Directed graph
graph with oriented edges
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.
Definition
In formal terms, a directed graph is an ordered pair G = (V, A) where
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.
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, links or lines.
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 multiset). Sometimes these entities are called directed multigraphs (or multidigraphs).
On the other hand, the aforementioned definition allows a directed graph to have loops (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 simple directed graphs, while directed graphs with loops may be called loop-digraphs (see section Types of directed graph).
Types of directed graphs
Subclasses
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.
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