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Topic: Eulerian path

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	Eulerian path - Encyclopedia, History, Geography and Biography
		Euler observed that a necessary condition for the existence of Eulerian cycles is that all vertices in the graph have an even degree, and that for an Eulerian path either all, or all but two, vertices have an even degree; this means the Königsberg graph is not Eulerian.
		An Eulerian path, Eulerian trail or Euler walk in an undirected graph is a path that uses each edge exactly once.
		The definition and properties of Eulerian paths, cycles and graphs are valid for multigraphs as well.
	www.arikah.com /encyclopedia/Eulerian_graph (766 words)

	Encyclopedia: Eulerian path (Site not responding. Last check: 2007-10-07)
		In mathematics, a path in a graph is a sequence of vertices such that from each of its vertices there is an edge to the successor vertex.
		An Eulerian path, Eulerian trail or Euler walk in a undirected graph is a path that uses each edge exactly once.
		A directed graph version of Eulerian cycle result and algorithm can be obtained in a similar fashion except that the even degree condition is replaced by an equal in- and out-degrees condition.
	www.nationmaster.com /encyclopedia/Eulerian-path (564 words)

	Glossary of graph theory
		Traditionally, a path is graph consisted of a sequence of successively incident edges and their endvertices, where the terminating vertices are distinct.
		An Eulerian path in a graph is a path that uses each edge precisely once.
		A directed path, or just a path when the context is clear, is an oriented simple path such that all arcs go the same direction, meaning all internal vertices have in- and out-degrees 1.
	www.sciencedaily.com /encyclopedia/glossary_of_graph_theory (4397 words)

	CLOSED EULERIAN PATH (Site not responding. Last check: 2007-10-07)
		A path is a sequence of vertices such that from each of its vertices there is an edge to the successor vertex.
		In the example graph, (1, 2, 5, 1, 2, 3) is a path with length 5, and (5, 2, 1) is a simple path of length 2.
		While determining whether a given graph has an Eulerian path or cycle is trivial, solving the same problem for Hamiltonian paths and cycles appears to be extremely hard.
	www.websters-online-dictionary.org /definition/CLOSED+EULERIAN+PATH (1234 words)

	\| International School of Photonics \| ISP Archives \| ISP Article Collection \|
		A path which connects all the nodes using all the edges only once was the one which was sought by the people of K. Such path,in the language of Graph Thoery (GT) is called the Eulerian path.
		Path is the walk in which all nodes are distinct(except for initial and final point which is a closed path).
		Euler found that The path envisaged by citizens of K is not possible (this path is now called Eulerian Path) since degree of each vertex (number of edges reaching the vertex) is odd number.
	www.photonics.cusat.edu /Article4.html (2329 words)

	Station Information - Eulerian path
		An Eulerian path (Euler walk) in a graph is a path that uses each edge precisely once.
		L.Euler showed that an Eulerian cycle exists if and only if all vertices in the graph are of even degree.
		An Eulerian path exists, if and only if at most two vertices in the graph are of odd degree.
	www.stationinformation.com /encyclopedia/e/eu/eulerian_path.html (125 words)

	Ideas, Concepts and Definitions (Site not responding. Last check: 2007-10-07)
		A hamiltonian path in a graph is a path that passes through every vertex in the graph exactly once.
		A hamiltonain path does not necessarily pass through all the edges of the graph, however.
		A hamiltonian path which ends in the same place in which it began is called a hamiltonian circuit or a hamiltonain cycle.
	www.c3.lanl.gov /mega-math/gloss/graph/grham.html (56 words)

	Eulerian Paths and Cycles (Site not responding. Last check: 2007-10-07)
		There can be an Eulerian path only if the graph is connected, meaning that for any pair of distinct nodes, there is a path from one to the other (see Figure 4).
		A graph has a Eulerian path if and only if the graph is connected and the number of vertices with odd degree is either 0 or 2.
		If the number is 0, then any Eulerian path is an Eulerian cycle; if it is 2, then an Eulerian path must begin at one of the vertices of odd degree and end at the other.
	www.webpearls.com /products/demos/pap/comap/chapter5/sec1/node3.html (692 words)

	Eulerian Cycle / Chinese Postman
		Finally, a directed graph contains an Eulerian path iff (1) it is connected and (2) all but two vertices have the same in-degree as out-degree, and these two vertices have their in-degree and out-degree differ by one.
		Given this characterization of Eulerian graphs, it is easy to test in linear time whether such a cycle exists: test whether the graph is connected using DFS or BFS, and then count the number of odd-degree vertices.
		Finding the best set of shortest paths to add to G reduces to identifying a minimum-weight perfect matching in a graph on the odd-degree vertices, where the weight of edge (i,j) is the length of the shortest path from i to j.
	www2.toki.or.id /book/AlgDesignManual/BOOK/BOOK4/NODE165.HTM (1161 words)

	Talk:Eulerian path - Wikipedia, the free encyclopedia
		There should be an explanation of how unicursality (?) relates to an Eulerian path so that the link doesn't seem confusing.
		Furthermore, G has an Euler path iff every vertex has even degree except for two distinct vertices, which have odd degree.
		When this is the case, the Euler path starts at one and ends at the other of these two vertices of odd degree." Is this contradicting the article?
	en.wikipedia.org /wiki/Talk:Eulerian_path (346 words)

	A Cell Generator for Static CMOS Circuits
		Such a path is called a dual Eulerian path and the next section provides a brief introduction to the graph theoretic background required to understand how the cell generator works.
		An Eulerian path through a graph is one which visits each edge of the graph once and only once.
		The relevance of an Eulerian path to the cell generation problem is that maximal run of abutting transistors for any one of the stacks is an Eulerian path of its TTSPMG.
	www.cs.utah.edu /~mbinu/coursework/ee6740 (3027 words)

	[No title]
		However the graph would have up to 4^8=65,536 vetices and probably more arcs (the simplest Hamiltonian path requires n-1 arcs, where n is the number of vertices, to join all the vertices together, and the real case of SBH would be much more complicated).
		Hamiltonian path is proved to be NP-complete, so it is not possible to apply for fragments that have hundreds of nucleotides.
		The algorithm to find an Eulerian path is also well studied and it is linear with respect to the number of arcs.
	www.ocf.berkeley.edu /~dywong/be142/experiment.txt (719 words)

	Marrakech (Site not responding. Last check: 2007-10-07)
		That is, a path through the network that visits each vertex only once, except possibly the starting/ending vertex which could be the same.
		Then he proved that such a path through a network does not exist if there are more than two vertices with an odd number of paths converging to them.
		That is, the Eulerian path cannot be found through networks with more than two odd vertices.
	home.austin.rr.com /cbergman/poetry/morocco.html (252 words)

	[No title] (Site not responding. Last check: 2007-10-07)
		As we must have two paths the maximal number of edges in the cycles can be 4 but the smalles cycle is a C_3.
		Classification of the Eulerian path pairs: (WLoG connect A-C and B-D) The classification is according the path length.
		Eulerian circuits of graph B: Dissect the graph by replacing the pair of edges AB, CD by the pair AD, BC.
	www.mathematik.uni-bielefeld.de /~sillke/PUZZLES/nikolaus (2994 words)

	Counting paths (Site not responding. Last check: 2007-10-07)
		There is an interesting relation between the number of paths in a diagram and the number to be found in the dual diagram.
		A Hamiltonian path is one which visits each node of the diagram once (clearly, the diagram must be connected), while an Eulerian path is one which traverses every link once (in the proper direction) even though nodes are repeated during the voyage.
		A Hamiltonian path, one of which is usually hard to find, becomes an Eulerian path in the antidual diagram, criteria for which are generally easier to obtain.
	delta.cs.cinvestav.mx /~mcintosh/newweb/cf/node20.html (268 words)

	No Title
		Eulerian path/tour find a path/tour through the graph such that every edge is visited exactly once.
		Alternating path is a path from v (in V) to u (in U, where U and V are node types of bipartite graph) both of which are unmatched in M such that the edges of P are alternatively in E-M and M (M is the maximal match)
		An augmenting path with respect to a given flow is a directed path from s to t which consists of edges from G, but not necessarily in the same direction.
	www.cs.usu.edu /~allanv/cs5050/ch7/ch7.html (3395 words)

	Graph Magics
		An undirected graph has an eulerian circuit if and only if it is connected and each vertex has an even degree (degree is the number of edges that are adjacent to that vertex).
		An undirected graph has an eulerian path if and only if it is connected and all vertices except 2 have even degree.
		It is needed to find a path that starts and ends at the post-office, and that passes through each street (edge) exactly once.
	www.graph-magics.com /articles/euler.php (1280 words)

	Ideas, Concepts and Definitions (Site not responding. Last check: 2007-10-07)
		An eulerian path in a graph is a path that travels along every edge of the graph exactly once.
		An eulerian path might pass through individual vertices of the graph more than once.
		An eulerian path which begins and ends in the same place is called an eulerian circuit or an eulerian cycle
	www.c3.lanl.gov /mega-math/gloss/graph/greuler.html (53 words)

	No Title
		Euler's Theorem tells us the conditions under which an Eulerian path or Eulerian circuit exists in a connected graph (circuit: all even vertices; path: all even, or a single pair of odd vertices).
		If so, neatly draw the path or circuit on the graph, indicating clearly the starting vertex and the order in which the edges are traversed.
		There is a path (not a circuit, as there is a pair of odd vertices) for the graph at left; and there is a circuit (all vertices even) for the graph at right.
	www.nku.edu /~longa/classes/2000fall/mat115/chapter6/answers4/answers4.html (794 words)

	Graph Theory Lecture Notes 10 (Site not responding. Last check: 2007-10-07)
		Def: An eulerian closed chain is a closed chain in a multigraph which uses each edge exactly once.
		Theorem (Euler): A multigraph G has an eulerian chain if and only if G is connected up to isolated vertices and the number of vertices of odd degree is either 0 or 2.
		Theorem (Good): A multidigraph D has an eulerian path if and only if D is weakly connected up to isolated vertices and for all vertices with the possible exception of two, indegree equals outdegree, and for at most two vertices, indegree and outdegree differ by one.
	www-math.cudenver.edu /~wcherowi/courses/m4408/gtln10.html (263 words)

	Route inspection problem - RecipeFacts (Site not responding. Last check: 2007-10-07)
		The Chinese postman problem is to find a shortest closed path (circuit) that goes through all edges of a (connected) undirected graph.
		When the graph has an Eulerian circuit, that circuit is an optimal solution.
		A semi-Eulerian path (a path which is not closed but uses all edges of G and all vertices of G just once) exists if and only if G is connected and exactly two vertices have non-even valence.
	www.recipeland.com /encyclopaedia/index.php/Route_inspection_problem (209 words)

	Pavel Pevzner's Web Page at UCSD
		Biologists are well-aware of these errors errors and are forced to carry additional experiments to verify the assembled contigs.
		We abandon the classical "overlap - layout - consensus" approach in favor of a new Eulerian Superpath approach that, for the first time, resolves the problem of repeats in fragment assembly.
		Our main result is the reduction of the fragment assembly to a variation of the classical Eulerian path problem.
	www-cse.ucsd.edu /users/ppevzner/research1.html (288 words)

	Math 118 -- Information about Midterm #2
		An Eulerian path which is a circuit (that is it is closed) is called an
		A graph has an Eulerian path if and only if it is connected and has either no odd vertices or exactly two odd vertices.
		If two of its vertices are odd, then any Eulerian path must begin at one these vertices and end at the other one.
	www.math.sunysb.edu /~sorin/118/E2info.html (1826 words)

	MATHmaniaCS - Lesson 12: Eulerian Paths and Circuits
		Now the Bridges problem becomes: "Is it possible to pass over each edge exactly once and return to the starting vertex?" This is the same as asking: "Is there an Eulerian Circuit?" (after Leonhard Euler (pronounced "oiler"), who was a prolific mathematician, the "father" of graph theory, and the first to solve this puzzle).
		An Eulerian Path is almost exactly like an Eulerian Circuit, except you don't have to finish where you started.
		Any animal that is made from a single balloon has an Eulerian path; to see it, simply follow the balloon from one end to the other through the animal.
	www.mathmaniacs.org /lessons/12-euler (1577 words)

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