Multigraph

A multidigraph or quiver G is an ordered 4-tuple G := (V, A, s, t) with This notion might be used to model the possible flight connections offered by an airline.

In this case the multigraph would be a directed graph with pairs of directed parallel edges connecting cities to show that it is possible to fly both to and from these locations.

In category theory a small category can be defined as a multidigraph (with edges having their own identity) equipped with an associative composition law and a distinguished self-loop at each vertex serving as the left and right identity for composition.

For this reason, in category theory the term graph is standardly taken to mean "multidigraph", and the underlying multidigraph of a category is called its underlying digraph.

Multigraphs and multidigraphs also support the notion of graph labeling, in a similar way.