of voltage graphs in the derivation of genus formulas, explain the Ringel-Youngs theorem—a proof that revolutionized the field of graph theory—and examine the genus of a group, including imbeddings of Cayley graphs.
		A non-technical introduction to the field of graph theory and its applications.
	www.doverdirect.com /0486417417.html (122 words)

	Graph theory
		Graphtheory.com homepage for "Graph Theory and it Applications" by Gross and Yellen.
		The Theory of 2-Structures by Andrzej Ehrenfeucht, Tero Harju and Grzegorz Rozenberg.
		Graph Theory Page by Stephen C. Locke (Course notes and definitions).
	users.utu.fi /harju/graphtheory/graphtheory.html (238 words)

	Week 2 Abstracts
		A labeled graph is said to be weakly bipartite if the clutter of its odd cycles has the weak max-flow min-cut property (also known as ideal property).
		Most of the theory of knots and links generalises to these abstract code s, with sometimes very fascinating variations and relationships with knots and links that embed in three manifolds of the form MxI where M is an orientable surface and I is the unit interval.
		Every planar graph has a medial graph, which is obtained by placing a new vertex in the middle of each edge and joining two such new vertices by an edge whenever they are consecutive around an original face.
	dimacs.rutgers.edu /drei/1998/week2.html (4127 words)

	Wiley::Fractional Graph Theory: A Rational Approach to the Theory of Graphs
		Professors Scheinerman and Ullman begin by developing a general fractional theory of hypergraphs and move on to provide in-depth coverage of fundamental and advanced topics, including fractional matching, fractional coloring, and fractional edge coloring; fractional arboricity via matroid methods; and fractional isomorphism.
		The final chapter is devoted to a variety of additional issues, such as fractional topological graph theory, fractional cycle double covers, fractional domination, fractional intersection number, and fractional aspects of partially ordered sets.
		Supplemented with many challenging exercises in each chapter as well as an abundance of references and bibliographic material, Fractional Graph Theory is a comprehensive reference for researchers and an excellent graduate-level text for students of graph theory and linear programming.
	www.wiley.com /WileyCDA/WileyTitle/productCd-0471178640.html (297 words)

	Toroidal Snarks and such
		Technically, an embedding is just a particular drawing of a graph; excluding edge crossings isn't actually part of the definition, but in topological graph theory it would be rare to consider an embedding which wasn't cellular.
		A drawing of this graph is paired with a toroidal embedding (an embedding of the graph on a torus).
		The girth of a graph is the length of its shortest cycle.
	www.toroidalsnark.net /torsn.html (1291 words)

	Topological graph theory (Site not responding. Last check: 2007-10-09)
		Topological graph theory comprises a large number of topics which have the common elements of points, lines, and patches sitting in an ambient space of three or four dimensions.
		[PCK74] is an early survey of topological graph theory, which among other topics, includes a survey of the connection with piecewise-linear topology.
		Relevant cycles in biopolymers and random graphs, 4th Slovene Conference on Graph Theory, Bled, June 28-July 2, 1999 DMW04: V. Dujmovic, P. Morin, and D. Wood, Layout of graphs with bounded tree-width, arXiv xxx.sf.nchc.gov.tw/pdf/cs.DM/0406024; also see arXiv:math.CO/0503553 v1 24 Mar 2005 [RT72] R. TARJAN, Sorting using networks of queues and stacks.
	www.georgetown.edu /faculty/kainen/top-gr-th.html (540 words)

	Publisher description for Library of Congress control number 2001030789 (Site not responding. Last check: 2007-10-09)
		The book, suitable as both an introductory reference and as a text book in the rapidly growing field of topological graph theory, models both maps (as in map-coloring problems) and groups by means of graph imbeddings on sufaces.
		There are chapters on the emerging subfields of enumerative topological graph theory and random topological graph theory, as well as a chapter on the composition of English church-bell music.
		Throughout the emphasis is on Cayley maps: imbeddings of Cayley graphs for finite groups as (possibly branched) covering projections of surface imbeddings of loop graphs with one vertex.
	www.loc.gov /catdir/enhancements/fy0612/2001030789-d.html (285 words)

	ImageBeat Bonsai (Site not responding. Last check: 2007-10-09)
		Bonsai is a general graph editing and browsing system that lets you define graph diagrams and explore their topological properties.
		Facilities are provided for generating topological building blocks such as bouquets, cubes, paths, cycles, and complete graphs.
		Several operations related to topological graph theory are also supported, such as Cartesian product, suspension, face tracing, genus distribution, crosscap distribution, and converting between graphical rotation projections and combinatorial rotation systems.
	imagebeat.com /bonsai (192 words)

	Summer Tutorials, 2003
		We will then discuss major theorems of excluded minor theory, including a generalization of Kuratowski's theorem that is the gem of Robertson-Seymour theory: given any infinite set of graphs, there exist two such that one is a minor of the other (the graph minor theorem).
		The problem of determining the minimum genus of a surface on which a graph can be embedded is NP-complete, but there are many partial results, including a recent linear-time algorithm that given a surface, decides whether an arbitrary graph is embeddable.
		Lectures 6-8: The major theorems of excluded minor theory, including a generalization of Kuratowski's theorem that is the gem of Robertson-Seymour theory: given any infinite set of graphs, there exist two such that one is a minor of the other (the graph minor theorem).
	www.math.harvard.edu /tutorials/2003.html (1754 words)

	Graphs: Theory - Algorithms - Complexity (Site not responding. Last check: 2007-10-09)
		Groups and Graphs: a software package for graphs, digraphs, combinatorial designs, and their automorphism groups, by B.
		Scheinerman, E.R., Ullman, D.H.: Fractional graph theory: a rational approach to the theory of graphs, John Wiley and Sons, New York, 1997.
		Graph connections -- relationships between graph theory and other areas of mathematics, Eds.
	people.freenet.de /Emden-Weinert/graphs.html (1244 words)

	Amazon.com: Topological Graph Theory: Books: Jonathan L. Gross,Thomas W. Tucker (Site not responding. Last check: 2007-10-09)
		Graph Theory (Graduate Texts in Mathematics) by Reinhard Diestel
		This definitive treatment written by well-known experts emphasizes graph imbedding while providing thorough coverage of the connections between topological graph theory and other areas of mathematics: spaces, finite groups, combinatorial algorithms, graphical enumeration, and block design.
		It tends to use an older approach to graphs that has to be adapted to modern computer mathematical systems.
	www.amazon.com /Topological-Graph-Theory-Jonathan-Gross/dp/0486417417 (1142 words)

	The Graph Theorists' Home Page Guide
		First of all, if you're a graph theorist or some person with strong interest in graph theory (you need not to be a mathematician!), and if you have a homepage but don't find a link to it on this page, please contact me as described above.
		PIGALE is a graph editor with an interface to the LEDA library and with many algorithms implemented essentially concerning planar graphs.
		"Graph Theory and Its Applications" (together with Jay Yellen), "a comprehensive applications-driven textbook that provides material for several different courses in graph theory." This site also provides links to other graph theoretical and mathematical resources.
	www.joergzuther.de /math/graph/homes.html (8736 words)

	Open Directory - Science: Math: Combinatorics: Graph Theory: Open Problems (Site not responding. Last check: 2007-10-09)
		Graph Coloring Problems - Archive for the book "Graph Coloring Problems" by Tommy R. Jensen and Bjarne Toft (Wiley Interscience 1995), dedicated to Paul Erdös.
		Graph Theory Open Problems - Six problems suitable for undergraduate research projects.
		Problems in Topological Graph Theory - Web text by Dan Archdeacon with a list of open questions in topological graph theory.
	dmoz.org /Science/Math/Combinatorics/Graph_Theory/Open_Problems (227 words)

	Problems in Topological Graph Theory
		Graphs that quadrangulate both the torus and Klein bottle
		Orientable genus of graphs of bounded nonorientable genus
		Interpolation conjectures on separating cycles in embedded graphs
	www.emba.uvm.edu /~archdeac/problems/problems.html (283 words)

	Topics in topological graph theory
		I will be giving a research tutorial (math 302-??) in the Spring, with the title Topics in Topological Graph Theory.
		The course will roughly parallel a paper I wrote in 1973, Recent results in topological graph theory, pp.
		of the Capital Conference on Graph Theory and Combinatorics, at the George Washington University, June 18--22, 1973, Springer-Verlag, Berlin, 1974.
	www.georgetown.edu /faculty/kainen/ttgt.html (158 words)

	Amazon.com: The Foundations of Topological Graph Theory: Books: C. Paul Bonnington,Charles H. C. Little (Site not responding. Last check: 2007-10-09)
		This is a book on topological graph theory written from a purely combinatorial viewpoint.
		The basic tool used is the idea of a 3-graph which is a cubic graph endowed with a proper edge colouring in three colours.
		A special case of a 3-graph, called a gem, provides a model for a cellular imbedding of a graph in a surface.
	www.amazon.com /Foundations-Topological-Graph-Theory/dp/0387945571 (678 words)

	Graph Theory Books
		I do have a bias for the text by Bondy and Murty, which one might assume follows from the fact that Adrian Bondy was my Ph.D. supervisor.
		Many people who research in Graph Theory think it is the best.
		Combinatorics with Emphasis on the Theory of Graphs.
	www.math.fau.edu /locke/Graphstx.htm (273 words)

	Marisa Debowsky - research (Site not responding. Last check: 2007-10-09)
		I've been working on a project with Joanna Ellis-Monaghan at St Michael's College and Dan Archdeacon at UVM on the relationship between biomolecular computing and topological graph theory.
		Jo has recently written a nice summary of the work, and we are looking forward to submitting a joint paper later in 2005.
		Note: For a reasonably non-technical summary of the field, I recommend Dan's recent summary, Topological graph theory: a picture is worth a thousand words (ps file).
	homepages.nyu.edu /~mad464/research.html (291 words)

	Graph Theory WS 03-04
		This course is an introduction to the theory of graphs intended for students in mathematics and computer science/engineering students with an interest in theory.
		In lecture we will follow the textbook "Introduction to Graph Theory" by Doug West.
		Douglas B. West, Introduction to Graph Theory (2nd edition), Prentice Hall, 2001.
	www.ti.inf.ethz.ch /ew/courses/GT03 (371 words)

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