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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).
A directed graph is weakly connected (or just connected [9]) if the undirected underlying graph obtained by replacing all directed edges of the graph with undirected edges is a connected graph. A directed graph is strongly connected or strong if it contains a directed path from x to y (and from y to x) for every pair of vertices (x, y).
A directed graph with three vertices and four directed edges (the double arrow represents an edge in each direction). A directed graph or digraph is a graph in which edges have orientations. In one restricted but very common sense of the term, [5] a directed graph is an ordered pair = (,) comprising:
The different types of edge in a bidirected graph. In the mathematical domain of graph theory, a bidirected graph (introduced by Edmonds & Johnson 1970) [1] is a graph in which each edge is given an independent orientation (or direction, or arrow) at each end. Thus, there are three kinds of bidirected edges: those where the arrows point outward ...
An extension of Robbins' theorem to mixed graphs by Boesch & Tindell (1980) shows that, if G is a graph in which some edges may be directed and others undirected, and G contains a path respecting the edge orientations from every vertex to every other vertex, then any undirected edge of G that is not a bridge may be made directed without changing the connectivity of G.
An ancestral graph is a further extension, having directed, bidirected and undirected edges. [4] Random field techniques A Markov random field, also known as a Markov network, is a model over an undirected graph. A graphical model with many repeated subunits can be represented with plate notation.
A directed graph is called an oriented graph if none of its pairs of vertices is linked by two mutually symmetric edges. Among directed graphs, the oriented graphs are the ones that have no 2-cycles (that is at most one of (x, y) and (y, x) may be arrows of the graph). [1] A tournament is an orientation of a complete graph.
In some important cases (topological index calculation etc.) the following classical definition is sufficient: a molecular graph is a connected, undirected graph which admits a one-to-one correspondence with the structural formula of a chemical compound in which the vertices of the graph correspond to atoms of the molecule and edges of the ...