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Topic: Hamiltonian path

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Hamiltonian cycle

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Hamiltonian path problem

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	Hamiltonian path
		The Hamiltonian path problem for graph G is equivalent to the Hamiltonian cycle problem in a graph H obtained from G by adding a new vertex and connecting it to all vertices of G.
		The Hamiltonian cycle problem is a special case of the traveling salesman problem, obtained by setting the distance between two cities to unity if they are adjacent and infinity otherwise.
		Hamiltonian paths and cycles are named after William Rowan Hamilton who invented the icosian game, now also known as Hamilton's puzzle, which involves finding a Hamiltonian cycle in the edge graph of the dodecahedron.
	publicliterature.org /en/wikipedia/h/ha/hamiltonian_path.html (495 words)

	Hamiltonian path - Wikipedia, the free encyclopedia
		In the mathematical field of graph theory, a Hamiltonian path is a path in an undirected graph which visits each vertex exactly once.
		Determining whether such paths and cycles exist in graphs is the Hamiltonian path problem which is NP-complete.
		A Hamiltonian cycle, Hamiltonian circuit, vertex tour or graph cycle is a cycle that visits each vertex exactly once (except the vertex which is both the start and end, and so is visited twice).
	en.wikipedia.org /wiki/Hamiltonian_path (639 words)

	The Hamiltonian Page
		Hamiltonian cycles avoiding the arcs of prescribed subtournaments, by J. Bang-Jensen, G. Gutin, and A. Yeo.
		Paths and cycles in extended and decomposable digraphs, by J. Bang-Jensen and G. Gutin.
		Hamiltonian paths and cycles in hypertournaments, by G. Gutin and A.
	www.ing.unlp.edu.ar /cetad/mos/Hamilton.html (1335 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)

	Hamiltonian path problem - Wikipedia, the free encyclopedia
		In the mathematical field of graph theory the Hamiltonian path problem and the Hamiltonian cycle problem are problems of determining whether a Hamiltonian path or a Hamiltonian cycle exists in a given graph (whether directed or undirected).
		The Hamiltonian cycle problem is a special case of the traveling salesman problem, obtained by setting the distance between two cities to a finite constant if they are adjacent and infinity otherwise.
		Garey and Johnson showed shortly afterwards in 1974 that the directed Hamiltonian cycle problem remains NP-complete for planar graphs and the undirected Hamiltonian cycle problem remains NP-complete for cubic planar graphs.
	en.wikipedia.org /wiki/Hamiltonian_path_problem (369 words)

	Computing with light: A light-based device for solving the Hamiltonian path problem (Site not responding. Last check: 2007-10-07)
		The paper is organized as follows: The Hamiltonian path problem is briefly described in section 2.
		We need only the ray, which has traversed a Hamiltonian path, to arrive in the destination node at the moment equal to the sum of delays of each node (the moment when the ray has entered in the start node is considered moment 0).
		The rays which have passed once through the previously described cycle are not considered Hamiltonian paths because the moments when they arrive at the destination node are greater than the sum of delays introduced by each node.
	www.cs.ubbcluj.ro /~moltean/light_computation.htm (3857 words)

	distributed
		A problem that seems ideally suited for this demonstration is the directed Hamiltonian path problem (a restricted form of the traveling salesperson problem), one of the class of combinatorially hard problems.
		A Hamiltonian path exists in a graph only if there is a connected path from a designated starting node to a designated output node that visits each node of the graph exactly once.
		There are algorithms that can find a Hamiltonian path in an arbitrary graph, but their time complexity grows exponentially with the size of the graph.
	www.isi.edu /~lerman/etc/distributed.html (1740 words)

	Yuan Xie
		The Hamiltonian Path Problem is a member of a class of computational problems called NP-Complete Problems, which are a group of problems that have been proven to be among the most difficult problems that Computer Scientists have attempted to solve.
		The goal of the Directed Hamiltonian Path Problem is to decide if a path exists in a given directed graph that visits all of the vertices in the graph exactly once.
		After all of the possible paths – of all lengths – have been generated through the self assembly of the designed DNA sequences, the second step is to sort the vertices (Winfree et al, 1996).
	www.its.caltech.edu /~sciwrite/journal03/xie.html (3282 words)

	Graph Concepts (Site not responding. Last check: 2007-10-07)
		A Hamiltonian path is a path that visits every vertex once and only once, except that it might (or might not) begin and end at the same vertex.
		A Hamiltonian cycle is a Hamiltonian path that begins and ends on the same vertex and visits all the other vertices in the graph exactly once.
		On the surface, a Hamiltonian cycle may seem to be very much like an Eulerian circuit, but they are actually quite different.
	home.comcast.net /~lcopes/SciMathMN/concepts/chamil.html (211 words)

	[No title]
		Each path from s to t can be represented in many different ways involving the path from s to some vertex v plus the shortest path between v and t.
		Lemma 2 notes that the length of any path p can be calculated from the sum of the shortest path between s and t plus the sum of delta(e) for all edges in p that are also in sidetracks(p).
		This implicit representation of path is extremely important, as the use of heaps will take advantage of the idea that each path p can be represented as the union of prefpath(p) and lastsidetrack(p).
	www.cs.caltech.edu /~srainier/project/finalpaper.doc (2818 words)

	Graph Theory Lecture Notes 12
		Each vertex of the dodecahedron was labeled with the name of a city, and one was to find a circuit using the edges of the dodecahedron which visited each city once and only once.
		Corollary: G is Hamiltonian iff c(G) is Hamiltonian.
		If there is more than one Hamiltonian path in a tournament, the vertices do not have a unique ranking.
	www-math.cudenver.edu /~wcherowi/courses/m4408/gtln12.html (723 words)

	Islamset - Computing with DNA: Rediscovering Biology ,Hamiltonian Path Problem, Seven Days in a Lab, A New Field ... (Site not responding. Last check: 2007-10-07)
		Your job (the Hamiltonian Path Problem) is to determine if a sequence of connecting flights (a path) exists that starts in Atlanta (the start vertex) and ends in Detroit (the end vertex), while passing through each of the remaining cities (Boston and Chicago) exactly once.
		More generally, given a graph with directed edges and a specified start vertex and end vertex, one says there is a Hamiltonian path if and only if there is a path that starts at the start vertex, ends at the end vertex and passes though each remaining vertex exactly once.
		The Hamiltonian Path Problem is to decide for any given graph with specified start and end vertices whether a Hamiltonian path exists or not.
	www.islamset.com /healnews/dna/dna.html (4075 words)

	More on Hamiltonian
		In graph theory, a graph is Hamiltonian if it contains a path that starts and ends at the same vertex and includes each vertex exactly once.
		Such a path is called a Hamiltonian cycle.
		Both the Hamiltonian operator in physics and Hamiltonian cycles in graph theory are named after Sir William Rowan Hamilton.
	www.artilifes.com /hamiltonian.htm (298 words)

	DNA Computing
		Examples are the Hamiltonian path or the Shortest-Path in a grap.
		Since the directed Hamiltonian path problem has been proven to be NP-complete, it seem likely that no efficient (that is, polynomial time) algorithm exists for solving it.
		The real interesting thing on this DNA solution for the Hamiltonian path problems is that most input data grow just linear with the growth of the number of edges.
	www.casi.net /D.BioInformatics1/D.Fall2000ClassPage/DC1/dc.htm (2738 words)

	CS 260-01 Chapter 7 Notes (Site not responding. Last check: 2007-10-07)
		A strongly connected digraph with n vertices is a Hamiltonian digraph if deg u + deg v is at least 2n-1 for every pair of vertices u and v such that there is no arc from u to v and from v to u.
		A digraph with n vertices is Hamiltonian if out deg u + in degree v is at least n for every pair of vertices u an v such that there is no arc from u to v.
		A digraph with v vertices is Hamiltonian if both the outdegree and the indegree of each vertex is at least n/2.
	www.wpunj.edu /irt/courses/CS260rv/cs_notes7.htm (2004 words)

	Hamiltonian cycle problem : Hamiltonian path problem
		The Hamiltonian cycle or Hamiltonian circuit problem in graph theory is to find a path through a given graph which starts and ends at the same vertex and includes each vertex exactly once.
		It is a special case of the traveling salesman problem, obtained by setting the distance between two cities to unity if they are adjacent and infinity otherwise.
		The requirement that the path start and end at the same vertex distinguishes it from the Hamiltonian path problem.
	www.mik.fastload.org /ha/Hamiltonian_path_problem.html (138 words)

	Hamiltonian path (Site not responding. Last check: 2007-10-07)
		Definition: A simple path through a graph that includes every vertex in the graph.
		Links to papers on many aspects of Hamiltonian cycles and paths.
		Paul E. Black, "Hamiltonian path", in Dictionary of Algorithms and Data Structures [online], Paul E. Black, ed., U.S. National Institute of Standards and Technology.
	www.nist.gov /dads/HTML/hamiltonianPath.html (94 words)

	Solutions to Truncated Octahedron's Hamiltonian Path
		() thus these are also the eighteen (lexicographically) first Hamiltonian* paths of the total 344 possible such paths through the truncated octahedron.
		Note also, how, because of that property and because transpositions are their own inverses, it can be specified naturally as a conjugate of the transposition b or any other "centered subword" of ababacabcbababcbacababa, i.e.
		Furthermore, if we take any two permutations equally distant from the both ends of that path, then the other can be produced from the other by multiplying it (from the left) by transposition (3 4).
	ndirty.cute.fi /~karttu/matikka/permgraf/troctahe.htm (349 words)

	[No title]
		Since we chose G to have as many edges as possible and not have a Hamiltonian circuit, any such larger graph must have a Hamiltonian circuit.
		But this means the path v1...v{i-1}vn...viv1 is a Hamiltonian circuit in G. This is a contradiction.
		Similar remarks apply to a nonclosed path x1...xn except that n might be odd.
	noether.uoregon.edu /~dolan/lecturenotes/notes10.html (1012 words)

	An Algorithmic Construction of Fault Free Hamiltonian Cycles in Faulty Hypercubes.
		In the language of graph embedding, constructing a Hamiltonian circuit in a graph is the same a finding an optimal dilation-1 cycle embedding in that graph.
		When the path is constructed recursively from identical Hamiltonian paths in cubes of smaller dimensions as described above the path is called a reflected Gray code Hamiltonian path.
		Since a reflected Gray code Hamiltonian path has the property that the end vertices are adjacent, adding the edge between them gives a reflected Gray code Hamiltonian circuit.
	www.colostate.edu /dept/Mathematics/profiles/ochs/paper.html (1613 words)

	Hamiltonian cycle (Site not responding. Last check: 2007-10-07)
		Definition: A path through a graph that starts and ends at the same vertex and includes every other vertex exactly once.
		See also Hamiltonian path, Euler cycle, vehicle routing problem, perfect matching.
		A Hamiltonian cycle includes each vertex once; an Euler cycle includes each edge once.
	www.nist.gov /dads/HTML/hamiltonianCycle.html (126 words)

	The Hamiltonian Page
		Parallel algorithms for the Hamiltonian cycle and Hamiltonian problems in semi-complete bipartite digraphs, Jorgen Bang-Jensen, Mohamed El Haddad, Yannis Manoussakis, and Teresa M.Przytycka.
		Optimal Parallel Construction of Hamiltonian Cycles and Spanning Trees in Random Graphs, by Philip D. MacKenzie and Quentin F. Stout, (preliminary version), In Proc.
		Phase Transitions in the Properties of Random Graphs, by J. Frank and C.U. Martel, Aug. 25, 1995.
	www.densis.fee.unicamp.br /~moscato/Hamilton.html (1318 words)

	1.5.5 Hamiltonian Cycle (Site not responding. Last check: 2007-10-07)
		Excerpt from The Algorithm Design Manual: The problem of finding a Hamiltonian cycle or path in a graph is a special case of the traveling salesman problem, one where each pair of vertices with an edge between them has distance 1, while nonedge vertex pairs are separated by distance infinity.
		Closely related is the problem of finding the longest path or cycle in a graph, which occasionally arises in pattern recognition problems.
		The longest path through this graph is likely the correct interpretation.
	www.cs.sunysb.edu /~algorith/files/hamiltonian-cycle.shtml (195 words)

	Graph Magics
		Hamiltonian Path/Circuit - Finds a path/circuit that passes through each vertex exactly once.
		Graph Median - Finds the vertex for which the sum of lengths of shortest paths to all other vertices is the smallest.
		Graph Center - Finds the vertex for which the length of shortest path to the farthest vertex is the smallest.
	www.graph-magics.com /algorithms.php (269 words)

	[No title]
		In particular, if $G$ is 3-connected, then $G$ is Hamiltonian connected.", vol= 121, year= "1993", pages= "223-228") @article(BolBri93, author= "Bela Bollobas and Graham Brightwell", title= "Cycles through specified vertices", journal= "Combinatorica.
		Using the relationship between maximal cliques the all-pair shortest path problem is solved in $O(\log n)$ time using $O(n^2)$ processors.") %COLORING PROBLEMS @article(OL91,author= {Olariu, S.},title= {Optimal greedy heuristic to color interval graphs},annote= {A $O(n)$ algorithm for coloring interval graphs.
		Algorithms for the same problem are also given for directed path and circular-arc graphs.") %MATCHING PROBLEMS @inproceedings(AACL95,author= {Andrews,M.G. and Atallah,M.J. and Chen,D.Z. and Lee,D.T.},title= {Parallel algorithms for maximum matching in interval graphs},booktitle= "Proceedings 9th Parallel Processing Symposium",publisher= "IEEE Comput.
	www.cs.ualberta.ca /~stewart/GRAPH/search/all.bib (1767 words)

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