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Graph theory

Graph theory is the branch of mathematics that examines the properties of graphs.

A graph with 6 vertices and 7 edges.

A graph is a set of dots, called vertices or nodes, connected by links, called edges or arcs. Depending on the applications, edges may or may not have a direction; edges joining a vertex to itself may or may not be allowed, and vertices and/or edges may be assigned weights, i.e. numbers. If the edges have a direction associated with them (indicated by an arrow in the graphical representation) we have a directed graph.

Structures that can be represented as graphs are ubiquitous, and many problems of practical interest can be formulated as questions about certain graphs. For example, the link structure of Wikipedia could be represented by a directed graph: the vertices are the articles in Wikipedia, and there's a directed edge from article A to article B if and only if A contains a link to B. Directed graphs are also used to represent finite state machines. The development of algorithms to compute certain properties of graphs is therefore of major interest in computer science.

Table of contents
1 Graph Problems
2 Important Algorithms
3 Generalizations
4 Related areas of mathematics

Definitions of Graphs and Digraphs

The basic definitions in graph theory vary in the literature. Here are the conventions used in this encyclopedia.

A directed graph (also called digraph or quiver) consists of

a set V of vertices, and
a set E of edges, and
maps s, t : EV, where s(e) is the source and t(e) is the target of the directed edge e.

An undirected graph (or graph for short) is given by

a set V of vertices,
a set E of edges,
a function w : EP(V) which associates to each edge a two- or one-element subset of V, interpreted as the endpoints of the edge.

In a weighted graph or digraph, an additional function E → R associates a value with each edge, which can be considered its "cost"; such graphs arise in optimal route problems[?] such as the traveling salesman problem.

"Graphical" representation

Graphs are often represented "graphically" as follows: draw a dot for every vertex, and for every edge draw an arc connecting its endpoints. If the graph is directed, indicate the endpoint of an edge by an arrow.

Note that this graphical representation should not be confused with the graph itself, which is an abstract, non-graphical structure. Very different graphical representation can correspond to the same graph. All that matters is which vertices are connected to which others by how many edges.

Glossary of basic graph theory concepts

A loop in a graph or digraph is an edge e in E whose endpoints are the same vertex. A digraph or graph is called simple if there are no loops and there is at most one edge between any pair of vertices.

The example graph pictured to the right is a simple graph with vertex set V = {1, 2, 3, 4, 5, 6} and edge set E = {{1,2}, {1,5}, {2,3}, {2,5}, {3,4}, {4,5}, {4,6}} (with the map w being the identity).

An edge connects two vertices; these two vertices are said to be incident to the edge. The valency (or degree) of a vertex is the number of edges incident to it, with loops being counted twice. In the example graph vertices 1 and 3 have a valency of 2, vertices 2,4 and 5 have a valency of 3 and vertex 6 has a valency of 1. If E is finite, then the total valency of the vertices is equal to twice the number of edges. In a digraph, we distinguish the out degree (=the number of edges leaving a vertex) and the in degree (=the number of edges entering a vertex). The degree of a vertex is equal to the sum of the out degree and the in degree.

Two vertices are considered adjacent if an edge exists between them. In the above graph, vertices 1 and 2 are adjacent, but vertices 2 and 4 are not. The set of neighbors for a vertex consists of all vertices adjacent to it. In the example graph, vertex 1 has two neighbors: vertex 2 and node 5. For a simple graph, the number of neighbors that a vertex has coincides with its valency.

In computers, a finite directed or undirected graph (with n vertices, say) is often represented by its adjacency matrix: an n-by-n matrix whose entry in row i and column j gives the number of edges from the i-th to the j-th vertex.

A path is a sequence of vertices such that from each of its vertices there is an edge to the successor vertex. A path is considered simple if none of the vertices in the path are repeated. The length of a path is the number of edges that the path uses, counting multiple edges multiple times. The cost of a path in a weighted graph is the sum of the costs of the traversed edges. Two paths are independent if they do not have any vertex in common, except the first and last one.

In the example graph, (1, 2, 5, 1, 2, 3) is a path with length 5, and (5, 2, 1) is a simple path of length 2.

If it is possible to establish a path from any vertex to any other vertex of a graph, the graph is said to be connected. If it is always possible to establish a path from any vertex to any other vertex even after removing k-1 vertices, then the graph is said to be k-connected. Note that a graph is k-connected if and only if it contains k independent paths between any two vertices. The example graph above is connected (and therefore 1-connected), but not 2-connected.

A cycle (or circuit) is a path that begins and ends with the same vertex. Cycles of length 1 are loops. In the example graph, (1, 2, 3, 4, 5, 2, 1) is a cycle of length 6. A simple cycle is a cycle which has length at least 3 and in which the beginning vertex only appears once more, as the ending vertex, and the other vertices appear only once. In the above graph (1, 5, 2, 1) is a simple cycle. A graph is called acyclic if it contains no simple cycles.

An articulation point is a vertex whose removal disconnects a graph. A bridge is an edge whose removal disconnects a graph. A biconnected component is a maximal set of edges such that any two edges in the set lie on a common simple cycle. The girth of a graph is the length of the shortest simple cycle in the graph. The girth of an acyclic graph is defined to be infinity.

A tree is a connected acyclic simple graph. Sometimes, one vertex of the tree is distinguished, and called the root. Trees are commonly used as data structures in computer science (see tree data structure).

A forest is a set of trees; equivalently, a forest is any acyclic graph.

A subgraph of the graph G is a graph whose vertex set is a subset of the vertex set of G, whose edge set is a subset of the edge set of G, and such that the map w is the restriction of the map from G.

A spanning subgraph of a graph G is a subgraph with the same vertex set as G. A spanning tree is a spanning subgraph that is a tree. Every graph has a spanning tree.

A complete graph is a simple graph in which every vertex is adjacent to every other vertex. The example graph is not complete. The complete graph on n vertices is often denoted by Kn. It has n(n-1)/2 edges (corresponding to all possible choices of pairs of vertices).

A planar graph is one which can be drawn in the plane without any two edges intersecting. The example graph is planar; the complete graph on n vertices, for n> 4, is not planar.

An Eulerian path in a graph is a path that uses each edge precisely once. If such a path exists, the graph is called traversable. An Eulerian cycle is a cycle with uses each edge precisely once. There is a dual to this concept: a Hamiltonian path in a graph is a path that visits each vertex once and only once; and a Hamiltonian cycle is a cycle which visits each vertex once and only once. The example graph does not contain an Eulerian path, but it does contain a Hamiltonian path. While determining whether a given graph has an Eulerian path or cycle is trivial, the same problem for Hamiltonian paths and cycles is extremely hard.

The null graph is the graph whose edge set and vertex set are empty.

An independent set in a graph is a set of pairwise nonadjacent vertices. In the example above, vertices 1,3, and 6 form an independent set and 3,5, and 6 are another independent set.

A clique (pronounced "click") in a graph is a set of pairwise adjacent vertices. In the example graph above, vertices 1, 2 and 5 form a clique.

A bipartite graph is any graph whose vertices can be divided into two sets, such that there are no edges between vertices of the same set. A graph can be proved bipartite if there do not exist any circuits of odd length.

A k-partite graph or k-colorable graph is a graph whose vertices can be partitioned into k disjoint subsets such that there are no edges between vertices in the same subset. A 2-partite graph is the same as a bipartite graph.

Graph Problems

  • Route problems:

Important Algorithms

Generalizations

In a hypergraph an edge can connect more than two vertices.

An undirected graph can be seen as a simplicial complex consisting of 1-simplices (the edges) and 0-simplices (the vertices). As such, complexes are generalizations of graphs since they allow for higher-dimensional simplices.

Every graph gives rise to a matroid, but in general the graph cannot be recovered from its matroid, so matroids are not truly generalizations of graphs.

Related areas of mathematics



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