Directed multigraph (edges with own identity) A multidigraph or quiver G is an ordered 4-tuple G := (V, A, s, t) with V a set of vertices or nodes, A a set of edges or lines, , assigning to each edge its source node, , assigning to each edge its target node. 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. Labeling Multigraphs and multidigraphs also support the notion of graph labeling, in a similar way. However there is no unity in terminology in this case. The definitions of labeled multigraphs and labeled multidigraphs are similar, and we define only the latter ones here. Definition 1: A labeled multidigraph is a labeled graph with labeled arcs. Formally: A labeled multidigraph G is a multigraph with labeled vertices and arcs. Formally it is an 8-tuple where V is a set of vertices and A is a set of arcs. and are finite alphabets of the available vertex and arc labels, and are two maps indicating the source and target vertex of an arc, and are two maps describing the labeling of the vertices and arcs. Definition 2: A labeled multidigraph is a labeled graph with multiple labeled arcs, i.e. arcs with the same end vertices and the same arc label (note that this notion of a labeled graph is different from the notion given by the article graph labeling). See also Multidimensional network Glossary of graph theory terms Graph theory Notes References External links Extensions and generalizations of graphs de:Graph (Graphentheorie)#Multigraph