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Topic: Bipartite graph

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	Puzzles on graphs
		A graph is called bipartite if the set of its vertices can be represented as a union of two disjoint sets such that no two nodes of the same set are connected by an edge.
		A subgraph of a graph is a graph whose vertices and edges form subsets of the sets of vertices and edges, respectively, of the given graph that may be called a supergraph.
		A connected component of a node is the biggest subgraph of a graph that consists of all the nodes that serve as endpoints of walks starting at the given node.
	www.cut-the-knot.org /do_you_know/graphs2.shtml (1158 words)

	Graph theory
		Graph theory is the branch of mathematics that examines the properties of graphs.
		In computers, a finite directed or undirected graph (with n vertices, say) is often represented by its adjacency matrix: an n-by-n matrix whose entry in row i and column j gives the number of edges from the i-th to the j-th vertex.
		A subgraph of the graph G is a graph whose vertex set is a subset of the vertex set of G, whose edge set is a subset of the edge set of G, and such that the map w is the restriction of the map from G.
	www.ebroadcast.com.au /lookup/encyclopedia/bi/Bipartite.html (1632 words)

	PlanetMath: bipartite graph (Site not responding. Last check: 2007-10-25)
		One way to think of a bipartite graph is by partitioning the vertices into two disjoint sets where vertices in one set are adjacent only to vertices in the other set.
		This is easy to see intuitively: any path of odd length on a bipartite must end on a vertex of the opposite colour from the beginning vertex and hence cannot be a cycle.
		This is version 6 of bipartite graph, born on 2002-02-03, modified 2004-04-30.
	www.planetmath.org /encyclopedia/bipartitegraph.html (165 words)

	Bipartite graph - Wikipedia, the free encyclopedia
		Bipartite graphs are useful for modelling matching problems.
		For a connected bipartite graph the size of the minimum edge cover is equal to the size of the maximum independent set.
		For a connected bipartite graph the size of the minimum edge cover plus the size of the minimum vertex cover is equal to the number of vertices.
	en.wikipedia.org /wiki/Bipartite (511 words)

	Graph Theory Glossary
		For example, Figure 1.3.8 shows a simple graph which is also a bipartite graph because it may be divided into two parts, given by the subsets {1, 2} and {3, 4, 5}, where every edge in the graph goes from a vertex in one part to a vertex in the other part.
		In the graph shown in Figure 1.3.14 a, cycles are represented, for example, by sequences of vertices 1, 5, 4, 2, 3, 4, 1 and 1, 2, 3, 4, 5, 1.
		The vertices of the graph shown in Figure 1.3.29 may be properly colored in four colors: the first color for vertex 1, the second color for vertices 2, and 7, the third color for vertices 4, and 5, and the fourth color for vertices 3, and 6.
	exchange.manifold.net /manifold/manuals/5_userman/mfd50Graph_Theory_Glossary.htm (3582 words)

	PlanetMath: bipartite matching (Site not responding. Last check: 2007-10-25)
		A matching on a bipartite graph is called a bipartite matching.
		A system of distinct representatives is equivalent to a maximal matching on some bipartite graph.
		This is version 4 of bipartite matching, born on 2002-05-26, modified 2003-11-06.
	planetmath.org /encyclopedia/BipartiteMatching.html (364 words)

	BipartiteGraphs - PineWiki
		A graph (see GraphTheory) is a bipartite graph if its vertex set can be written as X∪Y and every edge is an element of X×Y. Alternatively, a graph is bipartite if it can be 2-colored (the vertices in the two color sets give X and Y).
		Bipartite graphs are often used to model assignment problems, where the vertices of the left-hand side X represent things that need to be assigned, the vertices of the right-hand side Y represent places to put them, and an edge indicates compatibility.
		A complete bipartite matching is a subset of the edges of a bipartite graph such that every node is the endpoint of exactly one edge: such a matching corresponds to an assignment that assigns every object and fills every niche (it also implies X=Y).
	pine.cs.yale.edu /pinewiki/BipartiteGraphs (527 words)

	Graphs Glossary
		A graph is bipartite if the vertices can be partitioned into two sets, X and Y, so that the only edges of the graph are between the vertices in X and the vertices in Y. Trees are examples of bipartite graphs.
		The closure of a graph G with n vertices, denoted by c(G), is the graph obtained from G by repeatedly adding edges between non-adjacent vertices whose degrees sum to at least n, until this can no longer be done.
		An induced (generated) subgraph is a subset of the vertices of the graph together with all the edges of the graph between the vertices of this subset.
	www-math.cudenver.edu /~wcherowi/courses/m4408/glossary.html (2135 words)

	14. Some Graph Theory
		A graph G is a collection, E, of distinct unordered pairs of distinct elements of a set V.
		Graphs are things that underlie many mathematical structures, and in fact anything that involves pairs of elements, and this includes any kind of relationship between pairs of individual entities.
		If the bridge graph is bipartite, we can draw one part of it inside the cycle without crossings and the other outside it without crossings, and G is then planar.
	www-math.mit.edu /18.310/some_graph_theory.html (3368 words)

	Graph theory glossary
		A coclique in a graph is a clique in its complementary graph (q.v.).
		girth (n.): The girth of a graph is the length of the shortest cycle(s) in the graph.
		When A,B are graphs, an isomorphism is a bijection from the vertices of A to the vertices of B such that any two vertices of A are adjacent if and only if their images in B are adjacent.
	www.math.harvard.edu /~elkies/FS23j.04/glossary_graph.html (1317 words)

	graph@Everything2.com
		A graph is a set V of vertices (or nodes) together with a set E of edges (pairs of vertices from V).
		If the pairs are unordered, the graph is undirected; such a graph is usually called a graph.
		The generalization of the graph with hyperedges is a hypergraph.
	everything2.com /index.pl?node=graph (294 words)

	Vertex Coloring
		Such a coloring of the vertices of a bipartite graph means that the graph can be drawn with the red vertices on the left and the blue vertices on the right such that all edges go from left to right.
		Bipartite graphs are fairly simple, yet they arise naturally in such applications as mapping workers to possible jobs.
		The most famous problem in the history of graph theory is the four-color problem, first posed in 1852 and finally settled in 1976 by Appel and Haken using a proof involving extensive computation.
	www2.toki.or.id /book/AlgDesignManual/BOOK/BOOK4/NODE178.HTM (1427 words)

	Graphs Glossary
		A chain in a graph is a sequence of vertices from one vertex to another using the edges.
		The diameter of a graph is the length of the longest chain you are forced to use to get from one vertex to another in that graph.
		A vertex is a 'dot' in a graph.
	www-math.cudenver.edu /~wcherowi/courses/m4408/glossary.htm (1926 words)

	Graph Theory Open Problems
		A graph which can be embedded in the plane so that vertices correspond to points in the plane and edges correspond to unit-length line segments is called a ``unit-distance graph.'' The question above is equivalent to asking what the chromatic number of unit-distance graphs can be.
		It is known that this is not true if you remove the "bipartite" condition, but the smallest known such graph which is not Hamiltonian has 38 vertices, as shown to the right.
		To get the square of an oriented graph (or any directed graph) you leave the vertex set the same, keep all the arcs, and for each pair of arcs of the form (u,v), (v,w), you add the arc (u,w) if that arc was not already present.
	dimacs.rutgers.edu /~hochberg/undopen/graphtheory/graphtheory.html (705 words)

	CHAPTER 3 Graph Theory
		graphs and see if you can deduce, in general, the number of vertices and the number of edges for any given complete graph on n vertices.
		Because of their positioning when the graph is drawn we will call these the left vertex set and the right vertex set.
		-graphs; wheel graphs; and bipartite graphs, and determine the number of vertices and edges, and come to some general conclusions about them.
	mathematics.gulfcoast.edu /mgf1107ll/Chap3Sec2Lesson6.htm (428 words)

	CG: 2.2 A CG is a bipartite graph
		The nodes of a bipartite graph can be divided into two nonempty sets A and B, with two different kinds of nodes.
		All arcs of a bipartite graph then connect exactly one node from A and one node from B. Therefore, all arcs cross the boundary between the two sets of the two kinds of nodes.
		Conceptual graphs are bipartite in that it has two kinds of nodes, concepts and relations, and every arc "crosses the border" between the two sets of the two kinds of nodes.
	www.huminf.aau.dk /cg/Module_I/1005.html (145 words)

	Graph Theory Lesson 9
		A non-null graph is bipartite if and only if its chromatic number is 2.
		Note that figure 12 is a bipartite graph that is not a complet bipartite graph.
		A bipartite graph is used in a certain college to model the relationship between students and courses.
	oneweb.utc.edu /~Christopher-Mawata/petersen/lesson9.htm (314 words)

	Graph Theory Concepts (Site not responding. Last check: 2007-10-25)
		In a bipartite graph, it is possible to partition the set of vertices into two sets such that none of the vertices in either set are adjacent to one another.
		In a tripartite graph, the vertices are partitioned into three sets (partitions) so that no two vertices contained in any one partition are adjacent.
		Thus K(3, 3) is a complete bipartite graph with 3 vertices in each partition and K(3, 4, 5) is a complete tripartite graph with partitions of 3, 4, and 5 vertices.
	home.comcast.net /~lcopes/SciMathMN/concepts/cbipar.html (145 words)

	Graph Theory
		A graph G is planar if it can be drawn in the plane in such a way that no two edges meet each other except at a vertex to which they are incident.
		Note that the sum of all the degrees of the faces is equal to twice the number of edges in the the graph, since each edge either borders two different faces (such as bg, cd, and cf) or occurs twice when walk around a single face (such as ab and gh).
		Assume that the result is true for all connected plane graphs with fewer than m edges, where m is greater than or equal to 1, and suppose that G has m edges.
	www.personal.kent.edu /~rmuhamma/GraphTheory/MyGraphTheory/planarity.htm (1615 words)

	Graph Theory
		So the emphasis for the final will be on using graph theory as a tool to formulate problems, asking only for you to be familiar with a reasonable proportion of the material we've covered in class, including at least one of the class presentations in addition to that of your own group.
		Every graph G has embeddings in surfaces S_k when k is large enough (e.g., k = number of edges) and so there is a _least_ such nonnegative integer k which is defined to be the _genus of G_ denoted gamma(G) (the book uses g for genus but we use g for girth).
		The radius of a graph is the minimum eccentricity of the vertices, while the diameter of a graph is the maximum eccentricity of the vertices.
	www.georgetown.edu /faculty/kainen/graphtheory.html (3531 words)

	Matching
		General graphs prove trickier because it is possible to have augmenting paths that are odd-length cycles, i.e.
		The standard algorithms for bipartite matching are based on network flow, using a simple transformation to convert a bipartite graph into an equivalent flow graph.
		Another common ``application'' of bipartite matching is in marrying off a set of boys to a set of girls such that each boy gets a girl he likes.
	www2.toki.or.id /book/AlgDesignManual/BOOK/BOOK4/NODE164.HTM (1293 words)

	Graph Generators (graph_gen)
		creates a random bipartite graph G with a nodes on side A, b nodes on side B, and m edges.
		For n = 1, the graph consists of a single isolated node, for n = 2, the graph consists of two nodes and one uedge, for n = 3 the graph consists of three nodes and three uedges.
		The embedding is given by xcoord[v] and ycoord[v] for every node v of G. The generator chooses n segments whose endpoints have random coordinates of the form x/K, where K is the smallest power of two greater or equal to n, and x is a random integer in 0 to K -1.
	www-graphics.stanford.edu /courses/cs368/LEDA/node110.html (926 words)

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