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# Topic: Reconstruction conjecture

###### In the News (Tue 23 Jul 19)

 Reconstruction conjecture - Wikipedia, the free encyclopedia Informally, the reconstruction conjecture in graph theory says that graphs are determined uniquely by their subgraphs. The conjecture has been verified for a number of infinite classes of graphs, such as regular graphs (graphs in which all vertices have the same number of edges attached to them). In fact, it was shown that not only are almost all graphs reconstructible, but in fact that the entire deck is not necessary to reconstruct them --- almost all graphs have the property that there exist three cards in their deck that uniquely determine the graph. en.wikipedia.org /wiki/Reconstruction_conjecture   (436 words)

 TPS Standards for Reconstruction Reconstruction will be used to depict vanished or non-surviving portions of a property when documentary and physical evidence is available to permit accurate reconstruction with minimal conjecture, and such reconstruction is essential to the public understanding of the property. Reconstruction of a landscape, building, structure, or object in its historic location will be preceded by a thorough archeological investigation to identify and evaluate those features and artifacts which are essential to an accurate reconstruction. Reconstruction will be based on the accurate duplication of historic features and elements substantiated by documentary or physical evidence rather than on conjectural designs or the availability of different features from other historic properties. www.cr.nps.gov /hps/TPS/standards/reconstruction.htm   (273 words)

 Reconstruction conjecture Indeed, there are exactly two graphs on two vertices, the single edge and two isolated vertices, each of which leads to the same deck of cards, namely to two cards showing each a single vertex. However, for graphs with at least three vertices it has been conjectured in 1942 by P. Kelly and S. Ulam that it is always possible to reconstruct the graph from its deck of cards. www.ebroadcast.com.au /lookup/encyclopedia/re/Reconstruction_conjecture.html   (173 words)

 Reconstruction archaeology Reconstruction archaeology is a term sometimes used for the increasingly popular practice of attempting to shed light on the past by re-enacting history or reconstructing objects (such as weapons, buildings, etc). At Vindolanda on Hadrian's Wall, a reconstruction of part of the wall was carried out in limited time by local volunteers. Greek trireme s have been reconstructed by skilled sailors from plans and archaeological remains and have been successfully tried out at sea. www.nebulasearch.com /encyclopedia/article/Reconstruction_archaeology.html   (196 words)

 NationMaster - Encyclopedia: Soul theorem   (Site not responding. Last check: 2007-10-31) The conjecture was open for about 20 years, and was solved by Grigori Perelman with a surprisingly short argument. Like any living being, it must have a soul and a body, which two constitutive principles are the two levels of the mediation between the purity of the model and the lack of determination of the receptacle. This is particularly clear in the platonic reconstruction of the constitution of the soul of the cosmos. www.nationmaster.com /encyclopedia/Soul-theorem   (336 words)

 The Reconstruction Conjecture Since, the number of edges of G is reconstructible, the degree sequence of G is reconstructible. In 1967, Tutte proved that the dichromatic (rank) and Tutte (dichromate) polynomials are reconstructible. Thus, N(1,s) is reconstructible and, hence, all coefficients of Q(G;t,z) and Q(G;t,z) itself are reconstructible. www.math.fau.edu /locke/Recon.htm   (1083 words)

 Reconstruction conjecture Informally, the reconstruction conjecture in graph theory suggests that a graph is determined uniquely by it's sub-structures. Reconstruction Conjecture Let [G] and [H] be finite graphs with at least three vertices, and let there be a bijection [\sigma:V(G) \rightarrow V(H)] such that [G-v] is isomorphic to [H-\sigma(v)] for every [v \in V(G)]. In fact, it was shown that not only are almost all graphs reconstructible, but in fact that the entire deck is not necessary to reconstruct them. encycl.opentopia.com /term/Reconstruction_conjecture   (530 words)

 [No title] The Graph Reconstruction Conjecture was proposed by P.J. Kelly and S.M. Ulam in 1942. The existential reconstruction number of graph G, (rn(G), is defined as the minimum number of vertex-deleted subgraphs of G required to uniquely reconstruct G up to isomorphism. The deliverables of this project will be statistics on reconstruction numbers organized by numbers of vertices, a list of graphs with notable reconstruction numbers (at least all graphs with (rn(G)>3), a detailed write-up and analysis of the results, and the source code of the program used to calculate the results. www.cs.rit.edu /~bmm3056/PreProposal.doc   (589 words)

 Accident Reconstruction Software --> Info and Comparisons   (Site not responding. Last check: 2007-10-31) In the history of the United States, Reconstruction was the period after the American Civil War when the southern states of the breakaway Confederacy were reintegrated into the United States of America. Abraham Lincoln had endorsed a lenient plan for reconstruction, which neither aided the recently freed slaves, nor imposed a Northern agenda on the restoration of the Southern economy. However, for graphs with at least three vertices it has been conjectured in 1942 by P. Kelly and S. Ulam that it is always possible to reconstruct the graph from its deck of cards. www.crashdatabase.com /computers/2/accident-reconstruction-software.html   (1171 words)

 Week 2 Abstracts Adrian Bondy conjectured that if G has at least 2d vertices then every cycle in Z(G) can be written as the sum of an odd number of cycles, each having at least 2d-1 edges. Locke proved Conjecture 1 when G is not hamiltonian, exploiting relationships between longest cycles of G and their bridges. We conjecture that the construction is possible except for a small number of cases. dimacs.rutgers.edu /drei/1998/week2.html   (4127 words)

 Reconstruction The 11 Confederate states somehow had to be restored to their positions in the Union and provided with loyal governments, and the role of the emancipated slaves in Southern society had to be defined. Reconstruction: The Reconstruction Acts - The Reconstruction Acts On Mar. 2, 1867, Congress enacted the Reconstruction Act, which,... Reconstruction: Bibliography - Bibliography The literature on the Reconstruction is extensive and has shown sharp changes in... www.infoplease.com /ce6/history/A0841309.html   (261 words)

 TCS Daily - Conjecture vs. Science And yet, in producing the reconstructed global average temperature, Oerlemans gives the combined temperature history of these 15 glaciers the same weight as the combined history of the 154 glaciers from the Northern Hemisphere (which are a weighted combination of smaller regions within the Northern Hemisphere). Given these limitations, it's clear that producing a temperature reconstruction that is representative of the historical conditions across the Southern Hemisphere cannot be done from Oerlemans' existing dataset. Emphasis should be placed on the temperature reconstruction of the mid-to-high latitudes of the Northern Hemisphere where a greater number of locations are represented by the existing data. www.tcsdaily.com /article.aspx?id=033005G   (1229 words)

 Unsolved Problems Julio Subocz notes that this is also called Berge's conjecure or the Berge-Sauer conjecture and, in a conference in Lisboa (November 1995), Y. Hamidoune cited a proof (approximately 65 pages) by Taskinov of the conjecture. The conjecture is true for every tree on k vertices having a vertex adjacent to at least (k-2)/2 leaves. Goodey has verified this conjecture for plane graphs whose faces are all of degee four or six. www.math.fau.edu /locke/Unsolved.htm   (2911 words)

 Graph Reconstruction   (Site not responding. Last check: 2007-10-31) The reconstruction conjecture claims that every graph on at least three vertices is uniquely determined by its collection of vertex deleted subgraphs. Even though it is one of the foremost unsolved problems in Graph Theory, work on it has slowed down, may be due to the general feeling that existing techniques are not likely to lead to a complete solu­tion. Here we give the results that have appeared recently in some selected variations of the reconstruction problem like edge reconstruction, degree associated reconstruction, vertex switching reconstruction and reconstruc­tion numbers. www.akcejournal.org /contents/vol1no1/graph_reconstruction.htm   (99 words)

 Reconstructing patterns of reticulate evolution in plants -- Linder and Rieseberg 91 (10): 1700 -- American Journal of ...   (Site not responding. Last check: 2007-10-31) reconstruction, but they are not without their problems. Hein J. 1990 Reconstructing evolution of sequences subject to recombination using parsimony. Rieseberg L. Morefield 1995 Character expression, phylogenetic reconstruction, and the detection of reticulate evolution. www.amjbot.org /cgi/content/full/91/10/1700   (6555 words)

 RR-1660 : Towards the reconstruction of poset   (Site not responding. Last check: 2007-10-31) Abstract : The reconstruction conjecture for posets is the following : every finite poset P of more than three elements is uniquely determined - up to isomorphism - by its collection of (unlabelled) one-element-deleted subposets [ P - {x} : x V (P) ]. We show that the following parameters are reconstructible : the number of minimal (respectively, maximal) elements, the level structure, the ideal-size sequence of the maximal elements, the ideal size (respectively, filter-size) sequence of any fixed level of the HASSE-diagram and the number of edges of the HASSE-diagram. This is considered to be a first step towards a proof of the reconstruction conjecture for posets. www.inria.fr /rrrt/rr-1660.html   (256 words)

 Citations: Some open problems on permutation groups - Cameron (ResearchIndex)   (Site not responding. Last check: 2007-10-31) The reconstruction index ae(G; Omega Gamma of G is the least integer t such that j Deltaj t implies that Delta is G reconstructible. Applied to the edge reconstruction of graphs, for instance, it gives a simple proof of Muller s result [7] which shows that graphs on n vertices with more than n log 2 n edges are edge reconstructible. Conjecture 6 (Cameron, 1990) Let G be an almost simple primitive permutation group. citeseer.ist.psu.edu /context/121127/0   (677 words)

 U South AL Math-Stat Colloquia Schedule Abstract: The reconstruction conjecture is one of the most seductive open problems in mathematics today. The conjecture centers around reconstructing a graph based on knowledge of all its subgraphs. As another application, we conjecture necessary and sufficient conditions under which we may tile the sphere, hyperbolic or Euclidean plane by copies of a given triangle, and prove the conjecture on all but a measure-zero set in the space of all triangles. www.southalabama.edu /mathstat/colloquia/index.shtml   (2902 words)

 buch.de - bücher - versandkostenfrei - Eigenspaces of Graphs. Encyclopedia of Mathematics & Its Applications S. - ... The extension of spectral techniques proceeds at three levels: using eigenvectors associated with an arbitrary labelling of graph vertices, using geometrical invariants of eigenspaces such as graph angles and main angles, and introducing certain kinds of canonical eigenvectors by means of star partitions and star bases. One objective is to describe graphs by algebraic means as far as possible, and the book discusses the Ulam reconstruction conjecture and the graph isomorphism problem in this context. Further problems of graph reconstruction and identification are used to illustrate the importance of graph angles and star partitions in relation to graph structure. www.buch.de /buch/06683/599_eigenspaces_of_graphs__encyclopedia_of_mathematics__its_applications_s_.html   (299 words)

 Reconstruction conjecture - Encyclopedia, History, Geography and Biography Reconstruction conjecture - Encyclopedia, History, Geography and Biography This page was last modified 01:29, 21 August 2005. This encyclopedia, history, geography and biography article about Reconstruction conjecture contains research on www.arikah.com /encyclopedia/Reconstruction_conjecture   (234 words)

 Citebase - The Reconstruction of Graphs   (Site not responding. Last check: 2007-10-31) We proceed to show that reconstruction conjecture is true and show that the proof provides a systematic procedure to reconstruct uniquely the graph from its deck. The complexity of this algorithm for the desired reconstruction is obtained. A different approach for reconstruction of graphs is through a complete set of invariants. citebase.eprints.org /cgi-bin/citations?id=oai:arXiv.org:math/0503316   (149 words)

 Reconstructing Subsets of Reals (ResearchIndex)   (Site not responding. Last check: 2007-10-31) Abstract: We consider the problem of reconstructing a set of real numbers up to translation from the multiset of its subsets of fixed size, given up to translation. We therefore restrict our attention to subsets of # which are local ly finite; those which contain only finitely many translates of any given finite set of size at least 2. 1 Thek-orbit reconstruction and the orbit algebra (context) - Mnukhin - 1992 citeseer.ist.psu.edu /240733.html   (363 words)

 [No title]   (Site not responding. Last check: 2007-10-31) Abstract: In 1941, S. Ulam and P. Kelly made the conjecture that any two graphs with the same set of one-vertex deleted subgraphs must be isomorphic. The Reconstruction Conjecture is much like Goldbach's Conjecture in that it is so easy to state, yet so difficult to prove. Then, the conjecture states that, two graphs are isomorphic if and only if they have isomorphic decks. www.math.uconn.edu /Sigma/S05_8.htm   (134 words)

 Reconstruction: Early Congressional Legislation The newly created Joint Committee on Reconstruction reported (Apr. 28, 1866) that the ex-Confederate states were in a state of civil disorder, and hence, had not held valid elections. It also maintained that Reconstruction was a congressional, not an executive, function. Constitutional history and constitutional theory: reflections on Ackerman, Reconstruction, and the transformation of the American...... www.infoplease.com /ce6/history/a0860647.html   (298 words)

 DREI 1999 Research Program Abstracts - Week 2 However, it was published as an open problem in a paper by Alspach, Bermond, and Sotteau in 1987, and as a research problem by Alspach in 1989. In an attempt to prove this 33 years old conjecture we determine the automorphism groups of Walecki tournaments for all but those whose defining binary sequences are aperiodic. Part I: An Introduction A well-known problem in domination theory is the long standing conjecture of V.G. Vizing from 1963 that the domination number of the Cartesian product of two graphs is at least as large as the product of the domination numbers of the individual graphs. dimacs.rutgers.edu /drei/1999/abstractswk2.html   (3810 words)

 [No title] The interpretation of archaeological discoveries and written records of a certain time period is a reliable method to use to arrive at a reconstruction of historical events. In Florida?s Indians, author Jerald Milanich?s opinions might be highly credible, but in all fairness to the reader, he should clearly identify them to the reader as just that?opinions. Also troubling is how Milanich relates events as though someone had witnessed and recorded the occasion as they occurred 1,000 to 2,000 years ago and he had read that report. www.forewordmagazine.com /reviews/viewreviews.aspx?reviewID=39   (311 words)

 Edward J. Farrell - Mathematician of the African Diaspora [21] 98c:05118 Farrell, E. J.; Constantine, G. On the reconstruction of the subgraph polynomial and the reconstruction conjecture. [59] 87m:05129 Farrell, E. J.; Wahid, S. On the reconstruction of the matching polynomial and the reconstruction conjecture. [81] 85d:05181 Farrell, E. J.; Grell, J. On reconstructing the circuit polynomial of a graph. www.math.buffalo.edu /mad/PEEPS/farrell_edwardj.html   (1595 words)

 Paul Stockmeyer's Papers   (Site not responding. Last check: 2007-10-31) (with B. Manvel, A. Meyerowitz, A. Schwenk, and K. Smith), Reconstruction of Sequences, Discrete Mathematics 94 (1991) 209-219. Which reconstruction results are significant?, in The Theory and Applications of Graphs (G. Chartrand et al., eds), John Wiley and Sons, New York, 1981, 543-555. The falsity of the reconstruction conjecture for tournaments, Journal of Graph Theory 1 (1977), 19-25. www.cs.wm.edu /~pkstoc/papers.html   (344 words)

 SCU Mathematics Colloquium Series, Fall 2002 Finding a proof of the Reconstruction Conjecture is probably the most famous unsolved problem in graph theory today. Several variations of the reconstruction conjecture have evolved over the past 40 years and these will be presented. The speaker thought he had a constructive proof (how naive!) only to be shot out of the saddle by one of the fast guns (a referee) hired by the Journal of Graph Theory. math.scu.edu /colloquium/fall02.html   (554 words)

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