706:
31:
240:
918:
248:
1180:
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
1168:
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
259:
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
187:
The aforementioned definition does not allow a directed graph to have multiple arrows with the same source and target nodes, but some authors consider a broader definition that allows directed graphs to have such multiple arcs (namely, they allow the arc set to be a
1129:
330:
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.
614:, 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.
208:(that is, arcs that directly connect nodes with themselves), but some authors consider a narrower definition that does not allow directed graphs to have loops. Directed graphs without loops may be called
274:(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
1436:
578:
are directed graphs in which nodes represent system variables and branches (edges, arcs, or arrows) represent functional connections between pairs of nodes.
481:
are oriented trees in which all edges of the underlying undirected tree are directed either away from or towards the root (they are called, respectively,
1711:
380:
or two arcs in opposite directions. A semicomplete digraph is a quasi-transitive digraph. There are extensions of quasi-transitive digraphs called
1186:
1019:
1552:
1529:
568:
are rooted digraphs used in computer science as a representation of the paths that might be traversed through a program during its execution.
1622:
1169:
sequence does not, in general, uniquely identify a directed graph; in some cases, non-isomorphic digraphs have the same degree sequence.
290:
are simple directed graphs where each pair of vertices is joined by a symmetric pair of directed arcs (it is equivalent to an undirected
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1685:
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1610:
1599:
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55:
1182:
1216:
1198:
1173:
688:. Representations of a quiver label its vertices with vector spaces and its edges (and hence paths) compatibly with
1344:
1334:
461:
are DAGs in which there are no two distinct directed paths from the same starting vertex to the same ending vertex.
421:
1605:(the corrected 1st edition of 2007 is now freely available on the authors' site; the 2nd edition appeared in 2009
1369:
1364:
1731:
1319:
906:
582:
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may be arrows of the graph). It follows that a directed graph is an oriented graph if and only if it has no
693:
449:
300:
are simple digraphs in which the vertex set is partitioned into sets such that for every pair of vertices
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110:
105:
59:
1543:, Encyclopedia of Mathematics and Its Applications, vol. 108, Cambridge University Press, p.
705:
858:
623:
599:
1354:
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682:
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with the edges replaced by pairs of inverse arcs). It follows that a complete digraph is symmetric.
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205:
194:
63:
1671:
1324:
1224:
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is the problem of finding a directed graph with the degree sequence a given sequence of positive
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30:
1681:
1656:
1640:
1626:
1606:
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101:
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a directed graphic or directed graphical sequence. This problem can either be solved by the
935:
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642:
527:
261:
165:
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are DAGs formed by orienting the edges of trees (connected, acyclic undirected graphs).
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is a directed graph serving as the domain of, and thus characterizing the shape of, a
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are directed graphs having no opposite pairs of directed edges (i.e. at most one of
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552:
534:
129:
43:
1275:) is one where there exists a directed path to every vertex from a distinguished
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obtained by replacing all directed edges of the graph with undirected edges is a
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are digraphs associated with a set of linear algebraic or differential equations.
432:
are oriented graphs obtained by choosing a direction for each edge in undirected
204:
On the other hand, the aforementioned definition allows a directed graph to have
477:
39:
861:
with rows and columns corresponding to the vertices, where a nondiagonal entry
1309:
994:, as it is the origin of each of its outcoming arcs. Similarly, a vertex with
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689:
618:
457:
1124:{\displaystyle \sum _{v\in V}\deg ^{-}(v)=\sum _{v\in V}\deg ^{+}(v)=|E|.}
247:
1349:
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For a vertex, the number of head ends adjacent to a vertex is called the
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1585:
17:
1177:
638:
929:
of the vertex and the number of tail ends adjacent to a vertex is its
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560:) are digraphs in which a vertex has been distinguished as the root.
1700:
Structural Models: An
Introduction to the Theory of Directed Graphs
1691:(the electronic 3rd edition is freely available on author's site).
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704:
538:
are weighted directed graphs where two nodes are distinguished, a
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238:
29:
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921:
A directed graph with vertices labeled (indegree, outdegree)
1708:
Number of directed graphs (or directed graphs) with n nodes
1707:
1570:
p. 19 in the 2007 edition; p. 20 in the 2nd edition (2009).
898:
Another matrix representation for a directed graph is its
420:. (This is not the only meaning of "oriented graph"; see
895:, and is unique up to permutation of rows and columns.
1022:
78:
In formal terms, a directed graph is an ordered pair
1006:, since it is the end of each of its incoming arcs.
857:of a multidigraph with loops is the integer-valued
230:
Graph (discrete mathematics) § Types of graphs
1648:
1486:, Chapter 8 by Galeana-Sanchez and Hernandez-Cruz.
1467:
1465:
1123:
709:Oriented graph with corresponding incidence matrix
212:, while directed graphs with loops may be called
1698:; Norman, Robert Z.; Cartwright, Dorwin (1965),
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1520:Satyanarayana, Bhavanari; Prasad, Kuncham Syam,
1507:
1483:
1471:
1456:
1393:
1389:
891:. The adjacency matrix of a directed graph is a
526:(which are also known as undirected networks or
280:. Some authors describe digraphs with loops as
1264:are the maximal strongly connected subgraphs.
1587:Digraphs: Theory, Algorithms and Applications
217:
8:
1617:Bang-Jensen, Jørgen; Gutin, Gregory (2018),
1584:Bang-Jensen, Jørgen; Gutin, Gregory (2000),
168:, in that the latter is defined in terms of
1414:
1412:
1410:
336:are simple digraphs where for every triple
308:in different sets, there is an arc between
1401:
1712:On-Line Encyclopedia of Integer Sequences
1113:
1105:
1084:
1068:
1043:
1027:
1021:
436:. A tournament is a semicomplete digraph.
447:. The usual name for such a digraph is
1524:, PHI Learning Pvt. Ltd., p. 460,
1495:
1397:
1382:
522:assigned to their arrows, similarly to
192:). Sometimes these entities are called
677:is the category of finite-dimensional
506:Digraphs with supplementary properties
172:of vertices, which are usually called
1522:Discrete Mathematics and Graph Theory
1474:, Chapter 2 by Bang-Jensen and Havet.
372:. There can be just one arc between
7:
1232:if it contains a directed path from
518:) are (simple) directed graphs with
348:of distinct vertices with arcs from
1013:states that, for a directed graph,
975:) and its outdegree is denoted deg(
324:or two arcs in opposite directions.
1174:directed graph realization problem
870:is the number of arcs from vertex
298:Semicomplete multipartite digraphs
25:
887:is the number of loops at vertex
270:are directed graphs that have no
144:with the corresponding set named
692:between them, and transform via
641:, specifically an object of the
1439:from the original on 2023-02-04
316:. There can be one arc between
243:A simple directed acyclic graph
164:It differs from an ordinary or
1651:Graph Theory with Applications
1568:Bang-Jensen & Gutin (2000)
1508:Bang-Jensen & Gutin (2018)
1484:Bang-Jensen & Gutin (2018)
1472:Bang-Jensen & Gutin (2018)
1457:Bang-Jensen & Gutin (2018)
1394:Bang-Jensen & Gutin (2018)
1390:Bang-Jensen & Gutin (2000)
1300:Directed Graph Markup Language
1114:
1106:
1099:
1093:
1058:
1052:
1:
1248:) for every pair of vertices
1187:FulkersonâChenâAnstee theorem
725:is considered to be directed
1540:Combinatorial Matrix Classes
1537:Brualdi, Richard A. (2006),
58:that is made up of a set of
1199:Connectivity (graph theory)
1193:Directed graph connectivity
384:-quasi-transitive digraphs.
42:, and more specifically in
1758:
1670:Diestel, Reinhard (2005),
1619:Classes of Directed Graphs
1335:Graph (abstract data type)
1196:
422:Orientation (graph theory)
364:, there is an arc between
251:A tournament on 4 vertices
227:
1422:Introductory Graph Theory
1370:Zero-weight cycle problem
1365:Vertical constraint graph
878:, and the diagonal entry
334:Quasi-transitive digraphs
257:Symmetric directed graphs
27:Graph with oriented edges
1419:Chartrand, Gary (1977).
1402:Bondy & Murty (1976)
1320:Glossary of graph theory
1156:, the graph is called a
512:Weighted directed graphs
288:Complete directed graphs
224:Types of directed graphs
1425:. Courier Corporation.
1183:KleitmanâWang algorithm
1158:balanced directed graph
694:natural transformations
667:consisting of paths in
218:Types of directed graph
34:A simple directed graph
1125:
922:
913:Indegree and outdegree
909:for more definitions.
710:
690:linear transformations
553:Rooted directed graphs
450:directed acyclic graph
268:Simple directed graphs
252:
244:
210:simple directed graphs
62:connected by directed
35:
1742:Graph data structures
1510:, Chapter 3 by Gutin.
1126:
920:
708:
610:are digraphs used in
600:finite state machines
328:Semicomplete digraphs
260:with the meaning for
250:
242:
33:
1222:A directed graph is
1211:) if the undirected
1203:A directed graph is
1134:If for every vertex
1020:
607:Commutative diagrams
596:directed multigraphs
439:A directed graph is
195:directed multigraphs
132:of vertices, called
1459:, Chapter 7 by Yeo.
1355:Topological sorting
565:Control-flow graphs
277:directed multigraph
1641:Bondy, John Adrian
1325:Graph Style Sheets
1225:strongly connected
1121:
1079:
1038:
1011:degree sum formula
967:. The indegree of
923:
772:direct predecessor
711:
575:Signal-flow graphs
253:
245:
140:(sometimes simply
36:
1702:, New York: Wiley
1655:, North-Holland,
1554:978-0-521-86565-4
1531:978-81-203-3842-5
1262:strong components
1064:
1023:
701:Basic terminology
617:In the theory of
528:weighted networks
516:directed networks
262:bidirected graphs
16:(Redirected from
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1703:
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1676:(3rd ed.),
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1405:
1400:, Section 1.10.
1387:
1305:DRAKON flowchart
1259:
1213:underlying graph
1205:weakly connected
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936:branching factor
900:incidence matrix
855:adjacency matrix
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833:
814:is said to be a
794:is said to be a
770:is said to be a
760:direct successor
758:is said to be a
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643:functor category
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166:undirected graph
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1164:Degree sequence
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971:is denoted deg(
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703:
676:
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612:category theory
598:that represent
556:(also known as
524:weighted graphs
514:(also known as
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445:directed cycles
434:complete graphs
405:
393:
389:Oriented graphs
237:
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170:unordered pairs
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66:, often called
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1718:External links
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1441:. Retrieved
1421:
1396:, Chapter 1.
1385:
1330:Graph theory
1315:Globular set
1295:Coates graph
1276:
1272:
1269:rooted graph
1267:A connected
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816:predecessor
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583:Flow graphs
558:flow graphs
429:Tournaments
148:instead of
108:are called
40:mathematics
1726:Categories
1578:References
1443:2020-10-02
1310:Flow chart
1273:flow graph
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874:to vertex
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671:and FinVct
619:Lie groups
458:Multitrees
235:Subclasses
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74:Definition
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