Multigraph

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A multigraph with multiple edges (red) and a loop (blue). Not all authors allow multigraphs to have loops.

A multigraph or pseudograph 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. Formally, a multigraph G is an ordered pair G:=(V, E) with

V a set of vertices or nodes,
E a multiset of unordered pairs of vertices, called edges or lines.

Multigraphs might be used to model the possible flight connections offered by an airline. In this case the pseudograph 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.

Some authors also allow multigraphs to have loops, that is, an edge that connects a vertex to itself.^[2]

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.

In category theory a small category can be defined as a multidigraph equipped with an associative composition law and a distinguished self-loop at each vertex serving as the left and right identity for composition. "Graph" is standardly taken to mean "multidigraph" in category theory for this reason, and the underlying graph of a category refers to its underlying multidigraph.

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

1 Labeling
2 Notes
3 References
4 External links

[edit] 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 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 nodes and arcs. Formally it is an 8-tuple $G=(\Sigma_V, \Sigma_A, V, A, s, t, \ell_V, \ell_A)$ where

V is a set of nodes and A is a multiset of arcs.
$Σ V$ and $Σ A$ are finite alphabets of the available node and arc labels,
$s\colon A\rightarrow\ V$ and $t\colon A\rightarrow\ V$ are two maps indicating the source and target node of an arc,
$\ell_V\colon V\rightarrow\Sigma_V$ and $\ell_A\colon A\rightarrow\Sigma_A$ are two maps describing the labeling of the nodes and edges.

Definition 2: A labeled multidigraph is a labeled graph with multiple labeled edges, i.e. edges with the same end nodes and the same edge label (note that this notion of a labeled graph is different from the notion given by the article graph labeling).

[edit] Notes

^ For example, see Balakrishnan, p. 1.
^ For example, see. Bollobas, p. 7 and Diestel, p. 25.

[edit] References

Balakrishnan, V. K.; Graph Theory, McGraw-Hill; 1 edition (February 1, 1997). ISBN 0-07-005489-4.
Bollobas, Bela; Modern Graph Theory, Springer; 1st edition (August 12, 2002). ISBN 0-387-98488-7.
Diestel, Reinhard; Graph Theory, Springer; 2nd edition (February 18, 2000). ISBN 0-387-98976-5.
Gross, Jonathan L, and Yellen, Jay; Graph Theory and Its Applications, CRC Press (December 30, 1998). ISBN 0-8493-3982-0.
Gross, Jonathan L, and Yellen, Jay; (eds); Handbook of Graph Theory. CRC (December 29, 2003). ISBN 1-58488-090-2.
Harary, Frank; Graph Theory, Addison Wesley Publishing Company (January 1995). ISBN 0-201-41033-8.
Zwillinger, Daniel; CRC Standard Mathematical Tables and Formulae, Chapman & Hall/CRC; 31st edition (November 27, 2002). ISBN 1-58488-291-3.
Svante Janson, Donald E. Knuth, Tomasz Luczak, and Boris Pittel; The Birth of the Giant Component, Random Structures Algorithms 4 (1993), no. 3, 231-358.