Multigraph Explained

In mathematics, and more specifically in graph theory, a multigraph is a graph which is permitted to have multiple edges (also called parallel edges[1]), that is, edges that have the same end nodes. Thus two vertices may be connected by more than one edge.

There are 2 distinct notions of multiple edges:

A multigraph is different from a hypergraph, which is a graph in which an edge can connect any number of nodes, not just two.

For some authors, the terms pseudograph and multigraph are synonymous. For others, a pseudograph is a multigraph that is permitted to have loops.

Undirected multigraph (edges without own identity)

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

Undirected multigraph (edges with own identity)

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

Some authors allow multigraphs to have loops, that is, an edge that connects a vertex to itself,[2] while others call these pseudographs, reserving the term multigraph for the case with no loops.[3]

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

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

s:A → V

, assigning to each edge its source node,

t:A → 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

G=(\SigmaV,\SigmaA,V,A,s,t,\ellV,\ellA)

where

V

is a set of vertices and

A

is a set of arcs.

\SigmaV

and

\SigmaA

are finite alphabets of the available vertex and arc labels,

s\colonA →  V

and

t\colonA →  V

are two maps indicating the source and target vertex of an arc,

\ellV\colonV → \SigmaV

and

\ellA\colonA → \SigmaA

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

References

Notes and References

  1. For example, see Balakrishnan 1997, p. 1 or Chartrand and Zhang 2012, p. 26.
  2. For example, see Bollobás 2002, p. 7 or Diestel 2010, p. 28.
  3. For example, see Wilson 2002, p. 6 or Chartrand and Zhang 2012, pp. 26-27.

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