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Topic: Homeomorphism graph theory

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Planar graph

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	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/PlaneGraph.html (542 words)

	Homeomorphism Summary
		Roughly speaking, a topological space is a geometric object and the homeomorphism is a continuous stretching and bending of the object into a new shape.
		Thus, a square and a circle are homeomorphic.
		Intuitively, a homeomorphism maps points in the first object that are "close together" to points in the second object that are close together, and points in the first object that are not close together to points in the second object that are not close together.
	www.bookrags.com /Homeomorphism (1591 words)

	NationMaster - Encyclopedia: Homeomorphism
		Roughly speaking a topological space is a geometric object and the homeomorphism is a continuous stretching and bending of the object into a new shape.
		Thus a square and a circle are homeomorphic.
		Intuitively a homeomorphism maps points in the first object that are "close together" to points in the second object that are close together, and points in the first object that are not close together to points in the second object that are not close together.
	www.nationmaster.com /encyclopedia/Homeomorphism (2468 words)

	Springer Online Reference Works
		Graph, connectivity of a) can be uniquely imbedded in the sphere (up to a homeomorphism of the sphere).
		An intensively studied subject in the theory of graphs is the colouring of planar graphs (cf.
		Graph colouring); for non-planar graphs one studies various numerical characteristics yielding the degree of non-planarity, including the genus, the thickness or coarseness of the graph, the number of crossings, etc. (cf.
	eom.springer.de /g/g044990.htm (504 words)

	PlanetMath: homeomorphism
		Adapted with permission of the author from Modern Graph Theory by Béla Bollobás, published by Springer-Verlag New York, Inc., 1998.
		This is version 3 of homeomorphism, born on 2002-03-07, modified 2002-07-17.
		Object id is 2773, canonical name is Homeomorphic.
	planetmath.org /encyclopedia/Homeomorphic.html (39 words)

	Homeomorphism: Making a donut into a coffee cup
		In graph theory (and group theory), this equivalence relation is called an isomorphism.
		, and are all homeomorphic to a circle.
		In this case, graphs that are not isomorphic produce topological graphs that are not homeomorphic.
	msl.cs.uiuc.edu /planning/node130.html (508 words)

	Category:Graph theory help – Wiki at Help.com (Site not responding. Last check: )
		Graph theory is the branch of mathematics that examines the properties of graphs.
		Informally, a graph is a set of objects called vertices (or nodes) connected by links called edges (or arcs), which can also have associated directions.
		Typically, a graph is depicted as a set of dots (i.e., vertices) connected by lines (i.e., edges), with an arrowhead on a line representing a directed arc.
	www.help.com /wiki/Category:Graph_theory (186 words)

	-edge-coloured graphs and 3-manifolds
		The homeomorphism problem on 3-manifolds attracts mathematicians since the beginnings of 20-th century.
		The most developed part of a combinatorial approach is knot theory based on a well-known fact that any 3-manifold can be constructed as a branched cover over the 3-sphere, where the set of branch points forms a knot.
		We first introduce a combinatorial counterpart of the classical homeomorphism problem on 3-manifolds using a theory built by Pezzana and his successors.
	cms.jcmf.cz /czech-catalan/Nedela (376 words)

	Graph Theory Glossary - h (Site not responding. Last check: )
		Graph obtained by successfully adding edges between vertices whose degree-sum is as large as the number of vertices.
		A connected graph consisting of two vertex-disjoint polygons and a minimal (not necessarily minimum-length) connecting path (this is a loose handcuff), or of two polygons that meet at a single vertex (a tight handcuff or figure eight).
		A non-traceable graph whose vertex-deleted subgraphs are all traceable.
	www.cc.ioc.ee /jus/gtglossary/gtglos_h.htm (2413 words)

	Mathematics
		Nonetheless mathematics is often imagined to be (as far as its formal content) nothing but set theory in some axiomatization, in the sense that every mathematical statement or proof could be cast into formulas within set theory.
		An important field in applied mathematics is statistics, which uses probability theory as a tool and allows the description, analysis, and prediction of phenomena where chance plays a role.
		However, theorems are elements of formal theories, and in some cases computers can generate proofs of these theorems more or less automatically, by means of automated theorem provers.
	www.brainyencyclopedia.com /encyclopedia/m/ma/mathematics.html (3081 words)

	Mathematics - ExampleProblems.com
		In order to clarify the foundations of mathematics, the fields of mathematical logic and set theory were developed, as well as category theory which is still in development.
		Complexity theory is the study of tractability by computer; some problems, although theoretically solvable by computer, are so expensive in terms of time or space that solving them is likely to remain practically unfeasible, even with rapid advance of computer hardware.
		Finally, information theory is concerned with the amount of data that can be stored on a given medium, and hence concepts such as compression and entropy.
	www.exampleproblems.com /wiki/index.php/Mathematics (4097 words)

	UC Davis Math: Glossary (Site not responding. Last check: )
		The study of general relativity as a quantum field theory; the theory of integration of quantities over the space of Riemannian or Minkowskian metrics on a manifold.
		A theory in physics that postulates a counterintuitive symmetric relationship between fermions, which are particles such as electrons that obey the Pauli exclusion principle, and bosons, which are particles such as photons that enjoy being in the same state as each other.
		A graph drawn on a surface such that every vertex has degree three, and such that all three edges meeting at a vertex have a common tangent, two edges on one side and one on the other.
	math.ucdavis.edu /profiles/glossary.html (9932 words)

	Homeomorphism - Qwika
		Homeomorphism One homeomorphism between two topological spaces is one bijection...
		Local homeomorphism In topology, a local homeomorphism is a map from one topological space...
		Homeomorphism (graph theory) In graph theory, a homeomorphism exists between two graphs G and G′...
	www.qwika.com /find/Homeomorphism (521 words)

	Mathematics - Crystalinks
		Complexity theory is the study of tractability by computer; some problems, although theoretically soluble by computer, are so expensive in terms of time or space that solving them is likely to remain practically unfeasible, even with rapid advance of computer hardware.
		Finally, information theory is concerned with the amount of data that can be stored on a given medium, and hence concepts such as compression and entropy.As a relatively new field, discrete mathematics has a number of fundamental open problems.
		An important field in applied mathematics is statistics, which uses probability theory as a tool and allows the description, analysis, and prediction of phenomena where chance plays a role.
	www.crystalinks.com /math.html (3179 words)

	[No title]
		category theory The study of abstracted collections of mathematical objects, such as the category of sets or the category of vector spaces, together with abstracted operations sending one object to another, such as the collection of functions from one set to another or linear transformations from one vector space to another.
		graph theory The study of graphs, either for their own sake, or as models of such diverse things as groups (in pure mathematics) or computer networks.
		skein theory An inductive definition of an invariant of knots or links which postulates a linear relation between the invariant of a given link and the invariant of the same link with a crossing switched or otherwise simplified.
	www.ornl.gov /sci/ortep/topology/defs.txt (5717 words)

	Homeomorphism: Making a donut into a coffee cup
		In graph theory (and group theory), this equivalence relation is called an isomorphism.
		, and are all homeomorphic to a circle.
		In this case, graphs that are not isomorphic produce topological graphs that are not homeomorphic.
	planning.cs.uiuc.edu /node130.html (508 words)

	Topology Summary
		Georg Cantor, the inventor of set theory, had begun to study the theory of point sets in Euclidean space, in the later part of the 19th century, as part of his study of Fourier series.
		Formally, a homeomorphism is defined as a continuous bijection with a continuous inverse, which is not terribly intuitive even to one who knows what the words in the definition mean.
		In pointless topology one considers instead the lattice of open sets as the basic notion of the theory, while Grothendieck topologies are certain structures defined on arbitrary categories which allow the definition of sheaves on those categories, and with that the definition of quite general cohomology theories.
	www.bookrags.com /Topology (6184 words)

	Omnipelagos.com ~ article "Homeomorphism"
		In the mathematical field of topology a homeomorphism or topological isomorphism (from the Greek words homoios = identical and morphe = shape) is a special isomorphism between topological spaces which respects topological properties.
		The open interval (−1, 1) is homeomorphic to the real numbers R.
		Any 2-sphere with a single point removed is homeomorphic to the set of all points in R
	www.omnipelagos.com /entry?n=homeomorphism (767 words)

	Algebraic Graph Theory (Site not responding. Last check: )
		Algebraic graph theory - Algebraic graph theory is a branch of mathematics.
		Algebraic number theory - Algebraic number theory is a branch of number theory in which the concept of a number is expanded to the algebraic numbers which are roots of polynomials with rational coefficients.
		Evolutionary graph theory - An area lying at the intersection of graph theory, probability theory, and mathematical biology, evolutionary graph theory is an approach to studying how topology affects evolution of a population.
	al56.mcechess.com (1194 words)

	Homeomorphism (graph theory)
		In graph theory, a '''homeomorphism''' exists between two graphs ''G'' and ''G''andprime; if there exists a graph ''H'' such that both ''G'' and ''G''andprime; are subdivisions of that graph.
		If the edges of a graph are thought of as lines drawn from one vertex to another (as they are usually depicted in illustrations), then two graphs are homeomorphic to each other in the graph-theoretic sense precisely if they are homeomorphismhomeomorphic in the sense in which the term is used in topology.
		The subdivision of some edge ''e'' with endpoints yields a graph containing one new vertex ''v'', and with an edge set replacing ''e'' by two new edges with endpoints and.
	www.territoriopc.com /eng/homeomorphism__graph_theory_.php (165 words)

	graph theory and its applications
		Graph Theory and Its Applications is a comprehensive applications-driven textbook that provides material for several different courses.
		Voltage graphs, algebraic specification of graphs (including the wrapped butterfly), and other topics that showcase the interplay between graph theory and algebra.
		At graph theory, you'll discover an easy to use, information packed web site.
	www.graphtheory.com /av.htm (202 words)

	math lessons - Category:Graph theory
		Graph theory is the branch of mathematics that examines the properties of graphs.
		See glossary of graph theory for common terms and their definition.
		Informally, a graph is a set of objects called vertices (or nodes) connected by links called edges (or arcs).
	www.mathdaily.com /lessons/Category:Graph_theory (86 words)

	NationMaster - Encyclopedia: Degree (mathematics)
		In graph theory, the degree of a vertex in a graph is the number of edges incident to that vertex — in other words, the number of lines coming out of the point.
		The simplest and most important case is the degree of a continuous map
		In a neighborhood of each the map f is a homeomorphism to its image, so it might be orientation preserving or orientation reversing.
	www.nationmaster.com /encyclopedia/Degree-%28mathematics%29 (658 words)

	Illiteracy psychoanalysis and Topologu-Knot theory-Lituraterre.org
		Knot theory is a branch of algebraic topology where one studies what is known as the placement problem, or the embedding of one topological space into another.
		Most of knot theory concerns only tame knots, and these are the only knots examined here.
		Also, the right and left handed versions of the trefoil are only equivalent if the homeomorphism mapping one into the other includes a reflection (other knots, such as the Figure-8 knot are equivalent to their mirror images, these knots are known as achiral knots).
	www.lituraterre.org /illiteracy-knot_theory.htm (764 words)

	ScienceDaily: Mathematics
		A "Unified Theory" For Calculus (January 29, 2003) -- A University of Missouri-Rolla mathematician's research into a "unified theory" of continuous and discrete calculus is gaining the attention of mathematicians worldwide for numerous applications,...
		Probability theory -- Probability theory is the mathematical study of phenomena characterized by randomness or uncertainty.
		This text is designed for the sophomore/junior level introduction to discrete mathematics taken by students preparing for future coursework in areas such as math, computer science and engineering.
	www.sciencedaily.com /encyclopedia/mathematics (1370 words)

	Read This: When Topology Meets Chemistry: A Topological Look At Molecular Chirality
		It turns out that an essential notion here is that of a finite-order homeomorphism, that is, a homeomorphism of three-dimensional space (or of the three-sphere) which has finite order, in the sense that if we take its composition with itself a finite number of times, we obtain the identity homeomorphism.
		Although the basic ideas of knot theory and topological graph theory are very intuitive, "there are genuine practical difficulties in attempting to give a totally self-contained introduction to knot theory," as Raymond Lickorish has observed.
		Knot theory in the context of algebraic topology.
	www.maa.org /reviews/topochem.html (2915 words)

	Topology
		Extending beyond the boundaries of Hilbert and Banach space theory, this text focuses on key aspects of functional analysis, particularly in regard to solving partial differential equations.
		Detailed theory of Fréchet (V) spaces and a comprehensive examination of their relevance to topological spaces, plus in-depth discussions of metric and complete spaces.
		This introduction emphasizes graph imbedding but also covers the connections between topological graph theory and other areas of mathematics.
	store.doverpublications.com /by-subject-science-and-mathematics-mathematics-topology.html (686 words)

	2-4. Series-Parallel Experiment
		Although it is intuitively understood that a graph having parallel elements turns to be a non-Hamiltonian graph, the concept of series-parallel homeomorphism cited in Graph Theory cannot separate non-Hamiltonian graph from Hamiltonian graph, then we developed such conversion method to judge Hamiltonial homeomorphism.
		If a reduced graph obtained in such way is one of the case bellow, the original graph is a non-Hamiltonian graph.
		In the case that 3 determined branches are disposed on the graph dispersively, it is considered that a graph obtained by series-parallel experiment comes to be one of the 6 kinds of primitive letter graphs below.
	www.aya.or.jp /~babalabo/Hamilton/Draft2-4.html (818 words)

	Alternative Axiomatic Set Theories (Stanford Encyclopedia of Philosophy)
		The theory of isomorphism classes of well-founded extensional relations with a top element looks like the theory of (an initial segment of) the usual cumulative hierarchy, because every set in Zermelo-style set theory is uniquely determined by the isomorphism type of the restriction of the membership relation to its transitive closure.
		In this theory, the class of natural numbers (considered as finite ordinals) is not closed and acquires an extra element "at infinity" (which happens to be the closure of the class of natural numbers itself).
		The theories motivated by essentially different views of the realm of mathematics (the constructive theories and the theories which support nonstandard analysis) we set to one side.
	plato.stanford.edu /entries/settheory-alternative (17276 words)

	19th LL-Seminar on Graph Theory / Abstracts
		A minimum cycle basis in an undirected graph G is a set of simple cycles whose incidence vectors span the cycle space of G and whose overall edge sum is minimal.
		As to the structure PN are bipartite graphs where one kind of vertices - places $p_1,\,p_2,\,...,\,p_n$ represent some subsystems, elementary operations, etc. and the second kind of vertices - transitions $t_1,\,t_2,\,...,\,t_m$ represent discrete events - e.g.
		An automorphism of a map $\cal M$ is an automorphism of the embedded (combinatorial) graph which extends to a self-homeomorphism of the surface.
	www.tbi.univie.ac.at /LL/abstracts.html (3851 words)

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