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| {{about|sets of vertices connected by edges|graphs of mathematical functions|Graph of a function|other uses|Graph (disambiguation)}}
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| [[Image:6n-graf.svg|thumb|250px|A [[graph drawing|drawing]] of a graph]]
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| In [[mathematics]] and [[computer science]], '''graph theory''' is the study of ''[[graph (mathematics)|graph]]s'', which are mathematical structures used to model pairwise relations between objects. A "graph" in this context is made up of "[[vertex (graph theory)|vertices]]" or "nodes" and lines called ''edges'' that connect them. A graph may be ''undirected'', meaning that there is no distinction between the two vertices associated with each edge, or its edges may be ''[[Directed graph|directed]]'' from one vertex to another; see [[graph (mathematics)]] for more detailed definitions and for other variations in the types of graph that are commonly considered. Graphs are one of the prime objects of study in [[discrete mathematics]].
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| Refer to the [[glossary of graph theory]] for basic definitions in graph theory.
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| ==Applications==
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| [[File:Wikipedia multilingual network graph July 2013.svg|thumb|The network graph formed by Wikipedia editors (edges) contributing to different Wikipedia language versions (nodes) during one month in summer 2013.<ref>{{Cite arXiv|eprint=1312.0976|last1=Hale|first1=Scott A.|title=Multilinguals and Wikipedia Editing|class=cs.CY|year=2013}}</ref>]]
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| Graphs can be used to model many types of relations and processes in physical, biological,<ref>{{cite journal |last=Mashaghi |first=A. |last2=''et al.'' |title=Investigation of a protein complex network |journal=European Physical Journal B |volume=41 |issue=1 |pages=113–121 |year=2004 |doi=10.1140/epjb/e2004-00301-0 }}</ref> social and information systems. Many practical problems can be represented by graphs.
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| In [[computer science]], graphs are used to represent networks of communication, data organization, computational devices, the flow of computation, etc. For instance, the link structure of a [[website]] can be represented by a directed graph, in which the vertices represent web pages and directed edges represent [[Hyperlink|links]] from one page to another. A similar approach can be taken to problems in travel, biology, computer chip design, and many other fields. The development of [[algorithm]]s to handle graphs is therefore of major interest in computer science. The [[Graph transformation|transformation of graph]]s is often formalized and represented by [[graph rewriting|graph rewrite system]]s. Complementary to [[graph transformation]] systems focusing on rule-based in-memory manipulation of graphs are [[graph database]]s geared towards [[Database transaction|transaction]]-safe, [[Persistence (computer science)|persistent]] storing and querying of [[Graph (data structure)|graph-structured data]].
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| Graph-theoretic methods, in various forms, have proven particularly useful in [[linguistics]], since natural language often lends itself well to discrete structure. Traditionally, [[syntax]] and compositional semantics follow tree-based structures, whose expressive power lies in the [[principle of compositionality]], modeled in a hierarchical graph. More contemporary approaches such as [[head-driven phrase structure grammar]] model the syntax of natural language using [[feature structure|typed feature structure]]s, which are [[directed acyclic graph]]s. Within [[lexical semantics]], especially as applied to computers, modeling word meaning is easier when a given word is understood in terms of related words; [[semantic network]]s are therefore important in [[computational linguistics]]. Still other methods in phonology (e.g. [[optimality theory]], which uses [[lattice graph]]s) and morphology (e.g. finite-state morphology, using [[finite-state transducer]]s) are common in the analysis of language as a graph. Indeed, the usefulness of this area of mathematics to linguistics has borne organizations such as [http://www.textgraphs.org TextGraphs], as well as various 'Net' projects, such as [[WordNet]], [[VerbNet]], and others.
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| Graph theory is also used to study molecules in [[chemistry]] and [[physics]]. In [[condensed matter physics]], the three dimensional structure of complicated simulated atomic structures can be studied quantitatively by gathering statistics on graph-theoretic properties related to the topology of the atoms. In chemistry a graph makes a natural model for a molecule, where vertices represent [[atom]]s and edges [[Chemical bond|bond]]s. This approach is especially used in computer processing of molecular structures, ranging from [[Molecule editor|chemical editor]]s to database searching. In statistical physics, graphs can represent local connections between interacting parts of a system, as well as the dynamics of a physical process on such
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| systems. Graphs are also used to represent the micro-scale channels of [[Porous medium |porous media]], in which the vertices represent the pores and the edges represent the smaller channels connecting the pores.
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| Graph theory is also widely used in [[sociology]] as a way, for example, to [[Six Degrees of Kevin Bacon|measure actors' prestige]] or to explore [[Rumor spread in social network|rumor spreading]], notably through the use of [[social network analysis]] software. Under the umbrella of social networks are many different types of graphs:<ref>{{cite book|last=Rosen|first=Kenneth H.|title=Discrete mathematics and its applications|publisher=McGraw-Hill|location=New York|isbn=978-0-07-338309-5|edition=7th}}</ref> Acquaintanceship and friendship graphs describe whether people know each other. Influence graphs model whether certain people can influence the behavior of others. Finally, collaboration graphs model whether two people work together in a particular way, such as acting in a movie together.
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| Likewise, graph theory is useful in [[biology]] and conservation efforts where a vertex can represent regions where certain species exist (or habitats) and the edges represent migration paths, or movement between the regions. This information is important when looking at breeding patterns or tracking the spread of disease, parasites or how changes to the movement can affect other species.
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| In mathematics, graphs are useful in geometry and certain parts of topology such as [[knot theory]]. [[Algebraic graph theory]] has close links with [[group theory]].
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| A graph structure can be extended by assigning a weight to each edge of the graph. Graphs with weights, or [[weighted graph]]s, are used to represent structures in which pairwise connections have some numerical values. For example if a graph represents a road network, the weights could represent the length of each road.
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| ==History==
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| [[Image:Konigsberg bridges.png|thumb|200px|The Königsberg Bridge problem]]
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| The paper written by [[Leonhard Euler]] on the ''[[Seven Bridges of Königsberg]]'' and published in 1736 is regarded as the first paper in the history of graph theory.<ref name="Biggs">{{Citation|author=Biggs, N.; Lloyd, E. and Wilson, R.|title=Graph Theory, 1736-1936|publisher=Oxford University Press|year=1986}}</ref> This paper, as well as the one written by [[Alexandre-Théophile Vandermonde|Vandermonde]] on the ''[[Knight's tour|knight problem]],'' carried on with the ''analysis situs'' initiated by [[Gottfried Leibniz|Leibniz]]. Euler's formula relating the number of edges, vertices, and faces of a convex polyhedron was studied and generalized by [[Augustin Louis Cauchy|Cauchy]]<ref name="Cauchy">{{Citation|author=Cauchy, A.L.|year=1813|title=Recherche sur les polyèdres - premier mémoire|journal=[[Journal de l'École Polytechnique]]|volume= 9 (Cahier 16)|pages=66–86|postscript=.}}</ref> and [[Simon Antoine Jean L'Huillier|L'Huillier]],<ref name="Lhuillier">{{Citation|author=L'Huillier, S.-A.-J.|title=Mémoire sur la polyèdrométrie|journal=Annales de Mathématiques|volume=3|year=1861|pages=169–189|postscript=.}}</ref> and is at the origin of [[topology]].
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| More than one century after Euler's paper on the bridges of [[Königsberg]] and while [[Johann Benedict Listing|Listing]] introduced topology, [[Arthur Cayley|Cayley]] was led by the study of particular analytical forms arising from [[differential calculus]] to study a particular class of graphs, the ''[[tree (graph theory)|tree]]s''.<ref>{{citation|first=A.|last=Cayley|authorlink=Arthur Cayley|year=1857|title=On the theory of the analytical forms called trees|journal=[[Philosophical Magazine]]|series=Series IV|volume=13|issue=85|pages=172–176|doi=10.1017/CBO9780511703690.046}}.</ref> This study had many implications in theoretical [[chemistry]]. The involved techniques mainly concerned the [[enumeration of graphs]] having particular properties. Enumerative graph theory then rose from the results of Cayley and the fundamental results published by [[George Pólya|Pólya]] between 1935 and 1937 and the generalization of these by [[Nicolaas Govert de Bruijn|De Bruijn]] in 1959. Cayley linked his results on trees with the contemporary studies of chemical composition.<ref name="Cayley1">{{Citation|author=Cayley, A.|year=1875|journal=Berichte der deutschen Chemischen Gesellschaft|title=Ueber die Analytischen Figuren, welche in der Mathematik Bäume genannt werden und ihre Anwendung auf die Theorie chemischer Verbindungen|volume=8|pages=1056–1059|doi=10.1002/cber.18750080252|postscript=.|issue=2}}</ref> The fusion of the ideas coming from mathematics with those coming from chemistry is at the origin of a part of the standard terminology of graph theory.
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| In particular, the term "graph" was introduced by [[James Joseph Sylvester|Sylvester]] in a paper published in 1878 in ''[[Nature (journal)|Nature]]'', where he draws an analogy between "quantic invariants" and "co-variants" of algebra and molecular diagrams:<ref name="Sylvester">
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| John Joseph Sylvester (1878), ''Chemistry and Algebra''. Nature, volume 17, page 284. {{doi|10.1038/017284a0}}. [http://www.archive.org/stream/nature15unkngoog#page/n312/mode/1up Online version]. Retrieved 2009-12-30.</ref>
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| :"[...] Every invariant and co-variant thus becomes expressible by a ''graph'' precisely identical with a [[Friedrich August Kekulé von Stradonitz|Kekuléan]] diagram or chemicograph. [...] I give a rule for the geometrical multiplication of graphs, ''i.e.'' for constructing a ''graph'' to the product of in- or co-variants whose separate graphs are given. [...]" (italics as in the original).
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| The first textbook on graph theory was written by [[Dénes Kőnig]], and published in 1936.<ref>{{citation|last1=Tutte|first1=W.T.|authorlink=W. T. Tutte|title=Graph Theory|publisher=Cambridge University Press|year=2001|isbn=978-0-521-79489-3|page=30|url=http://books.google.com/books?id=uTGhooU37h4C&pg=PA30}}.</ref> Another book by [[Frank Harary]], published in 1969, was "considered the world over to be the definitive textbook on the subject",<ref>{{citation|page=203|title=Fractal Music, Hypercards, and more...Mathematical Recreations from Scientific American|first=Martin|last=Gardner|authorlink=Martin Gardner|publisher=W. H. Freeman and Company|year=1992}}.</ref> and enabled mathematicians, chemists, electrical engineers and social scientists to talk to each other. Harary donated all of the royalties to fund the [[Pólya Prize (SIAM)|Pólya Prize]].<ref>{{citation|contribution=The George Polya Prize|url=http://www.siam.org/about/more/siam50.pdf|page=26|title=Looking Back, Looking Ahead: A SIAM History|author=[[Society for Industrial and Applied Mathematics]]|year=2002}}.</ref>
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| One of the most famous and stimulating problems in graph theory is the [[four color problem]]: "Is it true that any map drawn in the plane may have its regions colored with four colors, in such a way that any two regions having a common border have different colors?" This problem was first posed by [[Francis Guthrie]] in 1852 and its first written record is in a letter of [[Augustus De Morgan|De Morgan]] addressed to [[William Rowan Hamilton|Hamilton]] the same year. Many incorrect proofs have been proposed, including those by Cayley, [[Alfred Bray Kempe|Kempe]], and others. The study and the generalization of this problem by [[Peter Guthrie Tait|Tait]], [[Percy John Heawood|Heawood]], [[Frank P. Ramsey|Ramsey]] and [[Hugo Hadwiger|Hadwiger]] led to the study of the colorings of the graphs embedded on surfaces with arbitrary [[Genus (mathematics)|genus]]. Tait's reformulation generated a new class of problems, the ''factorization problems'', particularly studied by [[Julius Petersen|Petersen]] and [[Dénes Kőnig|Kőnig]]. The works of Ramsey on colorations and more specially the results obtained by [[Pál Turán|Turán]] in 1941 was at the origin of another branch of graph theory, ''[[extremal graph theory]]''.
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| The four color problem remained unsolved for more than a century. In 1969 [[Heinrich Heesch]] published a method for solving the problem using computers.<ref>Heinrich Heesch: Untersuchungen zum Vierfarbenproblem. Mannheim: Bibliographisches Institut 1969.</ref> A computer-aided proof produced in 1976 by [[Kenneth Appel]] and [[Wolfgang Haken]] makes fundamental use of the notion of "discharging" developed by Heesch.<ref name="AA1">{{Citation|author=Appel, K. and Haken, W.|title=Every planar map is four colorable. Part I. Discharging|journal=Illinois J. Math.|volume=21|year=1977|pages=429–490|postscript=.}}</ref><ref name="AA2">{{Citation|author=Appel, K. and Haken, W.|title=Every planar map is four colorable. Part II. Reducibility|journal=Illinois J. Math.|volume=21|year=1977|pages=491–567|postscript=.}}</ref> The proof involved checking the properties of 1,936 configurations by computer, and was not fully accepted at the time due to its complexity. A simpler proof considering only 633 configurations was given twenty years later by [[Neil Robertson (mathematician)|Robertson]], [[Paul Seymour (mathematician)|Seymour]], [[Daniel Sanders (mathematician)|Sanders]] and [[Robin Thomas (mathematician)|Thomas]].<ref name="RSST">{{Citation|author=Robertson, N.; Sanders, D.; Seymour, P. and Thomas, R.|title=The four color theorem|journal=Journal of Combinatorial Theory Series B|volume=70|year=1997|pages=2–44|doi=10.1006/jctb.1997.1750|postscript=.}}</ref>
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| The autonomous development of topology from 1860 and 1930 fertilized graph theory back through the works of [[Camille Jordan|Jordan]], [[Kazimierz Kuratowski|Kuratowski]] and [[Hassler Whitney|Whitney]]. Another important factor of common development of graph theory and [[topology]] came from the use of the techniques of modern algebra. The first example of such a use comes from the work of the physicist [[Gustav Kirchhoff]], who published in 1845 his [[Kirchhoff's circuit law]]s for calculating the [[voltage]] and [[Electric current|current]] in [[electric circuit]]s.
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| The introduction of probabilistic methods in graph theory, especially in the study of [[Paul Erdős|Erdős]] and [[Alfréd Rényi|Rényi]] of the asymptotic probability of graph connectivity, gave rise to yet another branch, known as ''[[Random graph|random graph theory]]'', which has been a fruitful source of graph-theoretic results.
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| ==Drawing graphs==
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| {{main|Graph drawing}}
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| Graphs are represented graphically by drawing a dot or circle for every vertex, and drawing an arc between two vertices if they are connected by an edge. If the graph is directed, the direction is indicated by drawing an arrow.
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| A graph drawing should not be confused with the graph itself (the abstract, non-visual structure) as there are several ways to structure the graph drawing. All that matters is which vertices are connected to which others by how many edges and not the exact layout. In practice it is often difficult to decide if two drawings represent the same graph. Depending on the problem domain some layouts may be better suited and easier to understand than others.
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| The pioneering work of [[W. T. Tutte]] was very influential in the subject of graph drawing. Among other achievements, he introduced the use of linear algebraic methods to obtain graph drawings.
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| Graph drawing also can be said to encompass problems that deal with the [[Crossing number (graph theory)|crossing number]] and its various generalizations. The crossing number of a graph is the minimum number of intersections between edges that a drawing of the graph in the plane must contain. For a planar graph, the crossing number is zero by definition.
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| Drawings on surfaces other than the plane are also studied.
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| ==Graph-theoretic data structures==
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| {{main|Graph (data structure)}}
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| There are different ways to store graphs in a computer system. The [[data structure]] used depends on both the graph structure and the [[algorithm]] used for manipulating the graph. Theoretically one can distinguish between list and matrix structures but in concrete applications the best structure is often a combination of both. List structures are often preferred for [[sparse graph]]s as they have smaller memory requirements. [[Matrix(mathematics)|Matrix]] structures on the other hand provide faster access for some applications but can consume huge amounts of memory.
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| List structures include the [[incidence list]], an array of pairs of vertices, and the [[adjacency list]], which separately lists the neighbors of each vertex: Much like the incidence list, each vertex has a list of which vertices it is adjacent to.
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| Matrix structures include the [[incidence matrix]], a matrix of 0's and 1's whose rows represent vertices and whose columns represent edges, and the [[adjacency matrix]], in which both the rows and columns are indexed by vertices. In both cases a 1 indicates two adjacent objects and a 0 indicates two non-adjacent objects. The [[Laplacian matrix]] is a modified form of the adjacency matrix that incorporates information about the [[degree (graph theory)|degrees]] of the vertices, and is useful in some calculations such as [[Kirchhoff's theorem]] on the number of [[spanning tree]]s of a graph.
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| The [[distance matrix]], like the adjacency matrix, has both its rows and columns indexed by vertices, but rather than containing a 0 or a 1 in each cell it contains the length of a [[shortest path]] between two vertices.
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| ==Problems in graph theory==
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| ===Enumeration===
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| There is a large literature on [[graphical enumeration]]: the problem of counting graphs meeting specified conditions. Some of this work is found in Harary and Palmer (1973).
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| ===Subgraphs, induced subgraphs, and minors===
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| A common problem, called the [[subgraph isomorphism problem]], is finding a fixed graph as a [[Glossary of graph theory#Subgraphs|subgraph]] in a given graph. One reason to be interested in such a question is that many [[graph properties]] are ''hereditary'' for subgraphs, which means that a graph has the property if and only if all subgraphs have it too.
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| Unfortunately, finding maximal subgraphs of a certain kind is often an [[NP-complete problem]].
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| * Finding the largest [[complete graph]] is called the [[clique problem]] (NP-complete).
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| A similar problem is finding [[induced subgraph]]s in a given graph. Again, some important graph properties are hereditary with respect to induced subgraphs, which means that a graph has a property if and only if all induced subgraphs also have it. Finding maximal induced subgraphs of a certain kind is also often NP-complete. For example,
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| * Finding the largest edgeless induced subgraph, or [[Independent set (graph theory)|independent set]], called the [[independent set problem]] (NP-complete).
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| Still another such problem, the ''minor containment problem'', is to find a fixed graph as a minor of a given graph. A [[Minor (graph theory)|minor]] or '''subcontraction''' of a graph is any graph obtained by taking a subgraph and contracting some (or no) edges. Many graph properties are hereditary for minors, which means that a graph has a property if and only if all minors have it too. A famous example:
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| * A graph is [[planar graph|planar]] if it contains as a minor neither the [[complete bipartite graph]] <math>K_{3,3}</math> (See the [[Three-cottage problem]]) nor the complete graph <math>K_{5}</math>.
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| Another class of problems has to do with the extent to which various species and generalizations of graphs are determined by their ''point-deleted subgraphs'', for example:
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| * The [[reconstruction conjecture]].
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| ===Graph coloring===
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| Many problems have to do with various ways of [[graph coloring|coloring graphs]], for example:
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| * The [[four-color theorem]]
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| * The [[strong perfect graph theorem]]
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| * The [[Erdős–Faber–Lovász conjecture]](unsolved)
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| * The [[total coloring|total coloring conjecture]], also called [[Mehdi Behzad|Behzad]]'s conjecture) (unsolved)
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| * The [[list edge-coloring|list coloring conjecture]] (unsolved)
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| * The [[Hadwiger conjecture (graph theory)]] (unsolved)
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| ===Subsumption and unification===
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| Constraint modeling theories concern families of directed graphs related by a [[partial order]]. In these applications, graphs are ordered by specificity, meaning that more constrained graphs—which are more specific and thus contain a greater amount of information—are subsumed by those that are more general. Operations between graphs include evaluating the direction of a subsumption relationship between two graphs, if any, and computing graph unification. The unification of two argument graphs is defined as the most general graph (or the computation thereof) that is consistent with (i.e. contains all of the information in) the inputs, if such a graph exists; efficient unification algorithms are known.
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| For constraint frameworks which are strictly [[Principle of Compositionality|compositional]], graph unification is the sufficient satisfiability and combination function. Well-known applications include [[Automatic theorem prover|automatic theorem proving]] and modeling the [[Parsing|elaboration of linguistic structure]].
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| ===Route problems===
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| * [[Hamiltonian path problem|Hamiltonian path and cycle problem]]s
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| * [[Minimum spanning tree]]
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| * [[Route inspection problem]] (also called the "Chinese Postman Problem")
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| * [[Seven Bridges of Königsberg]]
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| * [[Shortest path problem]]
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| * [[Steiner tree]]
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| * [[Three-cottage problem]]
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| * [[Traveling salesman problem]] (NP-hard)
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| ===Network flow===
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| There are numerous problems arising especially from applications that have to do with various notions of [[flow network|flows in networks]], for example:
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| * [[Max flow min cut theorem]]
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| ===Visibility graph problems===
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| * [[Museum guard problem]]
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| ===Covering problems===
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| [[Covering problem]]s in graphs are specific instances of subgraph-finding problems, and they tend to be closely related to the [[clique problem]] or the [[independent set problem]].
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| * [[Set cover problem]]
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| * [[Vertex cover problem]]
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| ===Decomposition problems===
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| Decomposition, defined as partitioning the edge set of a graph (with as many vertices as necessary accompanying the edges of each part of the partition), has a wide variety of question. Often, it is required to decompose a graph into subgraphs isomorphic to a fixed graph; for instance, decomposing a complete graph into Hamiltonian cycles. Other problems specify a family of graphs into which a given graph should be decomposed, for instance, a family of cycles, or decomposing a complete graph ''K''<sub>''n''</sub> into ''n'' − 1 specified trees having, respectively, 1, 2, 3, ..., ''n'' − 1 edges.
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| Some specific decomposition problems that have been studied include:
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| *[[Arboricity]], a decomposition into as few forests as possible
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| *[[Cycle double cover]], a decomposition into a collection of cycles covering each edge exactly twice
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| *[[Edge coloring]], a decomposition into as few [[matching (graph theory)|matching]]s as possible
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| *[[Graph factorization]], a decomposition of a [[regular graph]] into regular subgraphs of given degrees
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| ===Graph classes===
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| Many problems involve characterizing the members of various classes of graphs. Some examples of such questions are below:
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| * [[Graph enumeration|Enumerating]] the members of a class
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| * Characterizing a class in terms of [[forbidden graph characterization|forbidden substructure]]s
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| * Ascertaining relationships among classes (e.g., does one property of graphs imply another)
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| * Finding efficient [[algorithm]]s to [[decision problem|decide]] membership in a class
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| * Finding [[representation (mathematics)|representation]]s for members of a class.
| |
| | |
| ==See also==
| |
| {{refbegin|colwidth=20em}}
| |
| * [[Gallery of named graphs]]
| |
| * [[Glossary of graph theory]]
| |
| * [[List of graph theory topics]]
| |
| * [[List of publications in mathematics#Graph theory|Publications in graph theory]]
| |
| | |
| ===Related topics===
| |
| * [[Algebraic graph theory]]
| |
| * [[Citation graph]]
| |
| * [[Conceptual graph]]
| |
| * [[Data structure]]
| |
| * [[Disjoint-set data structure]]
| |
| * [[Entitative graph]]
| |
| * [[Existential graph]]
| |
| * [[Graph algebras]]
| |
| * [[Graph automorphism]]
| |
| * [[Graph coloring]]
| |
| * [[Graph database]]
| |
| * [[Graph (data structure)|Graph data structure]]
| |
| * [[Graph drawing]]
| |
| * [[Graph equation]]
| |
| * [[Graph rewriting]]
| |
| * [[Graph sandwich problem]]
| |
| * [[Graph property]]
| |
| * [[Intersection graph]]
| |
| * [[Logical graph]]
| |
| * [[Loop (graph theory)|Loop]]
| |
| * [[Network theory]]
| |
| * [[Null graph]]
| |
| * [[Pebble motion problems]]
| |
| * [[Percolation]]
| |
| * [[Perfect graph]]
| |
| * [[Quantum graph]]
| |
| * [[Random regular graph]]s
| |
| * [[Semantic networks]]
| |
| * [[Spectral graph theory]]
| |
| * [[Strongly regular graph]]s
| |
| * [[Symmetric graph]]s
| |
| * [[Transitive reduction]]
| |
| * [[Tree (data structure)|Tree data structure]]
| |
| | |
| ===Algorithms===
| |
| * [[Bellman–Ford algorithm]]
| |
| * [[Dijkstra's algorithm]]
| |
| * [[Ford–Fulkerson algorithm]]
| |
| * [[Kruskal's algorithm]]
| |
| * [[Nearest neighbour algorithm]]
| |
| * [[Prim's algorithm]]
| |
| * [[Depth-first search]]
| |
| * [[Breadth-first search]]
| |
| | |
| ===Subareas===
| |
| * [[Algebraic graph theory]]
| |
| * [[Geometric graph theory]]
| |
| * [[Extremal graph theory]]
| |
| * [[Random graph|Probabilistic graph theory]]
| |
| * [[Topological graph theory]]
| |
| | |
| ===Related areas of mathematics===
| |
| * [[Combinatorics]]
| |
| * [[Group theory]]
| |
| * [[Knot theory]]
| |
| * [[Ramsey theory]]
| |
| | |
| ===Generalizations===
| |
| * [[Hypergraph]]
| |
| * [[Abstract simplicial complex]]
| |
| | |
| ===Prominent graph theorists===
| |
| * [[Noga Alon|Alon, Noga]]
| |
| * [[Claude Berge|Berge, Claude]]
| |
| * [[Béla Bollobás|Bollobás, Béla]]
| |
| * [[John Adrian Bondy|Bondy, Adrian John]]
| |
| * [[Graham Brightwell|Brightwell, Graham]]
| |
| * [[Maria Chudnovsky|Chudnovsky, Maria]]
| |
| * [[Fan Chung|Chung, Fan]]
| |
| * [[Gabriel Andrew Dirac|Dirac, Gabriel Andrew]]
| |
| * [[Paul Erdős|Erdős, Paul]]
| |
| * [[Leonhard Euler|Euler, Leonhard]]
| |
| * [[Ralph Faudree|Faudree, Ralph]]
| |
| * [[Martin Charles Golumbic|Golumbic, Martin]]
| |
| * [[Ronald Graham|Graham, Ronald]]
| |
| * [[Frank Harary|Harary, Frank]]
| |
| * [[Percy John Heawood|Heawood, Percy John]]
| |
| * [[Anton Kotzig|Kotzig, Anton]]
| |
| * [[Dénes Kőnig|Kőnig, Dénes]]
| |
| * [[László Lovász|Lovász, László]]
| |
| * [[U. S. R. Murty|Murty, U. S. R.]]
| |
| * [[Jaroslav Nešetřil|Nešetřil, Jaroslav]]
| |
| * [[Alfréd Rényi|Rényi, Alfréd]]
| |
| * [[Gerhard Ringel|Ringel, Gerhard]]
| |
| * [[Neil Robertson (mathematician)|Robertson, Neil]]
| |
| * [[Paul Seymour (mathematician)|Seymour, Paul]]
| |
| * [[Endre Szemerédi|Szemerédi, Endre]]
| |
| * [[Robin Thomas (mathematician)|Thomas, Robin]]
| |
| * [[Carsten Thomassen|Thomassen, Carsten]]
| |
| * [[Pál Turán|Turán, Pál]]
| |
| * [[W. T. Tutte|Tutte, W. T.]]
| |
| * [[Hassler Whitney|Whitney, Hassler]]
| |
| {{refend}}
| |
| | |
| ==Notes==
| |
| {{reflist|30em}}
| |
| | |
| ==References==
| |
| *{{citation|authorlink=Claude Berge|last=Berge|first=Claude|title=Théorie des graphes et ses applications|series=Collection Universitaire de Mathématiques|volume=II|publisher=Dunod|location=Paris|year=1958}}. English edition, Wiley 1961; Methuen & Co, New York 1962; Russian, Moscow 1961; Spanish, Mexico 1962; Roumanian, Bucharest 1969; Chinese, Shanghai 1963; Second printing of the 1962 first English edition, Dover, New York 2001.
| |
| *{{citation|last1=Biggs|first1=N.|last2=Lloyd|first2=E.|last3=Wilson|first3=R.|title=Graph Theory, 1736–1936|publisher=Oxford University Press|year=1986}}.
| |
| *{{citation|last1=Bondy|first1=J.A.|last2=Murty|first2=U.S.R.|title=Graph Theory|publisher=Springer|year=2008|isbn=978-1-84628-969-9}}.
| |
| *{{citation|last1=Bondy|first1=Riordan, O.M|title=Mathematical results on scale-free random graphs in "Handbook of Graphs and Networks" (S. Bornholdt and H.G. Schuster (eds)), Wiley VCH, Weinheim, 1st ed.|year=2003}}.
| |
| *{{citation|last=Chartrand|first=Gary|authorlink=Gary Theodore Chartrand|title=Introductory Graph Theory|publisher=Dover|isbn=0-486-24775-9|year=1985}}.
| |
| *{{citation|first=Alan|last=Gibbons|authorlink=|title=Algorithmic Graph Theory|year=1985|publisher=[[Cambridge University Press]]}}.
| |
| *{{citation|first=Shlomo Havlin|last=Reuven Cohen|title=Complex Networks: Structure, Robustness and Function|year=2010|publisher=Cambridge University Press}}
| |
| *{{citation|first=Martin|last=Golumbic|authorlink=Martin Charles Golumbic|title=Algorithmic Graph Theory and Perfect Graphs|year=1980|publisher=[[Academic Press]]}}.
| |
| *{{citation|authorlink=Frank Harary|last=Harary|first=Frank|title=Graph Theory|publisher=Addison-Wesley|location=Reading, MA|year=1969}}.
| |
| *{{citation|author1-link=Frank Harary|last1=Harary|first1=Frank|last2=Palmer|first2=Edgar M.|title=Graphical Enumeration|year=1973|publisher=Academic Press|location=New York, NY}}.
| |
| *{{citation|last1=Mahadev|first1=N.V.R.|last2=Peled|first2=Uri N.|title=Threshold Graphs and Related Topics|publisher=[[North-Holland Publishing Company|North-Holland]] |year=1995}}.
| |
| *{{citation|last1=Mark Newman|title=Networks: An Introduction|publisher=Oxford University Press|year=2010}}.
| |
| | |
| ==External links==
| |
| | |
| ===Online textbooks===
| |
| *[http://arxiv.org/pdf/cond-mat/0602129 Phase Transitions in Combinatorial Optimization Problems, Section 3: Introduction to Graphs] (2006) by Hartmann and Weigt
| |
| *[http://www.cs.rhul.ac.uk/books/dbook/ Digraphs: Theory Algorithms and Applications] 2007 by Jorgen Bang-Jensen and Gregory Gutin
| |
| *[http://diestel-graph-theory.com/index.html Graph Theory, by Reinhard Diestel]
| |
| | |
| ===Other resources===
| |
| * {{springer|title=Graph theory|id=p/g045010}}
| |
| * [http://www.utm.edu/departments/math/graph/ Graph theory tutorial]
| |
| * [http://www.gfredericks.com/main/sandbox/graphs A searchable database of small connected graphs]
| |
| * [http://web.archive.org/web/20060206155001/http://www.nd.edu/~networks/gallery.htm Image gallery: graphs]
| |
| * [http://www.babelgraph.org/links.html Concise, annotated list of graph theory resources for researchers]
| |
| * [http://www.kde.org/applications/education/rocs/ rocs] — a graph theory IDE
| |
| * [http://www.orgnet.com/SocialLifeOfRouters.pdf The Social Life of Routers] — non-technical paper discussing graphs of people and computers
| |
| * [http://graphtheorysoftware.com/ Graph Theory Software] — tools to teach and learn graph theory
| |
| * {{Library resources about
| |
| |onlinebooks=yes
| |
| |lcheading=Graph theory
| |
| |label=graph theory}}
| |
| | |
| {{Mathematics-footer}}
| |
| {{Computer science}}
| |
| | |
| {{DEFAULTSORT:Graph Theory}}
| |
| [[Category:Graph theory| ]]
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| | |
| {{link FA|nl}}
| |
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