directed graph

This entry is a disambiguation page, plus some commentary that would not fit well on either of the pages digraph and quiver, particularly on related technical terms such as oriented graph, bidirected graph, and signed graph, with their standard meanings in modern combinatorics.


The term directed graph is used in both graph theory and category theory. The definition varies – even within one of the two theories.

In graph theory, directed graph (often abbreviated to the contraction digraph) nowadays usually means a digraph, while in category theory, directed graph generally means a quiver. The basic difference is: quivers may have multiple arrows in the same direction (often called “parallel”), and also loops, while digraphs may not have any of those.

Overview of usual technical terms

We see one family, in the same room but hardly together. The family of signed, gain, and biased graphs is a lot like that. The Bibliography and Glossary are supposed to lead the family members to talk to one another. May they do so! (Thomas Zaslavsky: front page)

The following table may help distinguish the various terminological conventions in current use:

technical termdistinctive featureshort verbal definitionsignature; and axioms, of the theory (in a narrow sense of “theory”)comments
directed graphn/a (context-dependent umbrella-term)n/an/an/a
quiverloops allowed, parallel arcs allowedpresheaf on the walking quiver =, two function symbols; more than one axioms (how many is up to interpretation)
digraphno loops, no parallel arcs, but 2-cycles allowedbinary irreflexive relation =, one relation symbol; one axiom expressing irreflexivity; if constructive logic is required, signature might be larger, requiring an apartness relation expressing irreflexivity
oriented graphlooks essentially like oriented one-dimensional simplicial complexundirected simple graph whose edges were replaced by one arrow eachdependent on the chosen formalization of graph; if graph is formalized as a relation, and the orientedness by antisymmetry?, then the signature is: =, one relation symbol; and the axioms then are: one axiom expressing irreflexivity, one axiom expressing antisymmetry; if constructive logic is required, the signature might be larger, requiring an apartness relation expressing irreflexivity
signed graphundirected graph with a sign on each edgeundirected simple graph together with a function {\{edges}\} \rightarrow {,+}\{-,+\},not subject to any axiomsdependent on the chosen formalization of undirected simple graph; if undirected simple graph is formalized in one of the usual ways as a symmetric irreflexive relation, and the signs in the usual way as two constant symbols, then the signature is: =, one relation symbol, two constant symbols; and the axioms then are: one axiom expressing irreflexivity, one axiom expressing symmetry; note that no axiom to express distinctness of the two constant symbols is necessary, as this, in the usual approach, is built into the signature; if constructive logic is required, the signature might be larger, requiring an apartness relation to express irreflexivitybeware that this should not in a natural way be regarded a “directed graph”, though they are often used in contexts together with directed graphs and thus belong on this page; in a sense, “signed graphs” are a non-example, an odd-one-out, in particular in that they seem tricky to categorically construct
bidirected graphquiver with a sign on each vertex-arrow-incidence relation?quiver together with function {\{vertices}\} ×\times {\{arrows}\} \rightarrow {,+}\{-,+\}, subject to axioms

Note that there are two points of view in model theory: one regards = as a logical symbol of a meta-language (e.g., first-order logic with equality), the other regards = as a binary relation symbol defined within the language under investigation. In the table, as a reminder that equality plays an important role in these theories, the latter view was taken.

Table of definitions of directed graphs in various references

This is to give a compressed look-up table, quickly usable, telling you what definition of directed graph is being used in the reference you happen to be working with.

referencekind of directed graph emphasized in the referenceformalization used; and exact reference to place in referenceword used in referencemiscellaneous further comments
BangJensenGutin2nddigraphthe one in digraph, except for a needless assumption of a vertex-set being non-empty: an empty vertex-set implies that the binary relation is the empty relation, but this is a digraphdigraphbeware that that book sometimes treats quivers, but calls them directed pseudographs and formalizes those using multisets of ordered pairs
BondyMurtyt1stquiverunusually hybrid functional-relational formalization via families of ordered pairsdigraph; beware that Bondy-Murty-digraphs are not digraphsaversion to the void: needless assumption of vertex-sets being non-empty; essentially, Bondy-Murty-digraphs\simeqSchrijver-digraphs
ChartrandLesniakZhang6thdigraphthe one in digraph, except for a needless assumption of a digraph being non-empty set: the empty set is a digraphdigraphbeware that what the authors call “The First Theorem of Digraph Theory” is a not a theorem of “Digraph Theory” in the strictest technical sense of “Digraph Theory”(==the first-order theory of irreflexive binary relations) since it uses the language of arithmetic
CsabaEtAl2016digraphthe one in digraphdigraph
DiestelGraphTheory4thquiverthe usual one: cf. Section 1.10, which is the “nuts-and-bolts-definition” in quiverdirected graph; wisely, the contraction digraph is used but once in main text, avoiding a clash with digraphbeware that directed path in the book means something different, type-theoretically at least, from path in a digraph: in the book, a directed path is itself a digraph, not a sequence of vertices
HararyGraphTheory1969digraphthe one in digraph; cf. Chapter 16, provided the word “collection” is read “set”, which, as is abundantly clear from the context, is what Harary means; Harary is unambiguous that he does not allow digraphs to have loops; so Harary-digraphs are digraphsdigraphHarary uses p. 199, second paragraph the unusual term “semiwalk” what is called “wealk walk” in digraph; note p. 198, last paragraph that Harary’s “path” is (modulo “paths” in digraph not being digraphs but only vertex-sequences) precisely the usual meaning of “path”, although he only mentions that there be no point-repetitions: this evidently implies that there are no arc-repetitions; in summary, Harary-paths are paths
FlajoletSedgewick1stbinary relationbinary relations, formalized in material set theory-style, cf. V.5.1both digraph and directed graph used interchangeablybeware that Flajolet-Sedgwewick-digraphs are just binary relations, not digraphs
SchrijverComOpt1stquiverunusually hybrid formalization via families of ordered pairs; Bondy-Murty formalise Schrijver’s family using a function called ψ D\psi_Ddigraph; beware that Schrijver-digraphs are not digraphsno aversion to the void here; moreover, note that SchrijverDigraphs\simeqBondy-Murty-digraphs
West2002quiverunusually hybrid functional-relational formalization via families of ordered pairs: cf. Definition 1.4.2; West is very explicit about allowing both loops and multiple directed edgesdigraphbeware that, as usual in the combinatorial literature, a “path” in Definition 1.4.6 is itself a quiver, i.e. does not carry any parametrization-information, i.e. is not a map like paths are

Note that “kind of directed graph” is a used in a loose, undefined sense, sufficiently clear from the context.

Bidirected graphs

Roughly speaking, bidirected graphs are quivers in which each non-loop arrow gets decorated with two elements drawn from a two-element set fixed in advanced, one thought to be attached to the arrow’s tail, the other to its head. Like for signed graphs, the elements of this two-element set are sometimes interpreted to be signs and equipped with arithmetical structure (essentially, the two-element set is taken to be a two-element group).

(The treatment of loops is slightly controversial in the combinatorical literature, prompting some ad-hoc-ery, and this is explicitly commented on at least once in the literature.) The category-theoretic literature seems not to have picked up on this so far, and could clarify the issue.

Bidirected graphs are important in the theory of flows on graphs.

Connections to category theory

Herein some evident aspects are pointed out explicitly.

Underlying quivers

A basic connection is that for any category, its underlying quiver is a directed graph.

Another aspect is that a nonempty digraph never is isomorphic to the quiver underlying any category: such a quiver always has a loop at each vertex, which is already enough to rule out any isomorphism.

An application of directed graphs to category theory

Briefly, and from a slightly model theoretical point of view, the charm of this application is that A. J. Power's pasting theorem is an example of a transfer of knowledge from a theory without function symbols to a theory with function symbols. (The theory of digraphs, strictly construed, is purely relational, while higher category theory makes essential use of several function symbols).

An example of a use of digraph theory in category theory is giving a rigorous justification of the notational practice of pasting diagrams. This was achieved in the late 1980s (cf. e.g. Johnson 1987, Johnson 1989, Power 1990). In this approach, parallel arcs or loops are of secondary importance, and sometimes expressly forbidden, so the work of Johnson and Power can be seen as a genuine application of graph theory to category theory.

All quivers deemed irreflexive by Lawvere

There is an article of William Lawvere which is relevant in one precise respect to a disambiguation page such as the present one: an unusual use of the adjective “irreflexive”. Therein, one reads (cf. (Lawvere 1989, page 272))

[..] the elementary “parallel process” E=E= \bullet\overset{\rightarrow}{\rightarrow}\bullet is a reflexive graph which happens to admit only one definition of composition making it into a category P\mathbf{P}. [..] Its actions S P opS^{\mathbf{P}^{\mathrm{op}}} are the irreflexive graphs (the negative is in a way appropriate even for those objects which happen to have loops at some point pp, for morphisms are allowed to interchange any two such loops).

With “the negative is in a way appropriate” Lawvere explains the use of the strong negation irr-, despite such an “irreflexive graph” being permitted to contain loops at some of its vertices, which is obviously different from what one would expect from the usual sense of irreflexive relation. This is a usage in need of explanation, all the more against the backdrop of contemporary teaching where the term “simple undirected graph” is often literally defined as symmetric irreflexive relation (on the vertex set).

Some authors have adopted this interesting variant sense of “irreflexive”, prompting an additional use of the a modifier “strict” to form a composite “strict irreflexive graph”. (cf. (Brown et al 2008), which uses “digraph” as a synonym for “strict irreflexive graph”, and this sense of “digraph” is precisely the one in digraph). In a sense, a digraph is a “strictly-Lawvere-irreflexive graph”.

From a graph-theoretical point of view, every digraph is a quiver, but not every quiver is a digraph (again, in the sense of Gutin and Bang-Jensen 2009).


  • A. Bondy, U.S.R. Murty: Graph Theory with Applications. Fifth Printing. North Holland 1982.
  • André Bouchet: Nowhere-Zero Integral Flows on a Bidirected Graph. Journal of Combinatorial Theory, Series B 34, 279–292(1983)
  • Gary Chartrand, Linda Lesniak, Ping Zhang: Graphs & Digraphs. Sixth edition. CRC Press. 2016
  • B. Csaba, D. Kühn, A. Lo, D. Osthus, A. Treglown: Proof of the 1-factorization and Hamilton decomposition conjectures. Memoirs of the American Mathematical Society, 244 (2016), monograph 1154, 170 pages
  • Ronald Brown, I. Morris, J. Shrimpton, C.D. Wensley: Graphs of morphisms of graphs, The Electronic Journal of Combinatorics 15 (2008), A1
  • Richard Bumby and Dana Latch, Categorical constructions in graph theory, lnternat. J. Math. and Math. Sci., Vol. 9 No. l (1986), 1-16. (pdf)

  • R. Diestel: Graph Theory. Fourth Edition. Springer (2010)

  • Gregory Gutin?, Jørgen Bang-Jensen?: Digraphs: Theory, Algorithms and Applications. Springer Monographs in Mathematics. Second Edition (2009)
  • Frank Harary?: Graph Theory. Addison Wesley. First Edition. (1969)
  • Frank Harary?: The notion of balance of a signed graph.

  • Michael Johnson?: Pasting Diagrams in nn-Categories with Applications to Coherence Theorems and Categories of Paths, Doctoral Thesis, University of Sydney, 1987

  • Michael Johnson?: The Combinatorics of nn-Categorical Pasting, Journal of Pure and Applied Algebra 62 (1989)
  • William Lawvere: Qualitative Distinctions Between Some Toposes of Generalized Graphs, Contemporary Mathematics 92 (1989)
  • Eugene L. Lawler?: Combinatorial Optimization: Networks and Matroids, Holt, Rinehart and Winston. (1976)
  • John Power: A 2-Categorical Pasting Theorem, Journal of Algebra 129 (1990)
  • Alexander Schrijver?: Combinatorial Optimization. Volume A. Springer. First Edition (2003)
  • Douglas B. West: Introduction to Graph Theory. Second Edition. Pearson Education. (2002)
  • Thomas Zaslavsky: A Mathematical Bibliography of Signed and Gain Graphs and Allied Areas, Electronic Journal of Combinatorics, Dynamic Surveys in Combinatorics, #DS8.

Revised on August 4, 2017 09:38:03 by Todd Trimble (