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	Graph theory - Wikipedia, the free encyclopedia
		Informally, a graph is a set of objects called vertices (or nodes) connected by links called edges (or arcs) which can be directed (assigned a direction).
		Graphs are represented graphically by drawing a dot for every vertex, and drawing an arc between two vertices if they are connected by an edge.
		A graph drawing should not be confused with the graph itself (the abstract, non-graphical structure) as there are several ways to structure the graph drawing.
	en.wikipedia.org /wiki/Graph_theory (1209 words)

	Graph theory - Encyclopedia.WorldSearch (Site not responding. Last check: 2007-11-06)
		A graph structure can be extended by assigning a weight to each edge, or by making the edges to the graph directional (A links to B, but B does not necessarily link to A, as in webpages), technically called a digraph.
		The data structure used depends on the graph structure and the algorithm used for manipulating the graph.
		Incidence matrix - The graph is represented by a matrix of E (edges) by V (vertices), where contains the edge's data (simplest case: 1 - connected, 0 - not connected).
	encyclopedia.worldsearch.com /graph_theory.htm (996 words)

	Graph theory - Open Encyclopedia (Site not responding. Last check: 2007-11-06)
		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).
		Every graph gives rise to a matroid, but in general the graph cannot be recovered from its matroid, so matroids are not truly generalizations of graphs.
	open-encyclopedia.com /Graph_theory (881 words)

	Encyclopedia: Graph theory (Site not responding. Last check: 2007-11-06)
		Graphs are usually represented pictorially using dots to represent vertexes, and arcs representing the edges between connected vertexes.
		Given a connected, undirected graph, a spanning tree of that graph is a subgraph which is a tree and connects all the vertices together.
		In the mathematical discipline of graph theory a covering for a graph is a set of vertices (or edges) so that the elements of the set are close (adjacent) to all edges (or vertices) of the graph.
	www.nationmaster.com /encyclopedia/Graph-theory (3291 words)

	Graph theory (Site not responding. Last check: 2007-11-06)
		The basic approach is to use graph theory (in particular intersection graphs...
		Within graph theory, a digraph with weighted edges is called a network.
		However, it should be noted that within network analysis, the definition of network is much looser, and may often be a simple graph.
	hallencyclopedia.com /Graph_theory (1183 words)

	Clearing up the market cycle... best Perfect Graph (Site not responding. Last check: 2007-11-06)
		Perfect Graph Theorem -- from MathWorld Perfect Graph Theorem -- from MathWorld The graph complement of a perfect graph is itself perfect.
		The strong perfect graph theorem The strong perfect graph theorem A graph G is perfect if for every induced subgraph H, the chromatic number of H equals the size of the largest complete subgraph of H, and G is Berge if no induced subgraph of...
		The perfect graph conjecture states simply that a graph complement of a perfect graph is itself perfect...
	ascot.pl /th/Fourier5/Perfect-Graph.htm (1070 words)

	Perfect graph -- Facts, Info, and Encyclopedia article (Site not responding. Last check: 2007-11-06)
		A graph is perfect if and only if its (Number needed to make up whole force) complement is perfect.
		An (Click link for more info and facts about induced subgraph) induced subgraph that is a cycle of odd length at least 5 is called an odd hole.
		Efforts towards solving the problem have led to deep insights in the field of structural graph theory, where many related problems remain open.
	www.absoluteastronomy.com /encyclopedia/P/Pe/Perfect_graph.htm (405 words)

	Week 3 Abstracts
		The random bipartite model G(n,n,p) is a probability space, whose elements are labeled bipartite graphs with vertex classes A and B, both of size n, where a vertex from A and that from B are connected by an edge randomly and independently with probability p=p(n).
		We prove that every graph on the Klein Bottle which does not contain contractible cycles of length 3 or 4 is either 3-colorable or has a subgraph isomorphic to a member of a particular family of non-3-colorable graphs.
		The Kneser graph K(n,k) is the graph of the disjointness relation on the k-element subsets of an n-set.
	dimacs.rutgers.edu /drei/1998/week3.html (3950 words)

	The Strong Perfect Graph Theorem
		A hole is a chordless cycle of length at least four; an antihole is the complement of such a cycle; holes and antiholes are odd or even according to the parity of their number of vertices.
		since bipartite graphs and line-graphs of bipartite graphs are perfect and since minimal imperfect Berge graphs have neither of the two structural faults; it follows that every square-free Berge graph is perfect.
		Perfect Graphs (J.L. Ramírez-Alfonsín and B.A. Reed, eds.), Wiley, 2001, pp.
	www.cs.concordia.ca /~chvatal/perfect/spgt.html (756 words)

	The Strong Perfect Graph Theorem (Site not responding. Last check: 2007-11-06)
		A graph is perfect if for every induced subgraph, the chromatic number is equal to the maximum size of a complete subgraph.
		The Strong Perfect Graph Conjecture (SPGC) of Berge from 1960 asserts that a graph is perfect if and only if it has no induced subgraph isomorphic to an odd cycle of length at least five, or the complement of such a cycle.
		The class of perfect graphs is important for several reasons.
	www.maths.ox.ac.uk /notices/events/past2/mt02/colloq/thomas.shtml (188 words)

	MATHEMATICAL LINKS
		of graph theory could approach as mostly comprehensible.
		of graphs that are subsets of perfect graphs and
		perfect graphs, that one definitely is one that
	www.gettysburg.edu /~dweinrei/links/gtn.html (305 words)

	Graph theory
		A graph with only vertices and no edges is known as the empty graph, but is sometimes also known as the Null graph.
		In a hypergraph, an edge can connect more than two vertices.
		In model theory, a graph is just a structure.
	www.brainyencyclopedia.com /encyclopedia/g/gr/graph_theory.html (877 words)

	National Science Foundation: What Makes a Perfect Graph? Students of Math Get an Answer (Site not responding. Last check: 2007-11-06)
		The minimum number of colors, or chi, of a graph is at least as large as the number of points in its largest clique.
		According to the conjecture, a graph is perfect if, for the graph and any subgraph created by deleting some of the points, the chi equals the number of points in the largest clique.
		Thus, a phone network based on a perfect graph would run most efficiently with the minimum number of frequencies or channels (colors) assigned to its transmitters (points) and would continue to operate efficiently even if some of the transmitters were knocked out.
	calbears.findarticles.com /p/articles/mi_pfsf/is_200209/ai_2850916141 (378 words)

	Introduction
		A graph is perfect if the vertices of any induced subgraph H can be colored, in such a way that no two adjacent vertices receive the same color, with a number of colors (denoted by
		This definition leads to two interesting problems : to determine which graphs are perfect and to produce an optimal coloring of such perfect graphs.
		Theorem 1 (The Perfect Graph Theorem - Lovász [
	www.univ-lemans.fr /~barre/articles/b_cplt/node1.html (567 words)

	Open problems on perfect graphs
		A graph is called short-chorded or Raspail if each of its cycles whose length is odd and at least five has a ``short chord'', meaning a chord that along with two edges of the cycle forms a triangle.
		-perfect graphs are perfect and that the converse is false; it is known that parity graphs, balanced graphs, comparability graphs and cocomparability graphs are
		The odd stretcher in the first of them is any graph that consists of three vertex-disjoint triangles and three vertex-disjoint paths, each path having an odd number of edges and one endpoint in each of the two triangles.
	www.cs.concordia.ca /~chvatal/perfect/problems.html (4268 words)

	Perfect graph (Site not responding. Last check: 2007-11-06)
		chordal graph (every cycle of length at least 4 has a chord, which is an edge not on the cycle but its endvertices are)
		The first breakthrough was the affirmitive answer to the then perfect graph conjecture.
		This was known to be the strong perfect graph conjecture and was finally answered in the affirmitive in May, 2002.
	www.sciencedaily.com /encyclopedia/perfect_graph (410 words)

	USATODAY.com - Young scientists prove 'Popular' with magazine (Site not responding. Last check: 2007-11-06)
		Mathematics: Chudnovsky helped prove the perfect-graph conjecture, but perfect graphs aren't the x and y graphs most people are familiar with.
		These graphs are made of a bunch of dots, and some dots are connected by lines.
		Graph theory is used for urban planning, network design, molecular biology and similar organizational problems.
	www.usatoday.com /tech/science/discoveries/2004-09-28-brilliant10_x.htm (646 words)

	Rank-Perfect and Weakly Rank-Perfect Graphs (ResearchIndex) (Site not responding. Last check: 2007-11-06)
		Abstract: An edge e of a perfect graph G is critical if G e is imperfect.
		Via relaxations of the stable set polytope of a graph, we de ne two superclasses of perfect graphs: rank-perfect and weakly rank-perfect graphs.
		We study the cases, when a critical edge is removed from the line graph of a bipartite graph or from...
	citeseer.ist.psu.edu /503478.html (378 words)

	Graph Paper! (Site not responding. Last check: 2007-11-06)
		This graph paper is a must for any student doing extensive graphing.
		This graph is perfect for graphing class notes quickly.
		This graph paper is best when you have a lot of graphs to make.
	www.mathematicshelpcentral.com /graph_paper.htm (511 words)

	Jerusalem Mathematics Colloquium (Site not responding. Last check: 2007-11-06)
		Abstract: A graph is called perfect if for every induced subgraph the size of its largest clique equals the minimum number of colors needed to color its vertices.
		This conjecture is known as the Strong Perfect Graph Conjecture.
		A stronger conjecture was made recently by Conforti, Cornuejols and Vuskovic that any Berge graph either belongs to one of a few well understood basic classes or has a decomposition that can not occur in a minimal counterexample to Berge's Conjecture.
	www.ma.huji.ac.il /~colloq/2002-03/col.030501.html (217 words)

	The Math Forum - Math Library - Graph Theory (Site not responding. Last check: 2007-11-06)
		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.
		Unsolved problems on perfect graphs, a collection for people with at least a basic knowledge of the subject.
		Contents include: Perfection of special classes of Berge graphs; Recognition of special classes of Berge graphs; Decompositions of perfect graphs; Minimal imperfect graphs, partitionable graphs, and monsters; Parity problems; The P4-structure; Quantitative variations on the Strong Perfect Graph Conjecture; Intersection graphs; The Markosyan manoeuvre; Appendix: Odds and ends.
	mathforum.org /library/topics/graph_theory (2454 words)

	CS 672 (Site not responding. Last check: 2007-11-06)
		Due to the utility of graphs as models for a wide range of scientific problems, graph theory has become an important topic in the subject of computer algorithms.
		Indeed algorithmic graph theory, the study of graph theory problems from an algorithmic point of view, is now considered an integral subject in the fields of theoretical computer science, operations research, and discrete mathematics.
		The goal of this course is to provide a graduate level introduction to current algorithmic graph theory research problems related to the study of so-called "perfect" graphs.
	www.cs.ualberta.ca /~hayward/672/1999/oCS672.html (322 words)

	Amazon.com: Books: Graph Coloring Problems (Site not responding. Last check: 2007-11-06)
		Graph Theory and Applications: Proceedings (Lecture Notes in Mathematics (Springer-Verlag), 303.) by Conference on Graph Theory (Western Michigan University) 1972 western on page 75, and page 164
		To give you an idea of the level of the discussion in the text, here is an excerpt from page 1: After a terse definition of vertex coloring and "chromatic number", the authors state that "The existence of the chromatic number follows from the Well-Ordering Theorem of set theory...
		I would not recommend it to undergraduates in computer science or mathematics, nor to those seeking accessible discussions of classic graph algorithms; this is not an introductory text.
	www.amazon.com /exec/obidos/tg/detail/-/0471028657?v=glance (1053 words)

	A Generalization of the Perfect Graph Theorem Under the Disjunctive Index (Site not responding. Last check: 2007-11-06)
		In this paper, we relate antiblocker duality between polyhedra, graph theory, and the disjunctive procedure.
		In particular, we analyze the behavior of the disjunctive procedure over the clique relaxation, (G), of the stable set polytope in a graph ℛG, and the one associated to its complementary graph, ℛ(G).
		We obtain a generalization of the Perfect Graph Theorem, proving that the disjunctive indices of ℛ(G) and ℛ(G) always coincide.
	dx.doi.org /10.1287/moor.27.3.460.309 (120 words)

	Advanced Topics in Graph Algorithms
		The course emphasized algorithmic and structural aspects of "nice" graph families, in particular perfect graphs, interval graphs, chordal graphs and comparability graphs.
		In Fall 92 the course was based to a large extent on the classic book of Martin C. Golumbic "Algorithmic Graph Theory and Perfect Graphs' (Academic Press, 1980), and in some parts also on the manuscript "The Art of Combinatorics", by Douglas B. West.
		Interval graphs as a subset of tolerance graphs.
	www.math.tau.ac.il /~rshamir/atga/atga.html (262 words)

	Covering Orthogonal Polygons with Star Polygons: The Perfect Graph Approach (Site not responding. Last check: 2007-11-06)
		In this case, we show that the polygon covering problem can be reduced to the problem of covering a weakly triangulated graph with a minimum number of cliques.
		Since weakly triangulated graphs are perfect, we obtain the following duality relationship: the minimum number of star polygons needed to cover an orthogonal polygon P without holes is equal to the maximum number of points of P, no two of which can be contained together in a covering star polygon.
		In the case where the polygon has at most three dent orientations, we show that the polygon covering problem can be reduced to the problem of covering a triangulated (chordal) graph with a minimum number of cliques.
	sunsite.berkeley.edu /TechRepPages/CSD-87-384 (343 words)

	Atlas: Can the strong perfect graph conjecture be proved? by Robin Thomas (Site not responding. Last check: 2007-11-06)
		The Strong Perfect Graph Conjecture (SPGC) of Berge from 1960 asserts that a graph is perfect if and only it has no induced subgraph isomorphic to an odd cycle of length at least five (an `odd hole'), or the complement of such a cycle (an `odd antihole').
		We will discuss examples of perfect graphs and their importance.
		Then we will describe an approach to the SPGC based on decomposing graphs with no odd holes or odd antiholes, and will present a decomposition result for graphs with no odd holes and no K4 subgraph.
	atlas-conferences.com /cgi-bin/abstract/cafn-50 (167 words)

	Recognizing Perfect 2-Split Graphs
		A graph is a split graph if its vertices can be partitioned into a clique and a stable set.
		A graph is a k-split graph if its vertices can be partitioned into k sets, each of which induces a split graph.
		We show that the strong perfect graph conjecture is true for 2-split graphs and we design a polynomial algorithm to recognize a perfect 2-split graph.
	epubs.siam.org /sam-bin/dbq/article/32908 (110 words)

	Problems in Topological Graph Theory
		Graphs that quadrangulate both the torus and Klein bottle
		Perfect matchings in cubic graphs that have empty intersection
		Orientable genus of graphs of bounded nonorientable genus
	www.emba.uvm.edu /~archdeac/problems/problems.html (283 words)

	Cwikel (Site not responding. Last check: 2007-11-06)
		In 1961, Claude Berge proposed the conjecture that, in every graph with no odd hole or odd antihole, the number of colours needed to properly colour the graph equals the size of the largest complete subgraph.
		Most previous approaches to the conjecture were based on studying properties of a minimal counterexample, but our approach was different.
		We proved that every graph with no odd hole or antihole either falls into one of five well-understood classes, or admits a useful decomposition; and Berge's conjecture is a consequence.
	www.math.princeton.edu /~seminar/2002-03-sem/SeymourAbstract3-5-2003.html (166 words)

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