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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''
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:<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 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&nbsp;≠&nbsp;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]]

Latest revision as of 01:44, 26 July 2014

And how much of those feelings are caused by pressure from the society in general. Overall, most patients report long-term satisfaction with their implants even when they have to be re-operated for complications or aesthetic reasons. Rich anti oxident ftuits include mangoes , black grapes and currents. Sad to say that this behavior of hiding their embarrassing condition also affect their personal or romantic relationships. When it comes to breast enlargement you have a few choices: very expensive, potentially dangerous, ineffective or perhaps the only one out of all methods is the one mentioned above that's a lot cheaper and actually works for breast enlargement.

Saw palmetto berries can be purchased from any health food store for under $10. It is said to have obtained it's name from it's high reputation as a heal-all, being supposed even to cure the plague. Breast enlargement pills have natural composition which stimulate the growth of adipose tissues. Double lumen implants - silicon gel filled core with saline periphery. Breast Actives states their breasts improving cream to be lightly massaged thoroughly on both breast every morning, using your fingertips.

Sometimes you can even look a false doctor up on Google and see if they have been reported for any infractions or have had consumer complaints or lawsuits filed against them. Remember larger breasts are heavier and project (perky) less than tiny breasts. - Feel that your breasts lack in volume especially after pregnancy. It has even been tested in a manner breast enhancement with natural ingredients no need for silicone or anything like that. The breast implants are mainly made of silicon shell which is either filled with silicon gel or saline water.

Or you can simply increase your breast size to a level that compliments the rest of your body. They may look same in the appearance but they produce different results. Watercress contains gluconasturtin (only released when the leaves are chopped or chewed), which helps neutralize a carcinogen in tobacco. A woman massages her breasts, but it could happen to anyone. ) To gain maximum benefit and therapeutic effects, breast massage should be practiced daily, as part of a routine - perhaps when fresh out of the shower, since the skin is still somewhat moist and the room may still be steamy, for a smoother massage which may be more easily lubricated since there will still be moisture in the air.

Seatbelts ought to not be worn with Breast Implants. Pushup or padded bras are one option, but there are other seemingly possible ways, as well. In case you are interested in Breast Augmentation, below are guidelines to check out so that you understand what you need to do:. In warm infusions, it forms, in intermittent fevers, one of the most useful diaphoretics to which employment can be given. Sexual desires and personal interest enforce the women to adopt breast enlarging techniques.

wish to achieve, but doing your own research and getting an idea of. Enhancements that can have saline added or cleared after medical procedures to create any slight improvements that may be needed, give both the individual and the doctor some shake room for error. Herbal data bases indicate that Wild Yam contains some of the highest levels of natural estrogens, phyto-nutrients and diosgenin recommended by herbalists for breast enlargement. Different brands of Breast Enlargement Pills consist of different ingredients. Some earlier mammography machines did tend to compress the breast tissue and are therefore best avoided.

In case you liked this post along with you want to be given more information relating to how to make my breast bigger naturally (www.lucky-house.info) generously go to the webpage.

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