Directed graph

A simple directed graph.

In mathematics, and more specifically in graph theory, a directed graph (or digraph) is a graph (that is a 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) where[1]

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.

The aforementioned definition doesn't allow a directed graph to have multiple arrows with same source and target nodes, but some authors consider a broader definition that allow directed graphs to have such multiple arrows (namely, they allow the arrows set to be a multiset). More specifically, these entities are addressed as directed multigraphs (or multidigraphs).
On the other hand, the aforementioned definition allows a directed graph to have loops (that is, arrows that connect nodes with themselves), but some authors consider a narrower definition that doesn't allow directed graphs to have loops.[2] More specifically, directed graphs without loops are addressed as simple directed graphs, while directed graphs with loops are addressed as loop-digraphs (see section Types of directed graphs).

Types of directed graphs


A simple acyclic directed graph.
A tournament on 4 vertices.

Digraphs with supplementary properties

This list is incomplete; you can help by expanding it.

Basic terminology

Oriented graph with corresponding incidence matrix.

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).

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 aij is the number of arrows from vertex i to vertex j, and the diagonal entry aii 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, A) and vV. 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 outcoming arrows. Similarly, a vertex with deg+(v) = 0 is called a sink, since it is the end of each of its incoming arrows.

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,

If for every vertex vV, deg+(v) = deg(v), the graph is called a balanced directed graph.[4]

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

A directed graph is weakly connected (or just connected[5]) 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.

See also

Wikimedia Commons has media related to Directed graphs.


  1. Bang-Jensen & Gutin (2000). Diestel (2005), Section 1.10. Bondy & Murty (1976), Section 10.
  2. 1 2 3 Chartrand, Gary (1977). Introductory Graph Theory. Courier Corporation. ISBN 9780486247755.
  3. Diestel (2005), Section 1.10.
  4. 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.
  5. Bang-Jensen & Gutin (2000) p. 19 in the 2007 edition; p. 20 in the 2nd edition (2009).


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