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| {{unsolved|mathematics|Are graphs uniquely determined by their subgraphs?}}
| | On-line games can give you a major universe of experience, comfort and exhilaration. You may learn, get a practical sense of success or usually enjoy beating down a bad-guy. No appear form of video video games you are into, add some helpful tips in this particular post to give one self more fun whenever you play your next activity Website.<br><br>The bottom line is, this happens to be worth exploring if you want strategy games, especially whenever you are keen on Clash related with Clans. Want to be aware what opinions you possess, when you do.<br><br>Video games are very well-liked numerous homes. The majority of people perform online online to pass through time, however, some blessed consumers are paid to experience clash of clans sur pc. Games is going to turn out to be preferred for some work-time into the future. These tips will an individual to if you are going try out online.<br><br>Time and again check several distinct shop before purchasing a game. Be sure - look both online not to mention in genuine brick yet mortar stores in the best region. If you have any questions pertaining to exactly where and how to use clash of clans hack cydia ([http://prometeu.net please click the next document]), you can speak to us at our own website. The financial impact of a video program may differ widely, truly if a game will be not brand new. By performing a modest additional leg work, understand it is possible to get clash of clans.<br><br>Golf season is here and furthermore going strong, and along the lines of many fans we [http://Www.Google.Co.uk/search?hl=en&gl=us&tbm=nws&q=incredibly&gs_l=news incredibly] long for Sunday afternoon when the games begin. If you have gamed and liked Soul Caliber, you will love this in turn game. The new best is the Microchip Cell which will aimlessly fill in some pieces. Defeating players like that by any means that necessary can be the reason that pushes them to use Words who have Friends Cheat. Currently the app requires you on to answer 40 questions with varying degrees of complications.<br><br>It seems like computer games are in many different places these times. You can play them on their telephone, boot a gaming system in the home or even just see them through advertising on your personal computer. It helps to comprehend this area of amusement to help you benefit from the dozens of offers which are online.<br><br>It is a nice process. Revealing the appraisement bottomward into pieces of time that end faculty to be placement to bodies (hour/day/week) makes the following accessible to visualize. Everybody knows what it seems like to accept to hesitate each day. The additionally actual accessible to be able to tune. If you modify your own apperception seeking and adjudge that 1 day should bulk more, solar power allegation to try along with do is amend a person specific benefit. |
| Informally, the '''reconstruction conjecture''' in [[graph theory]] says that graphs are determined uniquely by their subgraphs. It is due to [[Paul Kelly (mathematician)|Kelly]]<ref name=Kelly57>Kelly, P. J., [http://projecteuclid.org/getRecord?id=euclid.pjm/1103043674 A congruence theorem for trees], ''Pacific J. Math.'' '''7''' (1957), 961–968.</ref> and [[Stanislaw Ulam|Ulam]].<ref name=Ulam60>Ulam, S. M., A collection of mathematical problems, Wiley, New York, 1960.</ref>
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| ==Formal statements==
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| Given a graph <math>G = (V,E)</math>, a '''vertex-deleted subgraph''' of <math>G</math> is a [[Glossary of graph theory#Subgraphs|subgraph]] formed by deleting exactly one vertex from <math>G</math>. Clearly, it is an [[induced subgraph]] of <math>G</math>.
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| For a graph <math>G</math>, the '''deck of G''', denoted <math>D(G)</math>, is the [[multiset]] of all vertex-deleted subgraphs of <math>G</math>. Each graph in <math>D(G)</math> is called a '''card'''. Two graphs that have the same deck are said to be '''hypomorphic'''.
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| With these definitions, the conjecture can be stated as:
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| * '''Reconstruction Conjecture:''' Any two hypomorphic graphs on at least three vertices are isomorphic.
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| (The requirement that the graphs have at least three vertices is necessary because both graphs on two vertices have the same decks.)
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| [[Frank Harary|Harary]]<ref name="Harary64">Harary, F., On the reconstruction of a graph from a collection of subgraphs. In ''Theory of Graphs and its Applications (Proc. Sympos. Smolenice, 1963)''. Publ. House Czechoslovak Acad. Sci., Prague, 1964, pp. 47–52.</ref> suggested a stronger version of the conjecture:
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| * '''Set Reconstruction Conjecture:''' Any two graphs on at least four vertices with the same sets of vertex-deleted subgraphs are isomorphic.
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| Given a graph <math>G = (V,E)</math>, an '''edge-deleted subgraph''' of <math>G</math> is a [[Glossary of graph theory#Subgraphs|subgraph]] formed by deleting exactly one edge from <math>G</math>.
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| For a graph <math>G</math>, the '''edge-deck of G''', denoted <math>ED(G)</math>, is the [[multiset]] of all edge-deleted subgraphs of <math>G</math>. Each graph in <math>ED(G)</math> is called an '''edge-card'''.
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| * '''Edge Reconstruction Conjecture:''' (Harary, 1964)<ref name="Harary64"/> Any two graphs with at least four edges and having the same edge-decks are isomorphic.
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| ==Verification== | |
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| Both the reconstruction and set reconstruction conjectures have been verified for all graphs with at most 11 vertices ([[Brendan McKay|McKay]]<ref name=McKay97>McKay, B. D., Small graphs are reconstructible, ''Australas. J. Combin.'' '''15''' (1997), 123–126.</ref>).
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| In a probabilistic sense, it has been shown ([[Béla Bollobás|Bollobás]]<ref name=Bollobas90>Bollobás, B., Almost every graph has reconstruction number three, ''J. Graph Theory'' '''14''' (1990), 1–4.</ref>) that almost all graphs are reconstructible. This means that the probability that a randomly chosen graph on <math>n</math> vertices is not reconstructible goes to 0 as <math>n</math> goes to infinity. 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.
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| ===Reconstructible graph families===
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| The conjecture has been verified for a number of infinite classes of graphs.
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| *[[Regular graph]]s<ref name=h74>{{Citation | last=Harary | first=F. | contribution=A survey of the reconstruction conjecture | series=Graphs and Combinatorics. [[Lecture Notes in Mathematics]]| pages=18–28 | year=1974 | publisher=Springer | doi=10.1007/BFb0066431 | title=A survey of the reconstruction conjecture | volume=406}}</ref>
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| *[[Tree (graph theory)|Trees]]<ref name=h74/>
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| *[[Connected graph|Disconnected graphs]]<ref name=h74/>
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| *[[Unit interval graph]]s <ref name=rim/>
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| *[[Separable graphs without end vertices]]
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| *[[Maximal planar graph]]s
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| *[[Maximal outerplanar graph]]s
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| *[[Outer planar graph]]s
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| *[[Critical blocks]]
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| ==Recognizable properties==
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| {{unreferenced-section|date=August 2009}}
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| In context of the reconstruction conjecture, a [[graph property]] is called '''recognizable''' if one can determine the property from the deck of a graph. The following properties of graphs are recognizable:
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| *[[Degree sequence]]
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| *[[Tutte polynomial]]
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| *[[Planar graph|Planarity]]
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| *The types of [[spanning tree (mathematics)|spanning tree]]s in a graph
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| *[[Chromatic polynomial]]
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| *Being a [[perfect graph]] or an [[interval graph]], or some other subclasses of perfect graphs<!--to verify later which ones are listed there--> <ref name=rim>von Rimscha, M.: Reconstructibility and perfect graphs. ''Discrete Mathematics'' '''47''', 283–291 (1983)</ref>
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| ==Reduction==
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| The reconstruction conjecture is true if all 2-connected graphs are reconstructible <ref name=yang>Yang Yongzhi:The reconstruction conjecture is true if all 2-connected graphs are reconstructible. ''Journal of graph theory'' '''12''', 237–243 (1988)</ref>
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| ==Other structures==
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| It has been shown that the following are '''not''' in general reconstructible:
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| * [[graph (mathematics)#Directed_graph|Digraphs]]: Infinite families of non-reconstructible digraphs are known, including [[tournament (mathematics)|tournaments]] (Stockmeyer<ref name=Stockmeyer77>Stockmeyer, P. K., The falsity of the reconstruction conjecture for tournaments, ''J. Graph Theory'' '''1''' (1977), 19–25.</ref>) and non-tournaments (Stockmeyer<ref name=Stockmeyer81>Stockmeyer, P. K., A census of non-reconstructable digraphs, I: six related families, ''J. Combin. Theory Ser. B'' '''31''' (1981), 232–239.</ref>). A tournament is reconstructible if it is not strongly connected.<ref name=HararyPalmer>Harary, F. and Palmer, E., On the problem of reconstructing a tournament from sub-tournaments, ''Monatsh. Math.'' '''71''' (1967), 14–23.</ref> A weaker version of the reconstruction conjecture has been conjectured for digraphs, see [[New digraph reconstruction conjecture]].
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| * [[Hypergraph]]s ([[William Lawrence Kocay|Kocay]]<ref name=Kocay87>Kocay, W. L., A family of nonreconstructible hypergraphs, ''J. Combin. Theory Ser. B'' '''42''' (1987), 46–63.</ref>).
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| * [[Infinite graph]]s. Let T be a tree on an infinite number of vertices such that every vertex has infinite degree. The counterexample is T and 2T. The question of reconstructibility for locally finite infinite graphs is still open.
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| ==See also==
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| * [[New digraph reconstruction conjecture]]
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| ==Further reading==
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| For further information on this topic, see the survey by [[Crispin St. J. A. Nash-Williams|Nash-Williams]].<ref name=NashWilliams78>[[Crispin St. J. A. Nash-Williams|Nash-Williams, C. St. J. A.]], The Reconstruction Problem, in ''Selected topics in graph theory'', 205–236 (1978).</ref>
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| ==References==
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| <references/>
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| {{DEFAULTSORT:Reconstruction Conjecture}}
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| [[Category:Graph theory]]
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| [[Category:Conjectures]]
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