Multigraph
Encyclopedia
In mathematics, a multigraph or pseudograph is a graph
which is permitted to have multiple edges
, (also called "parallel edges"), that is, edges that have the same end nodes
. Thus two vertices may be connected by more than one edge.
There are two distinct notions of multiple edges. One says that, as in graphs without multiple edges, the identity of an edge is defined by the nodes it connects, but the same edge can occur several times between these nodes. Alternatively, one defines edges to be first-class entities like nodes, each having its own identity independent of the nodes it connects.
G:=(V, E) with
Multigraphs 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.
Some authors also 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.
A mixed multigraph G:=(V,E, A) may be defined in the same way as a mixed graph.
G:=(V, A, s, t) with
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.
, 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
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
).
Lets think about a Travelling salesman problem
. Assume that he is travelling by train, and his trip duration between 2 cities (nodes) is the edge value.
But, train schedules vary and this brings us to:
Concluding, multigraph, in some cases may be a very practical tool for "plotting" the real life data.
Graph (mathematics)
In mathematics, a graph is an abstract representation of a set of objects where some pairs of the objects are connected by links. The interconnected objects are represented by mathematical abstractions called vertices, and the links that connect some pairs of vertices are called edges...
which is permitted to have multiple edges
Multiple edges
In graph theory, multiple edges , are two or more edges that are incident to the same two vertices...
, (also called "parallel edges"), that is, edges that have the same end nodes
Vertex (graph theory)
In graph theory, a vertex or node is the fundamental unit out of which graphs are formed: an undirected graph consists of a set of vertices and a set of edges , while a directed graph consists of a set of vertices and a set of arcs...
. Thus two vertices may be connected by more than one edge.
There are two distinct notions of multiple edges. One says that, as in graphs without multiple edges, the identity of an edge is defined by the nodes it connects, but the same edge can occur several times between these nodes. Alternatively, one defines edges to be first-class entities like nodes, each having its own identity independent of the nodes it connects.
Undirected multigraph (edges without own identity)
Formally, a multigraph G is an ordered pairOrdered pair
In mathematics, an ordered pair is a pair of mathematical objects. In the ordered pair , the object a is called the first entry, and the object b the second entry of the pair...
G:=(V, E) with
- V a set of vertices or nodes,
- E a multisetMultisetIn mathematics, the notion of multiset is a generalization of the notion of set in which members are allowed to appear more than once...
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 multigraph would be a directed graph
Directed graph
A directed graph or digraph is a pair G= of:* a set V, whose elements are called vertices or nodes,...
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
Loop (graph theory)
In graph theory, a loop is an edge that connects a vertex to itself. A simple graph contains no 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 G is an ordered 4-tupleTuple
In mathematics and computer science, a tuple is an ordered list of elements. In set theory, an n-tuple is a sequence of n elements, where n is a positive integer. There is also one 0-tuple, an empty sequence. An n-tuple is defined inductively using the construction of an ordered pair...
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.
In category theory
Category theory
Category theory is an area of study in mathematics that examines in an abstract way the properties of particular mathematical concepts, by formalising them as collections of objects and arrows , where these collections satisfy certain basic conditions...
a small category
Category (mathematics)
In mathematics, a category is an algebraic structure that comprises "objects" that are linked by "arrows". A category has two basic properties: the ability to compose the arrows associatively and the existence of an identity arrow for each object. A simple example is the category of sets, whose...
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 labelingGraph labeling
In the mathematical discipline of graph theory, a graph labeling is the assignment of labels, traditionally represented by integers, to the edges or vertices, or both, of a graph....
, 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
Graph labeling
In the mathematical discipline of graph theory, a graph labeling is the assignment of labels, traditionally represented by integers, to the edges or vertices, or both, of a graph....
).
Application
Multigraph may have a very practical application.Lets think about a Travelling salesman problem
Travelling salesman problem
The travelling salesman problem is an NP-hard problem in combinatorial optimization studied in operations research and theoretical computer science. Given a list of cities and their pairwise distances, the task is to find the shortest possible tour that visits each city exactly once...
. Assume that he is travelling by train, and his trip duration between 2 cities (nodes) is the edge value.
But, train schedules vary and this brings us to:
- 1) The travel time between the city of London and Paris is _not_ the same as from Paris to London.
- 2) There may be multiple travel options to consider. Like using a plane.
Concluding, multigraph, in some cases may be a very practical tool for "plotting" the real life data.