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Topic: Planar graph

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	Springer Online Reference Works
		A planar map each side of which is bounded by three edges is said to be a planar triangulation.
		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 (0 words)

	PlanetMath: planar graph (Site not responding. Last check: )
		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.
		A straight line drawing of a planar graph is a drawing in which each edge is drawn as a straight line segment.
		This is version 9 of planar graph, born on 2002-02-05, modified 2006-10-17.
	www.planetmath.org /encyclopedia/PlanarGraph.html (541 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.
		A straight line drawing of a planar graph is a drawing in which each edge is drawn as a straight line segment.
		This is version 9 of planar graph, born on 2002-02-05, modified 2006-10-17.
	planetmath.org /encyclopedia/Planar.html (540 words)

	Planar graph
		If, given the graphs A and B, and B which is an expansion of A, it is often described that A is homeomorphic to B. In practice, Kuratowski's criterion cannot be used to quickly decide whether a given graph is planar.
		Every planar graph is 4-partite, or 4-colorable; this is the graph-theoretical formulation of the four color theorem.
		For a planar graph G we may construct a graph whose vertices are the regions into which G divides the plane (including a single external region).
	www.ebroadcast.com.au /lookup/encyclopedia/pl/Planar_graph.html (417 words)

	Monadic Second-Order Logic and planar graph drawings with edge crossings: abstract
		This property is monadic second-order for the structures representing a planar drawing of a planar graph (called the frame) augmented with additional edges that may cross one another and that may cross the edges of the frame.
		Graphs can be represented by logical structures and their properties can be expressed by logical formulas.
		A framed predrawing is a structure representing a planar drawing of a planar graph (called the frame) augmented with additional edges that may cross one another and that may cross the edges of the frame.
	www.labri.fr /Perso/~courcell/Art13.html (0 words)

	Tom's Combinatorial Geometry Class
		A graph is called a planar graph if it is drawn in such a way that the edges never cross, except at where they meet at vertices.
		There is a direct connection between polyhedra and planar graphs, namely that we can take any polyhedron and "project" it down onto a flat piece of paper, turning it into a graph.
		Similarly, the degree of a face f in a planar graph is the number of edges going around the face.
	www.merrimack.edu /~thull/combgeom/graphnotes.html (1055 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.
		A chain in a graph is a sequence of vertices from one vertex to another using the edges.
		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.
	www-math.cudenver.edu /~wcherowi/courses/m4408/glossary.htm (1926 words)

	Embedding an outer-planar graph in a point set (Site not responding. Last check: )
		Planar graphs are all well and good, but we wish to consider the problem of straight-line embeddings in a given point set.
		Outer-planar graphs are a sub-class of planar graphs with one simple restriction: the graph must have a planar embedding in which every vertex lies on a single face.
		A theorem for planar graphs tells us that this is equivalent to the restriction that the graph must have a planar embedding in which every vertex lies on the unbounded face.
	www.cs.toronto.edu /~king/cgm/507planar.html (433 words)

	David Eppstein - Publications
		The complement of a minimum spanning tree is a maximum spanning tree in the dual graph.
		It was known that planar graphs have the diameter-treewidth property: there is a function f(D) such that any planar graph with diameter D has treewidth at most f(D).
		Some of these results were announced in the conference version of "subgraph isomorphism for planar graphs and related problems" but not included in the journal version.
	www.ics.uci.edu /~eppstein/pubs/graph-planar.html (568 words)

	Planar (Site not responding. Last check: )
		It is possible to detect whether or not a graph is planar in polynomial time (O(n)) [Hopcroft and Tarjan, 1974] as well as to draw a planar graph with no edge crossings, called a planar embedding [Chiba, et al.
		As discussed elsewhere, however, it may not even be desireable to draw a planar graph without edge crossings.
		Any planar graph admits multiple planar embeddings, since any face can be moved to the outside of the graph without changing its topology.
	catt.bus.okstate.edu /jones98/planar.htm (216 words)

	Graph Data Structures
		Planar graphs are those that can be drawn in the plane so that no two edges cross.
		GraphEd [Him94], written in C by Michael Himsolt, is a powerful graph editor that provides an interface for application modules and a wide variety of graph algorithms.
		Dynamic graph algorithms are essentially data structures that maintain quick access to an invariant (such as minimum spanning tree or connectivity) under edge insertion and deletion.
	www2.toki.or.id /book/AlgDesignManual/BOOK/BOOK3/NODE132.HTM (1450 words)

	3 Utilities Puzzle: Water, Gas, Electricity
		A graph is a collection of nodes (also called vertices) and edges each connecting a pair of nodes.
		To visualize a graph, nodes may be thought of as points in space, plane, or another surface, while edges are represented by curves connecting the nodes.
		A planar graph with a finite number of nodes may always be embedded into a bounded portion of the plane.
	www.cut-the-knot.org /do_you_know/3Utilities.html (1394 words)

	Graph Generators ( graph_gen )
		Planar graphs: Combinatorial Constructions A maximal planar map with n nodes, n > = 3, has 3n - 6 uedges.
		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 generators with the word map replaced by graph, first generate a map and then delete one edge from each uedge.
	www.cs.cmu.edu /afs/cs.cmu.edu/project/aladdin/LEDA/4.4.1/Manual/HTML/graph_gen.html (945 words)

	Graph Concepts (Site not responding. Last check: )
		When a planar graph is drawn with edges crossing, it is still a planar graph.
		In some contexts, a planar graph is called a map because of the resemblance that it bears to a geographic map.
		The crossing number of a graph is the smallest number of intersections needed to draw the graph.
	web.hamline.edu /~lcopes/SciMathMN/concepts/cplanr.html (200 words)

	The Artificial Unger Graph Coloring Applet
		The planar modifier in the previous sentence indicates a graph with no vertices intersecting another when the graph is drawn on a planar surface.
		The first graph is a planar bi-directional one, and thus the graph coloring problem becomes the map coloring problem.
		To solve the general graph coloring problem with either algorithm, the applet starts with a color palette of n, where n is the number of vertices in the graph.
	www.duke.edu /~jmu2/color/gc.html (1773 words)

	Planar Graphs
		A planar graph is one that can be drawn on a plane in such a way that there are no "edge crossings," i.e.
		A plane graph is one that has been drawn in the plane in such a way that its edges intersect only at their common end-vertices.
		Let G be a plane graph, and consider the regions bounded by the edges of G.
	www.math.lsa.umich.edu /mmss/coursesONLINE/graph/graph5 (734 words)

	The Four Color Theorem
		Also, Hamilton made contributions to graph theory (such as the idea of a Hamiltonian circuit, i.e., a path along the edges of a graph that visits each vertex exactly once), a subject that was developed largely through efforts to prove the four color conjecture.
		The graph of a set of three mutually adjoining regions is simply a topological triangle, and if we add a fourth region, it is represented by a fourth vertex in the graph, which must be located either inside or outside the triangle formed by the graph of the original three vertices.
		If we could reduce all the causal links in a graph to their bare minimums, it might be possible to reduce every n-colorable graph to a graph with just n mutually connected vertices, and hence n could never exceed four.
	www.mathpages.com /home/kmath266/kmath266.htm (4081 words)

	Embedding an outer-planar graph in a point set (Site not responding. Last check: )
		If the graph is disconnected, we can sort the point set by angle with some outside point (that is not in the point set), partition the point set appropriately, and embed the graph's connected components in the partitions of the point set.
		If the graph is connected but has a cutvertex v, we can sort the point set by angle with some point p, partition the point set (excluding an extreme point p) appropriately, and embed the cut components of in the partitions, joining them at p.
		In an outer-planar graph (remember, each bounded face is a triangle since we assume the graph is maximal), an (r,s)-triangle is a face in the graph with at least one edge on the unbounded face such that deleting the vertices of the triangle cuts the graph into two outer-planar graphs of size r and s.
	cgm.cs.mcgill.ca /~athens/cs507/Projects/2004/Andrew-King/507embedding.html (613 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)

	BackgroundMaterial (Site not responding. Last check: )
		A graph G is planar if it can be drawn on the ordinary euclidean plane without edge crossings.
		Now consider a non-crossing drawing of a planar graph G with n vertices and m edges.
		It is not at all clear that Kuratowski's theorem yields an efficient algorithm for planarity testing, but such algorithms exist and have running times which are linear in the number of edges in the graph being tested.
	www.math.gatech.edu /~trotter/Section3-planar.htm (353 words)

	Algorithms for Planar Graphs (plane_graph_alg)
		PLANAR takes as input a directed graph G(V,E) and performs a planarity test for it.
		If the second argument embed has value true and G is a planar graph it is transformed into a planar map (a combinatorial embedding such that the edges in all adjacency lists are in clockwise ordering).
		PLANAR takes as input a directed graph G(V,E) and performs a planarity test for G. PLANAR returns true if G is planar and false otherwise.
	graphics.stanford.edu /courses/cs368/LEDA/node126.html (586 words)

	Math 480- Planar Graphs
		Focus: Planar graphs are responsible for much of the early interest in Graph Theory.
		Our study of planar graphs will worry about several issues - the complete characterization of planar graphs (through Kuratowski's Theorem), vertex coloring bounds (hence face coloring bounds for the duals), and topological considerations of embeddings and for non-planar graphs, crossing number.
		Hence they are a good family of graph to test conjectures on (since there are more tools available for outerplanar graphs) but not particularly useful in the general scheme of things.
	faculty.oxy.edu /jquinn/home/Math480/module6.html (800 words)

	1.4.12 Planarity Detection and Embedding (Site not responding. Last check: )
		Planar graphs have a variety of nice properties, which can be exploited to yield faster algorithms for many problems on planar graphs.
		Since every subgraph of a planar graph is planar, this means that there is always a sequence of low-degree vertices whose deletion from G eventually leaves the empty graph.
		The best book available for this problem is Graph Drawing: Algorithms for the Visualization of Graphs by Giuseppe Di Battista, Peter Eades, Roberto Tamassia, and Ionnis G. Tollis.
	www.cs.sunysb.edu /~algorith/files/planar-drawing.shtml (293 words)

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