Graph Theory

In Mathematics and Computer Science , graph theory has for its subject matter the properties of '' Graphs ''. Informally speaking, a graph is a set of objects called ''points'' or ''vertices'' connected by links called ''lines'' or ''edges''. In a graph proper, which is by default ''undirected'', a line from point ''A'' to point ''B'' is considered to be the same thing as a line from point ''B'' to point ''A''. In a ''digraph'', short for ''directed graph'', the two directions are counted as being distinct ''arcs'' or ''directed edges''.

Structures that can be represented as graphs are ubiquitous, and many problems of practical interest can be represented by graphs. The link structure of a s to handle graphs is therefore of major interest in Computer Science .

A graph structure can be extended by assigning a weight to each edge of the graph. Graphs with weights can be used to represent many different concepts; for example if the graph represents a road network, the weights could represent the length of each roadThe only information a weighted graph provides as such is (a) the vertices, (b) the edges and (c) the weights. Therefore the example in which the weights represent the roads' lengths doesn't imply that the weights are merely redundant annotations: there is no actual topographical information associated with the graph, so unlike reading a map, measuring the distances between the vertices is completely meaningless -- without the weights, there would be no way of telling what the distance between the vertices is in real life.. A digraph with weighted edges is called a Network .

Networks have many uses in the practical side of graph theory, Network Analysis (for example, to model and analyze traffic networks or to discover the ''shape'' of the internet -- see Applications below). Within network analysis, the definition of the term "network" varies, and may often refer to a simple graph.

HISTORY

One of the first results in graph theory appeared in Leonhard Euler 's paper on '' Seven Bridges Of Königsberg '', published in 1736 . It is also regarded as one of the first topological results in geometry; that is, it does not depend on any measurements. This illustrates the deep connection between graph theory and Topology .

In 1845 Gustav Kirchhoff published his Kirchhoff's Circuit Laws for calculating the Voltage and Current in Electric Circuit s.

In 1852 Francis Guthrie posed the Four Color Problem which asks if it is possible to color, using only four colors, any map of countries in such a way as to prevent two bordering countries from having the same color. This problem, which was only solved a century later in 1976 by Kenneth Appel and Wolfgang Haken , can be considered the birth of graph theory. While trying to solve it mathematicians invented many fundamental graph theoretic terms and concepts.

DEFINITION

See Also: Graph (mathematics)

Definitions of graphs vary in style and substance, according to the level of abstraction that is approriate to a particular approach or application. For the sake of perspective, the following definitions present the same substance in two different styles:

First, a classic definition, that covers most of the essential ideas in a very short space:

A ''graph'' ''G'' consists of a finite nonempty set ''V'' = ''V''(''G'') of ''p'' ''points'' together with a prescribed set ''X'' of ''q'' unordered pairs of distinct points of ''V''. Each pair ''x'' = {''u'', ''v''} of points in ''X'' is a ''line'' of ''G'', and ''x'' is said to ''join'' ''u'' and ''v''. We write ''x'' = ''uv'' and say that ''u'' and ''v'' are ''adjacent points'' (sometimes denoted ''u'' adj ''v''); point ''u'' and line ''x'' are ''incident'' with each other, as are ''v'' and ''x''. If two distinct lines ''x'' and ''y'' are incident with a common point, then they are ''adjacent lines''. A graph with ''p'' points and ''q'' lines is called a (''p'', ''q'') ''graph''. The (1, 0) graph is ''trivial''. (Harary, ''Graph Theory'', p. 9).

Next, a style of definition that is preferred in some approaches and applications:

A graph or '''undirected graph''' ''G'' is an Ordered Triple ''G'':=(''V'', ''E'', ''f'') subject to the following conditions:

''V'' is a Set , whose elements are variously referred to as nodes, '''points''', or '''vertices'''.

''E'' is a set, whose elements are known as edges or '''lines'''.

''f'' is a function that maps each element of ''E'' to an unordered pair of distinct vertices in ''V'', referred to as the ends, '''endpoints''', or '''end vertices''' of the edge.

''V'' (and hence ''E'') are usually taken to be finite sets, and many of the well-known results are not true (or are rather different) for infinite graphs because many of the arguments fail in the infinite case.

A digraph or a '''directed graph''' ''G'' is an ordered pair ''G'':=(''V'', ''A'') subject to the following conditions:

''V'' is a Set , whose elements are variously referred to as nodes, '''points''', or '''vertices'''.

''A'' is a set of ordered pairs of vertices, called arcs, '''arrows''', or '''directed edges'''. An edge ''e'' = (''x'', ''y'') is said to be directed '''from''' ''x'' '''to''' ''y'', where ''x'' is the '''tail''' of ''e'' and ''y'' is the '''head''' of ''e''.

Alternatively, a digraph or a '''directed graph''' may be defined as an ordered triple ''G'':=(''V'', ''E'', ''f'') subject to the following conditions:

''V'' is a Set , whose elements are variously referred to as nodes, '''points''', or '''vertices'''.

''E'' is a set, whose elements are known as arcs, '''arrows''', or '''directed edges'''.

''f'' is a function that maps each element in ''E'' to an ordered pair of vertices in ''V''.

There are also some mixed type of graphs with undirected and directed edges.

DRAWING GRAPHS

See Also: Graph drawing

Graphs are represented graphically by drawing a dot for every vertex, and drawing an arc between two vertices if they are connected by an edge. If the graph is directed, the direction is indicated by drawing an arrow.

A graph drawing should not be confused with the graph itself (the abstract, non-graphical structure) as there are several ways to structure the graph drawing. All that matters is which vertices are connected to which others by how many edges and not the exact layout. In practise it is often difficult to decide if two drawings represent the same graph. Depending on the problem domain some layouts may be better suited and easier to understand than others.

GRAPH-THEORETIC DATA STRUCTURES

See Also: Graph (data structure)

There are different ways to store graphs in a computer system. The Data Structure used depends on both the graph structure and the Algorithm used for manipulating the graph. Theoretically one can distinguish between list and matrix structures but in concrete applications the best structure is often a combination of both. List structures are often preferred for Sparse Graph s as they have smaller memory requirements. Matrix structures on the other hand provide faster access but can consume huge amounts of memory if the graph is very large.

List structures

Incidence List - The edges are represented by an Array containing pairs (ordered if directed) of vertices (that the edge connects) and eventually weight and other data.

Adjacency List - Much like the incidence list, each node has a list of which nodes it is adjacent to. This can sometimes result in "overkill" in an undirected graph as node 3 may be in the list for node 2, then node 2 must be in the list for node 3. Either the programmer may choose to use the unneeded space anyway, or he/she may choose to list the adjacency once. This representation is easier to find all the nodes which are connected to a single node, since these are explicitly listed.

Matrix structures

Incidence Matrix - The graph is represented by a Matrix of E (edges) by V (vertices), where vertex contains the edge's data (simplest case: 1 - connected, 0 - not connected).

Adjacency Matrix - there is an N by N matrix, where N is the number of vertices in the graph. If there is an edge from some vertex x to some vertex y, then the element $M_{x, y}$ is 1, otherwise it is 0. This makes it easier to find subgraphs, and to reverse graphs if needed.

Admittance Matrix or ''' Kirchhoff Matrix ''' or '''Laplacian matrix''' - is defined as Degree Matrix minus Adjacency Matrix and thus contains adjacency information and degree information about the vertices

PROBLEMS IN GRAPH THEORY

Problems about subgraphs

A common problem, called Subgraph Isomorphism Problem , is finding Subgraph s in a given graph. Many Graph Properties are ''hereditary'', which means that a graph has a property if and only if all subgraphs have it too. For example a graph is Planar if it contains neither the Complete Bipartite Graph

K_{3,3}

(See Three Cottage Problem ) nor the Complete Graph

K_{5}

. Unfortunately, finding maximal subgraphs of a certain kind is often an NP-complete Problem .

Finding the largest Complete Graph is called the Clique Problem (NP-complete)

Finding the largest Independent Set is called the Independent Set Problem (NP-complete)

Another class of problems has to do with the extent to which various species and generalizations of graphs are determined by their ''point-deleted subgraphs'', for example:

Reconstruction Conjecture

Graph coloring

Many problems have to do with various ways of Coloring Graphs , for example:

The Four-color Theorem

The Strong Perfect Graph Theorem

The Erdős-Faber-Lovász Conjecture (unsolved)

The Total Coloring Conjecture (unsolved)

The List Coloring Conjecture (unsolved)

Route problems

Hamiltonian Path And Cycle Problems

Seven Bridges Of Königsberg

Minimum Spanning Tree

Steiner Tree

Shortest Path Problem

Route Inspection Problem (also called the "Chinese Postman Problem")

Traveling Salesman Problem (NP-Complete)

Network flow

There are numerous problems arising especially from applications that have to do with various notions of Flows In Networks , for example:

Max Flow Min Cut Theorem

Visibility graph problems

Museum Guard Problem

Covering problems

Covering Problems are specific instances of subgraph-finding problems, and they tend to be closely related to the Clique Problem or the Independent Set Problem .

Set Cover Problem

Vertex Cover Problem

APPLICATIONS

Many applications of graph theory exist in the form of Network Analysis . These split broadly into two categories. Firstly, analysis to determine structural properties of a network, such as whether or not it is a Scale-free Network , or a Small-world Network . Secondly, analysis to find a measurable quantity within the network, for example, for a Transportation Network , the level of vehicular flow within any portion of it.

Graph theory is also used to study molecules in science. In condensed matter physics, the three dimensional structure of complicated simulated atomic structures can be studied quantitatively by gathering statistics on graph-theoretic properties related to the topology of the atoms. For example, Franzblau's shortest-path (SP) rings.

NOTES

REFERENCES

Harary, Frank , ''Graph Theory'', Addison-Wesley, Reading, MA, 1969.

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Graph Theory