Fubini's theorem: Difference between revisions

Revision as of 19:16, 27 January 2014

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Here is my homepage ... new launch ec Informally, the reconstruction conjecture in graph theory says that graphs are determined uniquely by their subgraphs. It is due to Kelly^[1] and Ulam.^[2]

Formal statements

Given a graph $G = (V, E)$ , a vertex-deleted subgraph of $G$ is a subgraph formed by deleting exactly one vertex from $G$ . Clearly, it is an induced subgraph of $G$ .

For a graph $G$ , the deck of G, denoted $D (G)$ , is the multiset of all vertex-deleted subgraphs of $G$ . Each graph in $D (G)$ is called a card. Two graphs that have the same deck are said to be hypomorphic.

With these definitions, the conjecture can be stated as:

Reconstruction Conjecture: Any two hypomorphic graphs on at least three vertices are isomorphic.

(The requirement that the graphs have at least three vertices is necessary because both graphs on two vertices have the same decks.)

Harary^[3] suggested a stronger version of the conjecture:

Set Reconstruction Conjecture: Any two graphs on at least four vertices with the same sets of vertex-deleted subgraphs are isomorphic.

Given a graph $G = (V, E)$ , an edge-deleted subgraph of $G$ is a subgraph formed by deleting exactly one edge from $G$ .

For a graph $G$ , the edge-deck of G, denoted $E D (G)$ , is the multiset of all edge-deleted subgraphs of $G$ . Each graph in $E D (G)$ is called an edge-card.

Edge Reconstruction Conjecture: (Harary, 1964)^[3] Any two graphs with at least four edges and having the same edge-decks are isomorphic.

Verification

Both the reconstruction and set reconstruction conjectures have been verified for all graphs with at most 11 vertices (McKay^[4]).

In a probabilistic sense, it has been shown (Bollobás^[5]) that almost all graphs are reconstructible. This means that the probability that a randomly chosen graph on $n$ vertices is not reconstructible goes to 0 as $n$ 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.

Reconstructible graph families

The conjecture has been verified for a number of infinite classes of graphs.

Recognizable properties

Template:Unreferenced-section

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:

Degree sequence
Tutte polynomial
Planarity
The types of spanning trees in a graph
Chromatic polynomial
Being a perfect graph or an interval graph, or some other subclasses of perfect graphs ^[7]

Reduction

The reconstruction conjecture is true if all 2-connected graphs are reconstructible ^[8]

Other structures

It has been shown that the following are not in general reconstructible:

Digraphs: Infinite families of non-reconstructible digraphs are known, including tournaments (Stockmeyer^[9]) and non-tournaments (Stockmeyer^[10]). A tournament is reconstructible if it is not strongly connected.^[11] A weaker version of the reconstruction conjecture has been conjectured for digraphs, see New digraph reconstruction conjecture.
Hypergraphs (Kocay^[12]).
Infinite graphs. 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.

References

↑ Kelly, P. J., A congruence theorem for trees, Pacific J. Math. 7 (1957), 961–968.
↑ Ulam, S. M., A collection of mathematical problems, Wiley, New York, 1960.
↑ ^3.0 ^3.1 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.
↑ McKay, B. D., Small graphs are reconstructible, Australas. J. Combin. 15 (1997), 123–126.
↑ Bollobás, B., Almost every graph has reconstruction number three, J. Graph Theory 14 (1990), 1–4.
↑ ^6.0 ^6.1 ^6.2 Many property agents need to declare for the PIC grant in Singapore. However, not all of them know find out how to do the correct process for getting this PIC scheme from the IRAS. There are a number of steps that you need to do before your software can be approved.

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To achieve the very best outcomes, you must be always updated on market situations, including past transaction information and reliable projections. You could review and examine comparable homes that are currently available in the market, especially these which have been sold or not bought up to now six months. You'll be able to see a pattern of such report by clicking here It's essential to defend yourself in opposition to unscrupulous patrons. They are often very skilled in using highly unethical and manipulative techniques to try and lure you into a lure. That you must also protect your self, your loved ones, and personal belongings as you'll be serving many strangers in your home. Sign a listing itemizing of all of the objects provided by the proprietor, together with their situation. HSR Prime Recruiter 2010
↑ ^7.0 ^7.1 von Rimscha, M.: Reconstructibility and perfect graphs. Discrete Mathematics 47, 283–291 (1983)
↑ Yang Yongzhi:The reconstruction conjecture is true if all 2-connected graphs are reconstructible. Journal of graph theory 12, 237–243 (1988)
↑ Stockmeyer, P. K., The falsity of the reconstruction conjecture for tournaments, J. Graph Theory 1 (1977), 19–25.
↑ Stockmeyer, P. K., A census of non-reconstructable digraphs, I: six related families, J. Combin. Theory Ser. B 31 (1981), 232–239.
↑ Harary, F. and Palmer, E., On the problem of reconstructing a tournament from sub-tournaments, Monatsh. Math. 71 (1967), 14–23.
↑ Kocay, W. L., A family of nonreconstructible hypergraphs, J. Combin. Theory Ser. B 42 (1987), 46–63.
↑ Nash-Williams, C. St. J. A., The Reconstruction Problem, in Selected topics in graph theory, 205–236 (1978).

[Kelly57-1] Kelly, P. J., A congruence theorem for trees, Pacific J. Math. 7 (1957), 961–968.

[Ulam60-2] Ulam, S. M., A collection of mathematical problems, Wiley, New York, 1960.

[Harary64-3] 3.0 ^3.1 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.

[McKay97-4] McKay, B. D., Small graphs are reconstructible, Australas. J. Combin. 15 (1997), 123–126.

[Bollobas90-5] Bollobás, B., Almost every graph has reconstruction number three, J. Graph Theory 14 (1990), 1–4.

[h74-6] 6.0 ^6.1 ^6.2 Many property agents need to declare for the PIC grant in Singapore. However, not all of them know find out how to do the correct process for getting this PIC scheme from the IRAS. There are a number of steps that you need to do before your software can be approved.

Naturally, you will have to pay a safety deposit and that is usually one month rent for annually of the settlement. That is the place your good religion deposit will likely be taken into account and will kind part or all of your security deposit. Anticipate to have a proportionate amount deducted out of your deposit if something is discovered to be damaged if you move out. It's best to you'll want to test the inventory drawn up by the owner, which can detail all objects in the property and their condition. If you happen to fail to notice any harm not already mentioned within the inventory before transferring in, you danger having to pay for it yourself.

In case you are in search of an actual estate or Singapore property agent on-line, you simply should belief your intuition. It's because you do not know which agent is nice and which agent will not be. Carry out research on several brokers by looking out the internet. As soon as if you end up positive that a selected agent is dependable and reliable, you can choose to utilize his partnerise in finding you a home in Singapore. Most of the time, a property agent is taken into account to be good if he or she locations the contact data on his website. This may mean that the agent does not mind you calling them and asking them any questions relating to new properties in singapore in Singapore. After chatting with them you too can see them in their office after taking an appointment.

Have handed an trade examination i.e Widespread Examination for House Brokers (CEHA) or Actual Property Agency (REA) examination, or equal; Exclusive brokers are extra keen to share listing information thus making certain the widest doable coverage inside the real estate community via Multiple Listings and Networking. Accepting a severe provide is simpler since your agent is totally conscious of all advertising activity related with your property. This reduces your having to check with a number of agents for some other offers. Price control is easily achieved. Paint work in good restore-discuss with your Property Marketing consultant if main works are still to be done. Softening in residential property prices proceed, led by 2.8 per cent decline within the index for Remainder of Central Region

Once you place down the one per cent choice price to carry down a non-public property, it's important to accept its situation as it is whenever you move in – faulty air-con, choked rest room and all. Get round this by asking your agent to incorporate a ultimate inspection clause within the possibility-to-buy letter. HDB flat patrons routinely take pleasure in this security net. "There's a ultimate inspection of the property two days before the completion of all HDB transactions. If the air-con is defective, you can request the seller to repair it," says Kelvin.

15.6.1 As the agent is an intermediary, generally, as soon as the principal and third party are introduced right into a contractual relationship, the agent drops out of the image, subject to any problems with remuneration or indemnification that he could have against the principal, and extra exceptionally, against the third occasion. Generally, agents are entitled to be indemnified for all liabilities reasonably incurred within the execution of the brokers´ authority.

To achieve the very best outcomes, you must be always updated on market situations, including past transaction information and reliable projections. You could review and examine comparable homes that are currently available in the market, especially these which have been sold or not bought up to now six months. You'll be able to see a pattern of such report by clicking here It's essential to defend yourself in opposition to unscrupulous patrons. They are often very skilled in using highly unethical and manipulative techniques to try and lure you into a lure. That you must also protect your self, your loved ones, and personal belongings as you'll be serving many strangers in your home. Sign a listing itemizing of all of the objects provided by the proprietor, together with their situation. HSR Prime Recruiter 2010

[rim-7] 7.0 ^7.1 von Rimscha, M.: Reconstructibility and perfect graphs. Discrete Mathematics 47, 283–291 (1983)

[yang-8] Yang Yongzhi:The reconstruction conjecture is true if all 2-connected graphs are reconstructible. Journal of graph theory 12, 237–243 (1988)

[Stockmeyer77-9] Stockmeyer, P. K., The falsity of the reconstruction conjecture for tournaments, J. Graph Theory 1 (1977), 19–25.

[Stockmeyer81-10] Stockmeyer, P. K., A census of non-reconstructable digraphs, I: six related families, J. Combin. Theory Ser. B 31 (1981), 232–239.

[HararyPalmer-11] Harary, F. and Palmer, E., On the problem of reconstructing a tournament from sub-tournaments, Monatsh. Math. 71 (1967), 14–23.

[Kocay87-12] Kocay, W. L., A family of nonreconstructible hypergraphs, J. Combin. Theory Ser. B 42 (1987), 46–63.

[NashWilliams78-13] Nash-Williams, C. St. J. A., The Reconstruction Problem, in Selected topics in graph theory, 205–236 (1978).

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+{{unsolved|mathematics|Are graphs uniquely determined by their subgraphs?}}
+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&ndash;968.</ref> and [[Stanislaw Ulam|Ulam]].<ref name=Ulam60>Ulam, S. M., A collection of mathematical problems, Wiley, New York, 1960.</ref>
+==Formal statements==
+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>.
+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'''.
+With these definitions, the conjecture can be stated as:
+* '''Reconstruction Conjecture:''' Any two hypomorphic graphs on at least three vertices are isomorphic.
+(The requirement that the graphs have at least three vertices is necessary because both graphs on two vertices have the same decks.)
+[[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:
+* '''Set Reconstruction Conjecture:''' Any two graphs on at least four vertices with the same sets of vertex-deleted subgraphs are isomorphic.
+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>.
+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'''.
+* '''Edge Reconstruction Conjecture:''' (Harary, 1964)<ref name="Harary64"/> Any two graphs with at least four edges and having the same edge-decks are isomorphic.
+==Verification==
+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&ndash;126.</ref>).
+In a probabilistic sense, it has been shown ([[Béla Bollobás|Bollobás]]<ref name=Bollobas90>Bollob&aacute;s, B., Almost every graph has reconstruction number three, ''J. Graph Theory'' '''14''' (1990), 1&ndash;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 &mdash; almost all graphs have the property that there exist three cards in their deck that uniquely determine the graph.
+===Reconstructible graph families===
+The conjecture has been verified for a number of infinite classes of graphs.
+*[[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>
+*[[Tree (graph theory)|Trees]]<ref name=h74/>
+*[[Connected graph|Disconnected graphs]]<ref name=h74/>
+*[[Unit interval graph]]s <ref name=rim/>
+*[[Separable graphs without end vertices]]
+*[[Maximal planar graph]]s
+*[[Maximal outerplanar graph]]s
+*[[Outer planar graph]]s
+*[[Critical blocks]]
+==Recognizable properties==
+{{unreferenced-section|date=August 2009}}
+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:
+*[[Degree sequence]]
+*[[Tutte polynomial]]
+*[[Planar graph|Planarity]]
+*The types of [[spanning tree (mathematics)|spanning tree]]s in a graph
+*[[Chromatic polynomial]]
+*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>
+==Reduction==
+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>
+==Other structures==
+It has been shown that the following are '''not''' in general reconstructible:
+* [[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&ndash;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&ndash;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&ndash;23.</ref> A weaker version of the reconstruction conjecture has been conjectured for digraphs, see [[New digraph reconstruction conjecture]].
+* [[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&ndash;63.</ref>).
+* [[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.
+==See also==
+* [[New digraph reconstruction conjecture]]
+==Further reading==
+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&ndash;236 (1978).</ref>
+==References==
+<references/>
+{{DEFAULTSORT:Reconstruction Conjecture}}
+[[Category:Graph theory]]
+[[Category:Conjectures]]

Fubini's theorem: Difference between revisions

Revision as of 19:16, 27 January 2014

Contents

Formal statements

Verification

Reconstructible graph families

Recognizable properties

Reduction

Other structures

See also

Further reading

References

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Fubini's theorem: Difference between revisions

Revision as of 19:16, 27 January 2014

Formal statements

Verification

Reconstructible graph families

Recognizable properties

Reduction

Other structures

See also

Further reading

References

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