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| [[File:Directed.svg|125px|thumb|A directed graph.]]
| | I'm Santiago and I live with my husband and our 3 children in Bastia, in the south area. My hobbies are Conlanging, Kiteboarding and Gongoozling.<br><br>Feel free to surf to my weblog: [http://tinyurl.com/lxee56u http://tinyurl.com/lxee56u] |
| In [[mathematics]], and more specifically in [[graph theory]], a '''directed graph''' (or '''digraph''') is a [[graph (mathematics)|graph]], or set of nodes connected by edges, where the edges have a direction associated with them. In formal terms, a digraph is a pair <math>G=(V,A)</math> (sometimes <math>G=(V,E)</math>) of:<ref>{{harvtxt|Bang-Jensen|Gutin|2000}}. {{harvtxt|Diestel|2005}}, Section 1.10. {{harvtxt|Bondy|Murty|1976}}, Section 10.</ref>
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| * a [[Set (mathematics)|set]] ''V'', whose [[element (mathematics)|elements]] are called ''vertices'' or ''nodes'',
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| * a set ''A'' of [[ordered pair]]s of vertices, called ''arcs'', ''directed edges'', or ''arrows'' (and sometimes simply ''edges'' with the corresponding set named ''E'' instead of ''A'').
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| It differs from an ordinary or [[undirected graph]], in that the latter is defined in terms of [[unordered pair]]s of vertices, which are usually called [[edge (graph theory)|edges]].
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| Sometimes a digraph is called a ''simple digraph'' to distinguish it from a ''[[multigraph|directed multigraph]]'', in which the arcs constitute a [[multiset]], rather than a set, of ordered pairs of vertices. Also, in a simple digraph loops are disallowed. (A loop is an arc that pairs a vertex to itself.) On the other hand, some texts allow loops, multiple arcs, or both in a digraph.
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| ==Basic terminology==
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| An arc <math>e = (x, y)</math> is considered to be directed ''from'' <math>x</math> ''to'' <math>y</math>; <math>y</math> is called the ''head'' and <math>x</math> is called the ''tail'' of the arc; <math>y</math> is said to be a ''direct successor'' of <math>x</math>, and <math>x</math> is said to be a ''direct predecessor'' of <math>y</math>. If a [[path (graph theory)|path]] made up of one or more successive arcs leads from <math>x</math> to <math>y</math>, then <math>y</math> is said to be a ''successor'' of <math>x</math>, and <math>x</math> is said to be a ''predecessor'' of <math>y</math>. The arc <math>(y, x)</math> is called the arc <math>(x, y)</math> ''inverted''.
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| An [[Orientation (graph theory)|orientation]] of a [[simple graph|simple undirected graph]] is obtained by assigning a direction to each edge. Any directed graph constructed this way is called an "oriented graph". A directed graph is an oriented simple graph if and only if it has neither self-loops nor 2-cycles.<ref>{{harvtxt|Diestel|2005}}, Section 1.10.</ref>
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| A ''weighted digraph'' is a digraph with weights assigned to its arcs, similar to a [[weighted graph]].
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| In the context of graph theory a digraph with weighted edges is called a ''network''.
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| The [[adjacency matrix]] of a digraph (with loops and multiple arcs) is the integer-valued [[Matrix (mathematics)|matrix]] with rows and columns corresponding to the nodes, where a nondiagonal entry <math>a_{ij}</math> is the number of arcs from node ''i'' to node ''j'', and the diagonal entry <math>a_{ii}</math> is the number of loops at node ''i''. The adjacency matrix of a digraph is unique up to identical permutation of rows and columns.
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| Another matrix representation for a digraph is its [[incidence matrix]].
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| See [[Glossary of graph theory#Direction]] for more definitions.
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| ==Indegree and outdegree==
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| [[File:DirectedDegrees.svg|thumb|A digraph with vertices labeled (indegree, outdegree)]]
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| For a node, the number of head endpoints adjacent to a node is called the ''indegree'' of the node and the number of tail endpoints adjacent to a node is its ''outdegree'' (called "[[branching factor]]" in trees).
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| The indegree is denoted <math>\deg^-(v)</math> and the outdegree as <math>\deg^+(v).</math> A vertex with <math>\deg^-(v)=0</math> is called a ''source'', as it is the origin of each of its incident edges. Similarly, a vertex with <math>\deg^+(v)=0</math> is called a ''sink''.
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| The ''degree sum formula'' states that, for a directed graph,
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| :<math>\sum_{v \in V} \deg^+(v) = \sum_{v \in V} \deg^-(v) = |A|\, .</math>
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| If for every node {{math|''v'' ∈ ''V''}}, <math>\deg^+(v) = \deg^-(v)</math>, the graph is called a ''balanced digraph''.<ref>{{citation|page=460|title=Discrete Mathematics and Graph Theory|first1=Bhavanari|last1=Satyanarayana|first2=Kuncham Syam|last2=Prasad|publisher=PHI Learning Pvt. Ltd.|isbn=978-81-203-3842-5}}; {{citation|page=51|title=Combinatorial matrix classes|volume=108|series=Encyclopedia of mathematics and its applications|first=Richard A.|last=Brualdi|publisher=Cambridge University Press|year=2006|isbn=978-0-521-86565-4}}.</ref>
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| ==Digraph connectivity==
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| {{main|Connectivity (graph theory)}}
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| A digraph G is called ''weakly connected'' (or just ''connected''<ref>{{harvtxt|Bang-Jensen|Gutin|2000}} p. 19 in the 2007 edition; p. 20 in the 2nd edition (2009).</ref>) if the undirected ''underlying graph'' obtained by replacing all directed edges of G with undirected edges is a [[connected graph]]. A digraph is ''strongly connected'' or ''strong'' if it contains a directed path from ''u'' to ''v'' and a directed path from ''v'' to ''u'' for every pair of vertices ''u'',''v''. The ''strong components'' are the maximal strongly connected subgraphs.
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| ==Classes of digraphs==
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| A directed graph ''G'' is called '''symmetric''' if, for every arc that belongs to ''G'', the corresponding reversed arc also belongs to ''G''. A symmetric, loopless directed graph is equivalent to an undirected graph with the edges replaced by pairs of inverse arcs; thus the number of edges is equal to the number of arcs halved.
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| [[File:Directed acyclic graph 2.svg|right|100px|thumb|A simple acyclic directed graph]]
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| An '''acyclic''' directed graph, acyclic digraph, or [[directed acyclic graph]] is a directed graph with no [[directed cycle]]s. Special cases of acyclic directed graphs include [[multitree]]s (graphs in which no two directed paths from a single starting node meet back at the same ending node), [[oriented tree]]s or polytrees (digraphs formed by orienting the edges of undirected acyclic graphs), and [[rooted tree]]s (oriented trees in which all edges of the underlying undirected tree are directed away from the roots).
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| [[File:4-tournament.svg|thumb|right|100px|A tournament on 4 vertices]]
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| A '''[[tournament (mathematics)|tournament]]''' is an oriented graph obtained by choosing a direction for each edge in an undirected [[complete graph]].
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| In the theory of [[Lie group]]s, a '''[[quiver (mathematics)|quiver]]''' ''Q'' is a directed graph serving as the domain of, and thus characterizing the shape of, a ''representation'' ''V'' defined as a [[functor]], specifically an object of the [[functor category]] '''FinVct'''<sub>''K''</sub><sup>F(''Q'')</sup> where ''F''(''Q'') is the [[free category]] on ''Q'' consisting of paths in ''Q'' and '''FinVct'''<sub>''K''</sub> is the category of finite dimensional [[vector space]]s over a [[Field (mathematics)|field]] ''K''. Representations of a quiver label its vertices with vector spaces and its edges (and hence paths) compatibly with [[Linear map|linear transformations]] between them, and transform via [[natural transformation]]s.
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| ==See also==
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| * [[Flow chart]]
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| * [[Preorder]]
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| * [[Quiver (mathematics)|Quiver]]
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| * [[Transpose graph]]
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| * [[Vertical constraint graph]]
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| ==Notes==
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| {{reflist}}
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| ==References==
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| *{{Citation
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| | last1=Bang-Jensen | first1=Jørgen
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| | last2=Gutin | first2=Gregory
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| | title=Digraphs: Theory, Algorithms and Applications
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| | publisher=[[Springer Science+Business Media|Springer]]
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| | year=2000
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| | isbn=1-85233-268-9
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| | url=http://www.cs.rhul.ac.uk/books/dbook/
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| }}<br>(the corrected 1st edition of 2007 is now freely available on the authors' site; the 2nd edition appeared in 2009 ISBN 1-84800-997-6).
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| * {{citation
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| | last1=Bondy | first1=John Adrian | authorlink1=John Adrian Bondy
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| | last2=Murty | first2=U. S. R. | authorlink2=U. S. R. Murty
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| | title=Graph Theory with Applications
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| | year=1976
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| | publisher=North-Holland
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| | isbn=0-444-19451-7
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| | url=http://www.ecp6.jussieu.fr/pageperso/bondy/books/gtwa/gtwa.html
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| }}.
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| *{{Citation
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| | last=Diestel | first=Reinhard
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| | title=Graph Theory
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| | publisher=[[Springer Science+Business Media|Springer]]
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| | year=2005
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| | edition=3rd
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| | isbn=3-540-26182-6
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| | url=http://www.math.uni-hamburg.de/home/diestel/books/graph.theory/
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| }} (the electronic 3rd edition is freely available on author's site).
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| *{{citation|last1=Harary|first1=Frank|last2=Norman|first2=Robert Z.|last3=Cartwright|first3=Dorwin|title=Structural Models: An Introduction to the Theory of Directed Graphs|place=New York|publisher=Wiley|year=1965}}.
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| *{{citation | title=Number of directed graphs (or digraphs) with n nodes. | url=http://oeis.org/A000273}}
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| {{DEFAULTSORT:Directed Graph}}
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| [[Category:Directed graphs]]
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I'm Santiago and I live with my husband and our 3 children in Bastia, in the south area. My hobbies are Conlanging, Kiteboarding and Gongoozling.
Feel free to surf to my weblog: http://tinyurl.com/lxee56u