Multigraph

Undirected multigraph (edges with own identity)

A multigraph G is an ordered triple G := (V, E, r) with

• V a set of vertices or nodes,
• E a set of edges or lines,
• r : E → {{x,y} : x, y ∈ V}, assigning to each edge an unordered pair of endpoint nodes.

Some authors allow multigraphs to have loops, that is, an edge that connects a vertex to itself, while others call these pseudographs, reserving the term multigraph for the case with no loops.

Directed multigraph (edges without own identity)

A multidigraph is a directed graph which is permitted to have multiple arcs, i.e., arcs with the same source and target nodes. A multidigraph G is an ordered pair G := (V, A) with

• V a set of vertices or nodes,
• A a multiset of ordered pairs of vertices called directed edges, arcs or arrows.

A mixed multigraph G := (V, E, A) may be defined in the same way as a mixed graph.

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,
• ${\displaystyle s:A\to V}$, assigning to each edge its source node,
• ${\displaystyle t:A\to V}$, 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 ${\displaystyle G=({\mathrm{\Sigma }}_V,{\mathrm{\Sigma }}_A,V,A,s,t,{\ell }_V,{\ell }_A)}$ where

• ${\displaystyle V}$ is a set of vertices and ${\displaystyle A}$ is a set of arcs.
• ${\displaystyle {\mathrm{\Sigma }}_V}$ and ${\displaystyle {\mathrm{\Sigma }}_A}$ are finite alphabets of the available vertex and arc labels,
• ${\displaystyle s:A\to V}$ and ${\displaystyle t:A\to V}$ are two maps indicating the source and target vertex of an arc,
• ${\displaystyle {\ell }_V:V\to {\mathrm{\Sigma }}_V}$ and ${\displaystyle {\ell }_A:A\to {\mathrm{\Sigma }}_A}$ 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).