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In [[graph theory]], an '''isomorphism of [[graph (mathematics)|graph]]s''' ''G'' and ''H'' is a [[bijection]] between the vertex sets of ''G'' and ''H'' | |||
:<math> f \colon V(G) \to V(H) \,\!</math> | |||
such that any two vertices ''u'' and ''v'' of ''G'' are [[Adjacent (graph theory)|adjacent]] in ''G'' [[if and only if]] ƒ(''u'') and ƒ(''v'') are adjacent in ''H''. This kind of bijection is commonly called "edge-preserving bijection", in accordance with the general notion of [[isomorphism]] being a structure-preserving bijection. | |||
In | In the above definition, graphs are understood to be [[directed graph|undirected]] [[labeled graph|non-labeled]] [[weighted graph|non-weighted]] graphs. However, the notion of isomorphism may be applied to all other variants of the notion of graph, by adding the requirements to preserve the corresponding additional elements of structure: arc directions, edge weights, etc., with the following exception. When spoken about [[graph labeling]] with ''unique labels'', commonly taken from the integer range 1,...,''n'', where ''n'' is the number of the vertices of the graph, two labeled graphs are said to be isomorphic if the corresponding underlying unlabeled graphs are isomorphic. | ||
If an [[isomorphism]] exists between two graphs, then the graphs are called '''isomorphic''' and we write <math>G\simeq H</math>. In the case when the bijection is a mapping of a graph onto itself, i.e., when ''G'' and ''H'' are one and the same graph, the bijection is called an [[graph automorphism|automorphism]] of ''G''. | |||
The graph isomorphism is an [[equivalence relation]] on graphs and as such it partitions the [[class (set theory)|class]] of all graphs into [[equivalence class]]es. A set of graphs isomorphic to each other is called an '''[[isomorphism class]] of graphs'''. | |||
==Example== | |||
The two graphs shown below are isomorphic, despite their different looking [[graph drawing|drawings]]. | |||
{|class="wikitable" style="margin: 1em auto 1em auto" | |||
! Graph G | |||
! Graph H | |||
! An isomorphism<br />between G and H | |||
|- | |||
|style="padding-left:2em;padding-right:2em;"|[[Image:Graph isomorphism a.svg|100px]] | |||
|style="padding-left:1em;padding-right:1em;"|[[Image:Graph isomorphism b.svg|210px]] | |||
|align="center" style="background-color:white;"|''f''(''a'') = 1 | |||
''f''(''b'') = 6 | |||
''f''(''c'') = 8 | |||
''f''(''d'') = 3 | |||
''f''(''g'') = 5 | |||
''f''(''h'') = 2 | |||
''f''(''i'') = 4 | |||
''f''(''j'') = 7 | |||
|} | |||
==Motivation== | |||
The formal notion of "isomorphism", e.g., of "graph isomorphism", captures the informal notion that some objects have "the same structure" if one ignores individual distinctions of "atomic" components of objects in question, see the [[#Example|example above]]. Whenever individuality of "atomic" components (vertices and edges, for graphs) is important for correct representation of whatever is modeled by graphs, the model is refined by imposing additional restrictions on the structure, and other mathematical objects are used: [[digraph (mathematics)|digraph]]s, [[labeled graph]]s, [[colored graph]]s, [[rooted tree]]s and so on. The isomorphism relation may also be defined for all these generalizations of graphs: the isomorphism bijection must preserve the elements of structure which define the object type in question: [[arc (graph theory)|arc]]s, labels, vertex/edge colors, the root of the rooted tree, etc. | |||
The notion of "graph isomorphism" allows us to distinguish [[graph properties]] inherent to the structures of graphs themselves from properties associated with graph representations: [[graph drawing]]s, [[graph (data structure)|data structures for graphs]], [[graph labeling]]s, etc. For example, if a graph has exactly one [[cycle (graph theory)|cycle]], then all graphs in its isomorphism class also have exactly one cycle. On the other hand, in the common case when the vertices of a graph are (''represented'' by) the [[integer]]s 1, 2,... ''N'', then the expression | |||
:<math>\sum_{v \in V(G)} v\cdot\text{deg }v</math> | |||
may be different for two isomorphic graphs. | |||
==Recognition of graph isomorphism== | |||
=== Whitney theorem === | |||
{{main|Whitney graph isomorphism theorem}} | |||
[[Image:Whitneys theorem exception.svg|right|thumb|200px|The exception of Whitney's theorem: these two graphs are not isomorphic but have isomorphic line graphs.]] | |||
The '''Whitney graph isomorphism theorem''',<ref>{{cite journal|last=Whitney|first=Hassler|title=Congruent Graphs and the Connectivity of Graphs|journal=American Journal of Mathematics|date=January 1932|volume=54|issue=1|pages=150–168|url=http://www.jstor.org/stable/2371086|accessdate=17 August 2012|publisher=The Johns Hopkins University Press}}</ref> shown by [[H. Whitney]], states that two connected graphs are isomorphic if and only if their [[line graph]]s are isomorphic, with a single exception: ''K''<sub>3</sub>, the [[complete graph]] on three vertices, and the [[complete bipartite graph]] ''K''<sub>1,3</sub>, which are not isomorphic but both have ''K''<sub>3</sub> as their line graph. The Whitney graph theorem can be extended to [[hypergraph]]s.<ref>Dirk L. Vertigan, Geoffrey P. Whittle: A 2-Isomorphism Theorem for Hypergraphs. J. Comb. Theory, Ser. B 71(2): 215–230. 1997.</ref> | |||
=== Algorithmic approach === | |||
{{main|Graph isomorphism problem}} | |||
While graph isomorphism may be studied in a classical mathematical way, as exemplified by the Whitney theorem, it is recognized that it is a problem to be tackled with an algorithmic approach. The computational problem of determining whether two finite graphs are isomorphic is called the graph isomorphism problem. | |||
Its practical applications include primarily [[cheminformatics]], [[mathematical chemistry]] (identification of chemical compounds), and [[electronic design automation]] (verification of equivalence of various representations of the design of an [[electronic circuit]]). | |||
The graph isomorphism problem is one of few standard problems in [[computational complexity theory]] belonging to [[NP (complexity)|NP]], but not known to belong to either of its well-known (and, if [[P versus NP problem|P ≠ NP]], disjoint) subsets: [[P (complexity)|P]] and [[NP-complete]]. It is one of only two, out of 12 total, problems listed in {{harvtxt|Garey|Johnson|1979}} whose complexity remains unresolved, the other being [[integer factorization]]. It is however known that if the problem is NP-complete then the [[polynomial hierarchy]] collapses to a finite level.<ref>{{cite journal | title=Graph isomorphism is in the low hierarchy | first=Uwe | last=Schöning | journal=[[Journal of Computer and System Sciences]] | volume=37 | year=1988 | pages=312–323 }} </ref> | |||
Its generalization, the [[subgraph isomorphism problem]], is known to be NP-complete. | |||
The main areas of research for the problem are design of fast algorithms and theoretical investigations of its [[computational complexity]], both for the general problem and for special classes of graphs. | |||
==See also== | |||
* [[Graph homomorphism]] | |||
* [[Graph automorphism problem]] | |||
* [[Graph canonization]] | |||
==Notes== | |||
{{reflist|29em}} | |||
==References== | |||
* {{Garey-Johnson}} | |||
{{DEFAULTSORT:Graph Isomorphism}} | |||
[[Category:Graph theory]] | |||
[[Category:Graph algorithms]] | |||
[[Category:Morphisms]] | |||
Revision as of 02:40, 13 December 2013
In graph theory, an isomorphism of graphs G and H is a bijection between the vertex sets of G and H
such that any two vertices u and v of G are adjacent in G if and only if ƒ(u) and ƒ(v) are adjacent in H. This kind of bijection is commonly called "edge-preserving bijection", in accordance with the general notion of isomorphism being a structure-preserving bijection.
In the above definition, graphs are understood to be undirected non-labeled non-weighted graphs. However, the notion of isomorphism may be applied to all other variants of the notion of graph, by adding the requirements to preserve the corresponding additional elements of structure: arc directions, edge weights, etc., with the following exception. When spoken about graph labeling with unique labels, commonly taken from the integer range 1,...,n, where n is the number of the vertices of the graph, two labeled graphs are said to be isomorphic if the corresponding underlying unlabeled graphs are isomorphic.
If an isomorphism exists between two graphs, then the graphs are called isomorphic and we write . In the case when the bijection is a mapping of a graph onto itself, i.e., when G and H are one and the same graph, the bijection is called an automorphism of G.
The graph isomorphism is an equivalence relation on graphs and as such it partitions the class of all graphs into equivalence classes. A set of graphs isomorphic to each other is called an isomorphism class of graphs.
Example
The two graphs shown below are isomorphic, despite their different looking drawings.
| Graph G | Graph H | An isomorphism between G and H |
|---|---|---|
|
|
f(a) = 1
f(b) = 6 f(c) = 8 f(d) = 3 f(g) = 5 f(h) = 2 f(i) = 4 f(j) = 7 |
Motivation
The formal notion of "isomorphism", e.g., of "graph isomorphism", captures the informal notion that some objects have "the same structure" if one ignores individual distinctions of "atomic" components of objects in question, see the example above. Whenever individuality of "atomic" components (vertices and edges, for graphs) is important for correct representation of whatever is modeled by graphs, the model is refined by imposing additional restrictions on the structure, and other mathematical objects are used: digraphs, labeled graphs, colored graphs, rooted trees and so on. The isomorphism relation may also be defined for all these generalizations of graphs: the isomorphism bijection must preserve the elements of structure which define the object type in question: arcs, labels, vertex/edge colors, the root of the rooted tree, etc.
The notion of "graph isomorphism" allows us to distinguish graph properties inherent to the structures of graphs themselves from properties associated with graph representations: graph drawings, data structures for graphs, graph labelings, etc. For example, if a graph has exactly one cycle, then all graphs in its isomorphism class also have exactly one cycle. On the other hand, in the common case when the vertices of a graph are (represented by) the integers 1, 2,... N, then the expression
may be different for two isomorphic graphs.
Recognition of graph isomorphism
Whitney theorem
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The Whitney graph isomorphism theorem,[1] shown by H. Whitney, states that two connected graphs are isomorphic if and only if their line graphs are isomorphic, with a single exception: K3, the complete graph on three vertices, and the complete bipartite graph K1,3, which are not isomorphic but both have K3 as their line graph. The Whitney graph theorem can be extended to hypergraphs.[2]
Algorithmic approach
Mining Engineer (Excluding Oil ) Truman from Alma, loves to spend time knotting, largest property developers in singapore developers in singapore and stamp collecting. Recently had a family visit to Urnes Stave Church.
While graph isomorphism may be studied in a classical mathematical way, as exemplified by the Whitney theorem, it is recognized that it is a problem to be tackled with an algorithmic approach. The computational problem of determining whether two finite graphs are isomorphic is called the graph isomorphism problem.
Its practical applications include primarily cheminformatics, mathematical chemistry (identification of chemical compounds), and electronic design automation (verification of equivalence of various representations of the design of an electronic circuit).
The graph isomorphism problem is one of few standard problems in computational complexity theory belonging to NP, but not known to belong to either of its well-known (and, if P ≠ NP, disjoint) subsets: P and NP-complete. It is one of only two, out of 12 total, problems listed in Template:Harvtxt whose complexity remains unresolved, the other being integer factorization. It is however known that if the problem is NP-complete then the polynomial hierarchy collapses to a finite level.[3]
Its generalization, the subgraph isomorphism problem, is known to be NP-complete.
The main areas of research for the problem are design of fast algorithms and theoretical investigations of its computational complexity, both for the general problem and for special classes of graphs.
See also
Notes
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References
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In its statement, the singapore property listing - website link, government claimed that the majority citizens buying their first residence won't be hurt by the new measures. Some concessions can even be prolonged to chose teams of consumers, similar to married couples with a minimum of one Singaporean partner who are purchasing their second property so long as they intend to promote their first residential property. Lower the LTV limit on housing loans granted by monetary establishments regulated by MAS from 70% to 60% for property purchasers who are individuals with a number of outstanding housing loans on the time of the brand new housing purchase. Singapore Property Measures - 30 August 2010 The most popular seek for the number of bedrooms in Singapore is 4, followed by 2 and three. Lush Acres EC @ Sengkang
Discover out more about real estate funding in the area, together with info on international funding incentives and property possession. Many Singaporeans have been investing in property across the causeway in recent years, attracted by comparatively low prices. However, those who need to exit their investments quickly are likely to face significant challenges when trying to sell their property – and could finally be stuck with a property they can't sell. Career improvement programmes, in-house valuation, auctions and administrative help, venture advertising and marketing, skilled talks and traisning are continuously planned for the sales associates to help them obtain better outcomes for his or her shoppers while at Knight Frank Singapore. No change Present Rules
Extending the tax exemption would help. The exemption, which may be as a lot as $2 million per family, covers individuals who negotiate a principal reduction on their existing mortgage, sell their house short (i.e., for lower than the excellent loans), or take part in a foreclosure course of. An extension of theexemption would seem like a common-sense means to assist stabilize the housing market, but the political turmoil around the fiscal-cliff negotiations means widespread sense could not win out. Home Minority Chief Nancy Pelosi (D-Calif.) believes that the mortgage relief provision will be on the table during the grand-cut price talks, in response to communications director Nadeam Elshami. Buying or promoting of blue mild bulbs is unlawful.
A vendor's stamp duty has been launched on industrial property for the primary time, at rates ranging from 5 per cent to 15 per cent. The Authorities might be trying to reassure the market that they aren't in opposition to foreigners and PRs investing in Singapore's property market. They imposed these measures because of extenuating components available in the market." The sale of new dual-key EC models will even be restricted to multi-generational households only. The models have two separate entrances, permitting grandparents, for example, to dwell separately. The vendor's stamp obligation takes effect right this moment and applies to industrial property and plots which might be offered inside three years of the date of buy. JLL named Best Performing Property Brand for second year running
The data offered is for normal info purposes only and isn't supposed to be personalised investment or monetary advice. Motley Fool Singapore contributor Stanley Lim would not personal shares in any corporations talked about. Singapore private home costs increased by 1.eight% within the fourth quarter of 2012, up from 0.6% within the earlier quarter. Resale prices of government-built HDB residences which are usually bought by Singaporeans, elevated by 2.5%, quarter on quarter, the quickest acquire in five quarters. And industrial property, prices are actually double the levels of three years ago. No withholding tax in the event you sell your property. All your local information regarding vital HDB policies, condominium launches, land growth, commercial property and more
There are various methods to go about discovering the precise property. Some local newspapers (together with the Straits Instances ) have categorised property sections and many local property brokers have websites. Now there are some specifics to consider when buying a 'new launch' rental. Intended use of the unit Every sale begins with 10 p.c low cost for finish of season sale; changes to 20 % discount storewide; follows by additional reduction of fiftyand ends with last discount of 70 % or extra. Typically there is even a warehouse sale or transferring out sale with huge mark-down of costs for stock clearance. Deborah Regulation from Expat Realtor shares her property market update, plus prime rental residences and houses at the moment available to lease Esparina EC @ Sengkang - ↑ Dirk L. Vertigan, Geoffrey P. Whittle: A 2-Isomorphism Theorem for Hypergraphs. J. Comb. Theory, Ser. B 71(2): 215–230. 1997.
- ↑ One of the biggest reasons investing in a Singapore new launch is an effective things is as a result of it is doable to be lent massive quantities of money at very low interest rates that you should utilize to purchase it. Then, if property values continue to go up, then you'll get a really high return on funding (ROI). Simply make sure you purchase one of the higher properties, reminiscent of the ones at Fernvale the Riverbank or any Singapore landed property Get Earnings by means of Renting
In its statement, the singapore property listing - website link, government claimed that the majority citizens buying their first residence won't be hurt by the new measures. Some concessions can even be prolonged to chose teams of consumers, similar to married couples with a minimum of one Singaporean partner who are purchasing their second property so long as they intend to promote their first residential property. Lower the LTV limit on housing loans granted by monetary establishments regulated by MAS from 70% to 60% for property purchasers who are individuals with a number of outstanding housing loans on the time of the brand new housing purchase. Singapore Property Measures - 30 August 2010 The most popular seek for the number of bedrooms in Singapore is 4, followed by 2 and three. Lush Acres EC @ Sengkang
Discover out more about real estate funding in the area, together with info on international funding incentives and property possession. Many Singaporeans have been investing in property across the causeway in recent years, attracted by comparatively low prices. However, those who need to exit their investments quickly are likely to face significant challenges when trying to sell their property – and could finally be stuck with a property they can't sell. Career improvement programmes, in-house valuation, auctions and administrative help, venture advertising and marketing, skilled talks and traisning are continuously planned for the sales associates to help them obtain better outcomes for his or her shoppers while at Knight Frank Singapore. No change Present Rules
Extending the tax exemption would help. The exemption, which may be as a lot as $2 million per family, covers individuals who negotiate a principal reduction on their existing mortgage, sell their house short (i.e., for lower than the excellent loans), or take part in a foreclosure course of. An extension of theexemption would seem like a common-sense means to assist stabilize the housing market, but the political turmoil around the fiscal-cliff negotiations means widespread sense could not win out. Home Minority Chief Nancy Pelosi (D-Calif.) believes that the mortgage relief provision will be on the table during the grand-cut price talks, in response to communications director Nadeam Elshami. Buying or promoting of blue mild bulbs is unlawful.
A vendor's stamp duty has been launched on industrial property for the primary time, at rates ranging from 5 per cent to 15 per cent. The Authorities might be trying to reassure the market that they aren't in opposition to foreigners and PRs investing in Singapore's property market. They imposed these measures because of extenuating components available in the market." The sale of new dual-key EC models will even be restricted to multi-generational households only. The models have two separate entrances, permitting grandparents, for example, to dwell separately. The vendor's stamp obligation takes effect right this moment and applies to industrial property and plots which might be offered inside three years of the date of buy. JLL named Best Performing Property Brand for second year running
The data offered is for normal info purposes only and isn't supposed to be personalised investment or monetary advice. Motley Fool Singapore contributor Stanley Lim would not personal shares in any corporations talked about. Singapore private home costs increased by 1.eight% within the fourth quarter of 2012, up from 0.6% within the earlier quarter. Resale prices of government-built HDB residences which are usually bought by Singaporeans, elevated by 2.5%, quarter on quarter, the quickest acquire in five quarters. And industrial property, prices are actually double the levels of three years ago. No withholding tax in the event you sell your property. All your local information regarding vital HDB policies, condominium launches, land growth, commercial property and more
There are various methods to go about discovering the precise property. Some local newspapers (together with the Straits Instances ) have categorised property sections and many local property brokers have websites. Now there are some specifics to consider when buying a 'new launch' rental. Intended use of the unit Every sale begins with 10 p.c low cost for finish of season sale; changes to 20 % discount storewide; follows by additional reduction of fiftyand ends with last discount of 70 % or extra. Typically there is even a warehouse sale or transferring out sale with huge mark-down of costs for stock clearance. Deborah Regulation from Expat Realtor shares her property market update, plus prime rental residences and houses at the moment available to lease Esparina EC @ Sengkang