Directed graph

A directed graph.

Directed graph with corresponding incidence matrix.

In mathematics, and more specifically in graph theory, a directed graph (or digraph) is a graph, or set of vertices connected by edges, where the edges have a direction associated with them. In formal terms, a directed graph is an ordered pair G = (V, A) (sometimes G = (V, E)) where^[1]

V is a set whose elements are called vertices, nodes, or points;
A is a set of ordered pairs of vertices, called arrows, directed edges (sometimes simply edges with the corresponding set named E instead of A), directed arcs, or directed lines.

It differs from an ordinary or undirected graph, in that the latter is defined in terms of unordered pairs of vertices, which are usually called edges, arcs, or lines.

A directed graph is called a simple digraph if it has no multiple arrows (two or more edges that connect the same two vertices in the same direction) and no loops (edges that connect vertices to themselves). A directed graph is called a directed multigraph or multidigraph if it may have multiple arrows (and sometimes loops). In the latter case the arrow set forms a multiset, rather than a set, of ordered pairs of vertices.

Basic terminology

An arrow (x, y) is considered to be directed from x to y; y is called the head and x is called the tail of the arrow; 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 arrow (y, x) is called the inverted arrow of (x, y).

An orientation of an undirected graph is an assignment of a direction to each edge, turning the initial graph into a directed graph. Any directed graph constructed this way is called an oriented graph. Among directed graphs, the oriented graphs are the ones that have no 2-cycles (that is at most one of (x, y) and (y, x) may be arrows of the graph).^[2]

A weighted directed graph is a directed graph with weights assigned to its arrows, similarly to a weighted graph. In the context of graph theory, a weighted directed graph is often called a network.

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 arrows 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 unique up to identical permutation of rows and columns.

Another matrix representation for a directed graph is its incidence matrix.

See direction for more definitions.

Indegree and outdegree

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 G = (V, E) and v∈V. The indegree of v is denoted deg⁻(v) and its outdegree is denoted deg⁺(v). A vertex with deg⁻(v) = 0 is called a source, as it is the origin of each of its incident arrows.

Similarly, a vertex with deg⁺(v) = 0 is called a sink.

If a vertex is neither a source nor a sink, it is called an internal.

The degree sum formula states that, for a directed graph,

\sum_{v \in V} \deg^+(v) = \sum_{v \in V} \deg^-(v) = |A|.

If for every vertex v∈V, deg⁺(v) = deg⁻(v), the graph is called a balanced directed graph.^[3]

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 Kleitman–Wang algorithm or by the Fulkerson–Chen–Anstee theorem.

Directed graph connectivity

Main article: Connectivity (graph theory)

A directed graph is weakly connected (or just connected^[4]) 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 a directed path from y to x for every pair of vertices {x, y}. The strong components are the maximal strongly connected subgraphs.

Classes of directed graphs

A symmetric digraph is a directed graph in which for every arrow that belongs to it, the corresponding inversed arrow also belongs to it. A symmetric, loopless directed graph is equivalent to an undirected graph with the edges replaced by pairs of inverse arrows; thus the number of edges is equal to the number of arrows halved.

A simple acyclic directed graph.

A directed acyclic graph is a directed graph with no directed cycles. Special cases of directed acyclic graphs include multitrees (graphs in which no two directed paths from a single starting vertex meet back at the same ending vertex), oriented trees or polytrees (directed graphs formed by orienting the edges of undirected acyclic graphs), and rooted trees (oriented trees in which all edges of the underlying undirected tree are directed away from the roots).

A tournament on 4 vertices.

A tournament is an oriented graph obtained by choosing a direction for each edge in an undirected complete graph.

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 FinVct_K^F(Q) where F(Q) is the free category on Q consisting of paths in Q and FinVct_K 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.

Notes

↑ Bang-Jensen & Gutin (2000). Diestel (2005), Section 1.10. Bondy & Murty (1976), Section 10.
↑ Diestel (2005), Section 1.10.
↑ Satyanarayana, Bhavanari; Prasad, Kuncham Syam, Discrete Mathematics and Graph Theory, PHI Learning Pvt. Ltd., p. 460, ISBN 978-81-203-3842-5 ; Brualdi, Richard A. (2006), Combinatorial Matrix Classes, Encyclopedia of Mathematics and Its Applications 108, Cambridge University Press, p. 51, ISBN 978-0-521-86565-4 .
↑ Bang-Jensen & Gutin (2000) p. 19 in the 2007 edition; p. 20 in the 2nd edition (2009).

References

Bang-Jensen, Jørgen; Gutin, Gregory (2000), directed graphs: Theory, Algorithms and Applications, Springer, ISBN 1-85233-268-9
(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).
Bondy, John Adrian; Murty, U. S. R. (1976), Graph Theory with Applications, North-Holland, ISBN 0-444-19451-7 .
Diestel, Reinhard (2005), Graph Theory (3rd ed.), Springer, ISBN 3-540-26182-6 (the electronic 3rd edition is freely available on author's site).
Harary, Frank; Norman, Robert Z.; Cartwright, Dorwin (1965), Structural Models: An Introduction to the Theory of Directed Graphs, New York: Wiley .
Number of directed graphs (or directed graphs) with n nodes.

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