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| In [[theoretical computer science]], the '''subgraph isomorphism''' problem is a computational task in which two [[undirected graph|graphs]] ''G'' and ''H'' are given as input, and one must determine whether ''G'' contains a [[Glossary of graph theory#Subgraphs|subgraph]] that is [[graph isomorphism|isomorphic]] to ''H''.
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| Subgraph isomorphism is a generalization of both the [[clique problem|maximum clique problem]] and the problem of testing whether a graph contains a [[Hamiltonian cycle]], and is therefore [[NP-complete]].<ref>The original {{harvtxt|Cook|1971}} paper that proves the [[Cook–Levin theorem]] already showed subgraph isomorphism to be NP-complete, using a reduction from [[3-SAT]] involving cliques.</ref> However certain other cases of subgraph isomorphism may be solved in polynomial time.<ref name="e99"/>
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| Sometimes the name '''subgraph matching''' is also used for the same problem. This name puts emphasis on finding such a subgraph as opposed to the bare decision problem.
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| ==Decision problem and computational complexity==
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| To prove subgraph isomorphism is NP-complete, it must be formulated as a [[decision problem]]. The input to the decision problem is a pair of graphs ''G'' and ''H''. The answer to the problem is positive if ''H'' is isomorphic to a subgraph of ''G'', and negative otherwise.
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| Formal question:
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| Let <math>G=(V,E)</math>, <math>H=(V^\prime,E^\prime)</math> be graphs. Is there a subgraph <math>G_0=(V_0,E_0): V_0\subseteq V, E_0=E\cap(V_0\times V_0)</math> such that <math>G_0\cong H</math>? I.e., does there exist an <math>f\colon V_0\rightarrow V^\prime</math> such that <math>(v_1,v_2)\in E_0\Leftrightarrow (f(v_1),f(v_2))\in E^\prime</math>?
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| The proof of subgraph isomorphism being NP-complete is simple and based on reduction of the [[clique problem]], an NP-complete decision problem in which the input is a single graph ''G'' and a number ''k'', and the question is whether ''G'' contains a [[complete graph|complete subgraph]] with ''k'' vertices. To translate this to a subgraph isomorphism problem, simply let ''H'' be the complete graph ''K''<sub>''k''</sub>; then the answer to the subgraph isomorphism problem for ''G'' and ''H'' is equal to the answer to the clique problem for ''G'' and ''k''. Since the clique problem is NP-complete, this [[polynomial-time many-one reduction]] shows that subgraph isomorphism is also NP-complete.<ref>{{citation|title=Complexity Theory: Exploring the Limits of Efficient Algorithms|first=Ingo|last=Wegener|publisher=Springer|year=2005|isbn=9783540210450|page=81|url=http://books.google.com/books?id=1fo7_KoFUPsC&pg=PA81}}.</ref>
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| An alternative reduction from the [[Hamiltonian path problem|Hamiltonian cycle]] problem translates a graph ''G'' which is to be tested for Hamiltonicity into the pair of graphs ''G'' and ''H'', where ''H'' is a cycle having the same number of vertices as ''G''. Because the Hamiltonian cycle problem is NP-complete even for [[planar graphs]], this shows that subgraph isomorphism remains NP-complete even in the planar case.<ref>{{citation
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| | last1 = de la Higuera | first1 = Colin
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| | last2 = Janodet | first2 = Jean-Christophe
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| | last3 = Samuel | first3 = Émilie
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| | last4 = Damiand | first4 = Guillaume
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| | last5 = Solnon | first5 = Christine
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| | doi = 10.1016/j.tcs.2013.05.026
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| | journal = Theoretical Computer Science
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| | mr = 3083515
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| | pages = 76–99
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| | title = Polynomial algorithms for open plane graph and subgraph isomorphisms
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| | url = https://www.ibisc.univ-evry.fr/~janodet/pub/hjsds13.pdf
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| | volume = 498
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| | year = 2013
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| | quotation = It is known since the mid-70’s that the isomorphism problem is solvable in polynomial time for plane graphs. However, it has also been noted that the subisomorphism problem is still N P-complete, in particular because the Hamiltonian cycle problem is NP-complete for planar graphs.}}</ref>
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| Subgraph isomorphism is a generalization of the [[graph isomorphism problem]], which asks whether ''G'' is isomorphic to ''H'': the answer to the graph isomorphism problem is true if and only if ''G'' and ''H'' both have the same number of vertices and the subgraph isomorphism problem for ''G'' and ''H'' is true. However the complexity-theoretic status of graph isomorphism remains an open question.
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| In the context of the [[Aanderaa–Karp–Rosenberg conjecture]] on the [[query complexity]] of monotone graph properties, {{harvtxt|Gröger|1992}} showed that any subgraph isomorphism problem has query complexity Ω(''n''<sup>3/2</sup>); that is, solving the subgraph isomorphism requires an algorithm to check the presence or absence in the input of Ω(''n''<sup>3/2</sup>) different edges in the graph.<ref>Here Ω invokes [[Big O notation|Big Omega notation]].</ref>
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| ==Algorithms==
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| {{harvtxt|Ullmann|1976}} describes a recursive backtracking procedure for solving the subgraph isomorphism problem. Although its running time is, in general, exponential, it takes polynomial time for any fixed choice of ''H'' (with a polynomial that depends on the choice of ''H''). When ''G'' is a [[planar graph]] and ''H'' is fixed, the running time of subgraph isomorphism can be reduced to [[linear time]].<ref name="e99">{{harvtxt|Eppstein|1999}}</ref>
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| ==Applications==
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| As subgraph isomorphism has been applied in the area of [[cheminformatics]] to find similarities between chemical compounds from their structural formula; often in this area the term '''substructure search''' is used.<ref>{{harvtxt|Ullmann|1976}}</ref> Typically a [[Query language|query structure]] is defined as [[Smiles arbitrary target specification|SMARTS]], a [[SMILES]] extension.
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| The closely related problem of counting the number of isomorphic copies of a graph ''H'' in a larger graph ''G'' has been applied to pattern discovery in databases,<ref>{{harvtxt|Kuramochi|Karypis|2001}}.</ref> the [[bioinformatics]] of protein-protein interaction networks,<ref>{{harvtxt|Pržulj|Corneil|Jurisica|2006}}.</ref> and in [[exponential random graph]] methods for mathematically modeling [[social network]]s.<ref>{{harvtxt|Snijders|Pattison|Robins|Handcock|2006}}.</ref>
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| {{harvtxt|Ohlrich|Ebeling|Ginting|Sather|1993}} describe an application of subgraph isomorphism in the [[computer-aided design]] of [[electronic circuits]]. Subgraph matching is also a substep in [[graph rewriting]] (the most runtime-intensive), and thus offered by [[Graph rewriting#Implementations and applications|graph rewrite tools]].
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| ==See also==
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| *[[Induced subgraph isomorphism problem]]
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| *[[Maximum common subgraph isomorphism problem]]
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| *[[Maximum common edge subgraph problem]]
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| ==Notes==
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| {{Reflist}}
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| ==References==
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| *{{Citation|first=S. A.|last=Cook|authorlink=Stephen Cook|contribution=The complexity of theorem-proving procedures|year=1971|title=[[Symposium on Theory of Computing|Proc. 3rd ACM Symposium on Theory of Computing]]|pages=151–158|url=http://4mhz.de/cook.html|doi=10.1145/800157.805047}}.
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| *{{Citation|last=Eppstein|first=David|authorlink=David Eppstein|title=Subgraph isomorphism in planar graphs and related problems|journal=[[Journal of Graph Algorithms and Applications]]|volume=3|issue=3|pages=1–27|year=1999|url=http://www.cs.brown.edu/publications/jgaa/accepted/99/Eppstein99.3.3.pdf|arxiv=cs.DS/9911003}}.
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| *{{Citation|author1-link = Michael R. Garey|last1=Garey|first1=Michael R.|author2-link = David S. Johnson | last2=Johnson | first2 = David S. | year = 1979 | title = [[Computers and Intractability: A Guide to the Theory of NP-Completeness]] | publisher = W.H. Freeman | isbn = 0-7167-1045-5}}. A1.4: GT48, pg.202.
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| *{{Citation
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| | volume = 10
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| | issue = 3
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| | pages = 119–127
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| | last = Gröger
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| | first = Hans Dietmar
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| | title = On the randomized complexity of monotone graph properties
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| | journal = Acta Cybernetica
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| | year = 1992
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| | url = http://www.inf.u-szeged.hu/actacybernetica/edb/vol10n3/pdf/Groger_1992_ActaCybernetica.pdf
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| }}.
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| *{{Citation|first1=Michihiro|last1=Kuramochi|first2=George|last2=Karypis|contribution=Frequent subgraph discovery|title=1st IEEE International Conference on Data Mining|year=2001|page=313|doi=10.1109/ICDM.2001.989534|isbn=0-7695-1119-8}}.
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| *{{Citation|first1=Miles|last1=Ohlrich|first2=Carl|last2=Ebeling|first3=Eka|last3=Ginting|first4=Lisa|last4=Sather|contribution=SubGemini: identifying subcircuits using a fast subgraph isomorphism algorithm|title=Proceedings of the 30th international Design Automation Conference|year=1993|pages=31–37|doi=10.1145/157485.164556|isbn=0-89791-577-1}}.
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| *{{Citation|first1=N.|last1=Pržulj|first2=D. G.|last2=Corneil|author2-link=Derek Corneil|first3=I.|last3=Jurisica|title=Efficient estimation of graphlet frequency distributions in protein–protein interaction networks|journal=Bioinformatics|volume=22|issue=8|pages=974–980|year=2006|doi=10.1093/bioinformatics/btl030|pmid=16452112}}.
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| *{{Citation|first1=T. A. B.|last1=Snijders|first2=P. E.|last2=Pattison|first3=G.|last3=Robins|first4=M. S.|last4=Handcock|title=New specifications for exponential random graph models|journal=Sociological Methodology|volume=36|issue=1|pages=99–153|year=2006|doi=10.1111/j.1467-9531.2006.00176.x}}.
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| *{{Citation|first=Julian R.|last=Ullmann|year=1976|title=An algorithm for subgraph isomorphism|journal=[[Journal of the ACM]]|volume=23|issue=1|pages=31–42|doi=10.1145/321921.321925}}.
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| *{{Citation|first=Hasan|last=Jamil|contribution=Computing Subgraph Isomorphic Queries using Structural Unification and Minimum Graph Structures|year=2011|title=26th ACM Symposium on Applied Computing|pages=1058–1063}}.
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| {{DEFAULTSORT:Subgraph Isomorphism Problem}}
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| [[Category:NP-complete problems]]
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| [[Category:Graph algorithms]]
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| [[Category:Computational problems in graph theory]]
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