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directed graph
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{{Short description|Graph with oriented edges}}(File:Directed graph no background.svg|upright|thumb|A simple directed graph)In mathematics, and more specifically in graph theory, a directed graph (or digraph) is a graph that is made up of a set of vertices connected by directed edges, often called arcs.- the content below is remote from Wikipedia
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Definition
In formal terms, a directed graph is an ordered pair {{nowrap|1=G = (V, A)}} where{{harvtxt|Bang-Jensen|Gutin|2000}}. {{harvtxt|Bang-Jensen|Gutin|2018}}, Chapter 1.{{harvtxt|Diestel|2005}}, Section 1.10. {{harvtxt|Bondy|Murty|1976}}, Section 10.- V is a set whose elements are called vertices, nodes, or points;
- A is a set of ordered pairs of vertices, called arcs, directed edges (sometimes simply edges with the corresponding set named E instead of A), arrows, or directed lines.
Types of directed graphs
{{see also|Graph (discrete mathematics)#Types of graphs}}Subclasses
(File:Directed acyclic graph 2.svg|right|upright=0.65|thumb|A simple directed acyclic graph)(File:4-tournament.svg|thumb|right|upright=0.4|A tournament on 4 vertices)- Symmetric directed graphs are directed graphs where all edges appear twice, one in each direction (that is, for every arrow that belongs to the digraph, the corresponding inverse arrow also belongs to it). (Such an edge is sometimes called "bidirected" and such graphs are sometimes called "bidirected", but this conflicts with the meaning for bidirected graphs.)
- Simple directed graphs are directed graphs that have no loops (arrows that directly connect vertices to themselves) and no multiple arrows with same source and target nodes. As already introduced, in case of multiple arrows the entity is usually addressed as directed multigraph. Some authors describe digraphs with loops as loop-digraphs.
- Complete directed graphs are simple directed graphs where each pair of vertices is joined by a symmetric pair of directed arcs (it is equivalent to an undirected complete graph with the edges replaced by pairs of inverse arcs). It follows that a complete digraph is symmetric.
- Semicomplete multipartite digraphs are simple digraphs in which the vertex set is partitioned into sets such that for every pair of vertices x and y in different sets, there is an arc between x and y. There can be one arc between x and y or two arcs in opposite directions.{{harvtxt|Bang-Jensen|Gutin|2018}}, Chapter 7 by Yeo.
- Semicomplete digraphs are simple digraphs where there is an arc between each pair of vertices. Every semicomplete digraph is a semicomplete multipartite digraph in a trivial way, with each vertex constituting a set of the partition.{{harvtxt|Bang-Jensen|Gutin|2018}}, Chapter 2 by Bang-Jensen and Havet.
- Quasi-transitive digraphs are simple digraphs where for every triple x, y, z of distinct vertices with arcs from x to y and from y to z, there is an arc between x and z. There can be just one arc between x and z or two arcs in opposite directions. A semicomplete digraph is a quasi-transitive digraph. There are extensions of quasi-transitive digraphs called k-quasi-transitive digraphs.{{harvtxt|Bang-Jensen|Gutin|2018}}, Chapter 8 by Galeana-Sanchez and Hernandez-Cruz.
- Oriented graphs are directed graphs having no opposite pairs of directed edges (i.e. at most one of {{nowrap|(x, y)}} and {{nowrap|(y, x)}} may be arrows of the graph). It follows that a directed graph is an oriented graph if and only if it has no 2-cycle.{{harvtxt|Diestel|2005}}, Section 1.10. (This is not the only meaning of "oriented graph"; see Orientation (graph theory).)
- Tournaments are oriented graphs obtained by choosing a direction for each edge in undirected complete graphs. A tournament is a semicomplete digraph.
- A directed graph is acyclic if it has no directed cycles. The usual name for such a digraph is directed acyclic graph (DAG).{{harvtxt|Bang-Jensen|Gutin|2018}}, Chapter 3 by Gutin.
Digraphs with supplementary properties
{{Incomplete list|date=August 2016}}- Weighted directed graphs (also known as directed networks) are (simple) directed graphs with weights assigned to their arrows, similarly to weighted graphs (which are also known as undirected networks or weighted networks).
- Flow networks are weighted directed graphs where two nodes are distinguished, a source and a sink.
- Rooted directed graphs (also known as flow graphs) are digraphs in which a vertex has been distinguished as the root.
- Control-flow graphs are rooted digraphs used in computer science as a representation of the paths that might be traversed through a program during its execution.
- Signal-flow graphs are directed graphs in which nodes represent system variables and branches (edges, arcs, or arrows) represent functional connections between pairs of nodes.
- Flow graphs are digraphs associated with a set of linear algebraic or differential equations.
- State diagrams are directed multigraphs that represent finite state machines.
- Commutative diagrams are digraphs used in category theory, where the vertices represent (mathematical) objects and the arrows represent morphisms, with the property that all directed paths with the same start and endpoints lead to the same result by composition.
- In the theory of Lie groups, a 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 FinVctKF(Q) where F(Q) is the free category on Q consisting of paths in Q and FinVctK is the category of finite-dimensional vector spaces over a field K. Representations of a quiver label its vertices with vector spaces and its edges (and hence paths) compatibly with linear transformations between them, and transform via natural transformations.
Basic terminology
(File:Incidence matrix - directed graph.svg|thumb|Oriented graph with corresponding incidence matrix)An arc {{nowrap|(x, y)}} is considered to be directed from x to y; y is called the head and x is called the tail of the arc; y is said to be a direct successor of x and x is said to be a direct predecessor of y. If a path leads from x to y, then y is said to be a successor of x and reachable from x, and x is said to be a predecessor of y. The arc {{nowrap|(y, x)}} is called the reversed arc of {{nowrap|(x, y)}}.The adjacency matrix of a multidigraph with loops is the integer-valued matrix with rows and columns corresponding to the vertices, where a nondiagonal entry a'ij is the number of arcs from vertex i to vertex j, and the diagonal entry a'ii is the number of loops at vertex i. The adjacency matrix of a directed graph is a logical matrix, and isunique up to permutation of rows and columns.Another matrix representation for a directed graph is its incidence matrix.See direction for more definitions.Indegree and outdegree
(File:DirectedDegrees.svg|thumb|A directed graph with vertices labeled (indegree, outdegree))For a vertex, the number of head ends adjacent to a vertex is called the indegree of the vertex and the number of tail ends adjacent to a vertex is its outdegree (called branching factor in trees).Let {{nowrap|1=G = (V, E)}} and {{nowrap|v â V}}. The indegree of v is denoted degâ(v) and its outdegree is denoted deg+(v).A vertex with {{nowrap|1=degâ(v) = 0}} is called a source, as it is the origin of each of its outcoming arcs. Similarly, a vertex with {{nowrap|1=deg+(v) = 0}} is called a sink, since it is the end of each of its incoming arcs.The degree sum formula states that, for a directed graph,
sum_{v in V} deg^-(v) = sum_{v in V} deg^+(v) = |E|.
If for every vertex {{nowrap|v â V}}, {{nowrap|1=deg+(v) = degâ(v)}}, the graph is called a balanced directed graph.{{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| url=https://archive.org/details/combinatorialmat0000brua/page/51}}.Degree sequence
The degree sequence of a directed graph is the list of its indegree and outdegree pairs; for the above example we have degree sequence ((2, 0), (2, 2), (0, 2), (1, 1)). The degree sequence is a directed graph invariant so isomorphic directed graphs have the same degree sequence. However, the degree sequence does not, in general, uniquely identify a directed graph; in some cases, non-isomorphic digraphs have the same degree sequence.The directed graph realization problem is the problem of finding a directed graph with the degree sequence a given sequence of positive integer pairs. (Trailing pairs of zeros may be ignored since they are trivially realized by adding an appropriate number of isolated vertices to the directed graph.) A sequence which is the degree sequence of some directed graph, i.e. for which the directed graph realization problem has a solution, is called a directed graphic or directed graphical sequence. This problem can either be solved by the KleitmanâWang algorithm or by the FulkersonâChenâAnstee theorem.Directed graph connectivity
A directed graph is weakly connected (or just connected{{harvtxt|Bang-Jensen|Gutin|2000}} p. 19 in the 2007 edition; p. 20 in the 2nd edition (2009).) if the undirected underlying graph obtained by replacing all directed edges of the graph with undirected edges is a connected graph.A directed graph is strongly connected or strong if it contains a directed path from x to y (and from y to x) for every pair of vertices {{nowrap|(x, y)}}. The strong components are the maximal strongly connected subgraphs.A connected rooted graph (or flow graph) is one where there exists a directed path to every vertex from a distinguished root vertex.See also
{hide}cmn|- Binary relation
- Coates graph
- Directed Graph Markup Language
- DRAKON flowchart
- Flow chart
- Globular set
- Glossary of graph theory
- Graph Style Sheets
- Graph theory
- Graph (abstract data type)
- Network theory
- Orientation
- Preorder
- Topological sorting
- Transpose graph
- Vertical constraint graph
Notes
{{Reflist}}References
- {{Citation
first1=Jørgen first2=Gregory| title=Digraphs: Theory, Algorithms and Applications Springer Science+Business Media>Springer| year=2000| isbn=1-85233-268-9| url=http://www.cs.rhul.ac.uk/books/dbook/}}(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}}). - {hide}Citation
first1=Jørgen first2=Gregory| title= Classes of Directed Graphs Springer International Publishing AG>Springer| year=2018| isbn=978-3319718408{edih}. - {{citation |last1 = Bondy
|first1 = John Adrian}}.
|author-link1 = John Adrian Bondy
|last2 = Murty
|first2 = U. S. R.
|author-link2 = U. S. R. Murty
|title = Graph Theory with Applications
|year = 1976
|publisher = North-Holland
|isbn = 0-444-19451-7
|url =weblink
|url-access = registration- {{Citation
first=Reinhard| title=Graph Theory Springer Science+Business Media>Springer| year=2005| edition=3rd| isbn=3-540-26182-6| url=http://www.math.uni-hamburg.de/home/diestel/books/graph.theory/}} (the electronic 3rd edition is freely available on author's site). - {{citation|last1=Harary|first1=Frank|author-link1=Frank Harary|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}}.
- Number of directed graphs (or directed graphs) with n nodes from On-Line Encyclopedia of Integer Sequences
External links
{{Commons category|Directed graphs}}{{Authority control}}
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