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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.
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| 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.
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| 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''.
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| 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'''. | |
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| ==Example==
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| The two graphs shown below are isomorphic, despite their different looking [[graph drawing|drawings]].
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| {|class="wikitable" style="margin: 1em auto 1em auto"
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| ! Graph G
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| ! Graph H
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| ! An isomorphism<br />between G and H
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| |-
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| |style="padding-left:2em;padding-right:2em;"|[[Image:Graph isomorphism a.svg|100px]]
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| |style="padding-left:1em;padding-right:1em;"|[[Image:Graph isomorphism b.svg|210px]]
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| |align="center" style="background-color:white;"|''f''(''a'') = 1
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| ''f''(''b'') = 6
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| ''f''(''c'') = 8
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| ''f''(''d'') = 3
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| ''f''(''g'') = 5
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| ''f''(''h'') = 2
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| ''f''(''i'') = 4
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| ''f''(''j'') = 7
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| |}
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| ==Motivation==
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| 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.
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| 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
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| :<math>\sum_{v \in V(G)} v\cdot\text{deg }v</math>
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| may be different for two isomorphic graphs.
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| ==Recognition of graph isomorphism==
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| === Whitney theorem ===
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| {{main|Whitney graph isomorphism theorem}}
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| [[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.]]
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| 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>
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| === Algorithmic approach ===
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| {{main|Graph isomorphism problem}}
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| 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.
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| 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]]).
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| 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>
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| Its generalization, the [[subgraph isomorphism problem]], is known to be NP-complete.
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| 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.
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| ==See also==
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| * [[Graph homomorphism]]
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| * [[Graph automorphism problem]]
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| * [[Graph canonization]]
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| ==Notes==
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| {{reflist|29em}}
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| ==References==
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| * {{Garey-Johnson}}
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| {{DEFAULTSORT:Graph Isomorphism}}
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| [[Category:Graph theory]]
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| [[Category:Graph algorithms]]
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| [[Category:Morphisms]]
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