Two edges of a directed graph are called consecutive if the head of the first one is the tail of the second one. Two edges of a graph are called adjacent if they share a common vertex. See also: Glossary of graph theory and Graph property A k-vertex-connected graph is often called simply a k-connected graph. Otherwise it is called a disconnected graph.Ī k-vertex-connected graph or k-edge-connected graph is a graph in which no set of k − 1 vertices (respectively, edges) exists that, when removed, disconnects the graph. Otherwise, it is called a weakly connected graph if every ordered pair of vertices in the graph is weakly connected. Otherwise, the ordered pair is called disconnected.Ī strongly connected graph is a directed graph in which every ordered pair of vertices in the graph is strongly connected. Otherwise, the ordered pair is called weakly connected if an undirected path leads from x to y after replacing all of its directed edges with undirected edges. In a directed graph, an ordered pair of vertices ( x, y) is called strongly connected if a directed path leads from x to y. Otherwise, it is called a disconnected graph. Otherwise, the unordered pair is called disconnected.Ī connected graph is an undirected graph in which every unordered pair of vertices in the graph is connected. The vertices x and y of an edge is called connected if a path leads from x to y. The following are some of the more basic ways of defining graphs and related mathematical structures.Ī graph with three vertices and three edgesĪ graph (sometimes called an undirected graph to distinguish it from a directed graph, or a simple graph to distinguish it from a multigraph) is a pair G = ( V, E), where V is a set whose elements are called vertices (singular: vertex), and E is a set of paired vertices, whose elements are called edges (sometimes links or lines). Definitions ĭefinitions in graph theory vary. Sylvester in 1878 due to a direct relation between mathematics and chemical structure (what he called a chemico-graphical image). The word "graph" was first used in this sense by J. Graphs are the basic subject studied by graph theory. In contrast, if an edge from a person A to a person B means that A owes money to B, then this graph is directed, because owing money is not necessarily reciprocated. For example, if the vertices represent people at a party, and there is an edge between two people if they shake hands, then this graph is undirected because any person A can shake hands with a person B only if B also shakes hands with A. Graphs are one of the objects of study in discrete mathematics. Typically, a graph is depicted in diagrammatic form as a set of dots or circles for the vertices, joined by lines or curves for the edges. The objects correspond to mathematical abstractions called vertices (also called nodes or points) and each of the related pairs of vertices is called an edge (also called link or line). In discrete mathematics, and more specifically in graph theory, a graph is a structure amounting to a set of objects in which some pairs of the objects are in some sense "related". A graph with six vertices and seven edges
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