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A multigraph with multiple edges (red) and several loops (blue). Not all authors allow multigraphs to have loops. 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 ...
A graph with three vertices and three edges. A graph (sometimes called an undirected graph to distinguish it from a directed graph, or a simple graph to distinguish it from a multigraph) [4] [5] is a pair G = (V, E), where V is a set whose elements are called vertices (singular: vertex), and E is a set of unordered pairs {,} of vertices, whose elements are called edges (sometimes links or lines).
Where graphs are defined so as to allow multiple edges and loops, a graph without loops or multiple edges is often distinguished from other graphs by calling it a simple graph. [1] Where graphs are defined so as to disallow multiple edges and loops, a multigraph or a pseudograph is often defined to mean a "graph" which can have multiple edges. [2]
Graphs as defined in the two definitions above cannot have loops, because a loop joining a vertex to itself is the edge (for an undirected simple graph) or is incident on (for an undirected multigraph) {,} = {} which is not in {{,},}. To allow loops, the definitions must be expanded.
Every Moore graph is a cage. multigraph A multigraph is a graph that allows multiple adjacencies (and, often, self-loops); a graph that is not required to be simple. multiple adjacency A multiple adjacency or multiple edge is a set of more than one edge that all have the same endpoints (in the same direction, in the case of directed graphs).
Where graphs are defined so as to disallow loops and multiple edges, a graph that does have loops or multiple edges is often distinguished from the graphs that satisfy these constraints by calling it a multigraph or pseudograph. In a graph with one vertex, all edges must be loops. Such a graph is called a bouquet.
The definition above generalizes from a directed graph to a directed hypergraph by defining the head or tail of each edge as a set of vertices (or ) rather than as a single vertex. A graph is then the special case where each of these sets contains only one element.
In this context, the term graph means multigraph. There are several ways to define series–parallel graphs. The following definition basically follows the one used by David Eppstein. [1] A two-terminal graph (TTG) is a graph with two distinguished vertices, s and t called source and sink, respectively.