Introduction to Graph Theory

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• Introduction to Graph Theory

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The graphs studied by graph theorists have
nothing to do with the wiggly-line charts that
plot stock prices. Here is a definition of a
graph, in all its glory of abstraction A graph
is a pair of sets, V and E, where V is a nonempty
finite set and every element of E is a two-member
set whose members are elements of V. For example,
this is a graph V a, b, c, E a, b,
a,c.  
Most of the time, a picture is worth at least a
thousand sets, and yet there are reasons for
retaining the more formal definition.
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• Consider a number of cities connected by
  highways .Given a map of the highways, we want to
  determine whether there is a highway route
  between two cities on the map.
• In this e.g of highway map,
• let Sa,b,c be the set of cities, and
• R be a binary relation on S such that (a, b) is
  in R if there is a highway from city a to city b.
  We want to know whether, for a given a and b in
  S, there exist c, d, e, h in S such that
• (a,c),(c,d ),(d,e),,(h,b) ? R.

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• A directed graph G is defined abstractly as an
  ordered pair (V,E),
• where V is a set and E is a binary relation on
  V.
• A directed graph can be represented geometrically
  as a set of marked points V with a set of arrows
  E between pairs of points.

The elements in V are called the vertices and,
and the ordered pairs In E are called the edges
of the directed graph.
An edge is said to be incident with the vertices
it joins.
To be more specific we say that the edge (a,b) is
incident from a and Incident into b.
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Vertex a is called the initial vertex and vertex
b is called the terminal Vertex.
An edge that is incident from and into the same
vertex, like (c,c) Is called a loop. Two
vertices are said to be adjacent if they are
joined by an edge Corresponding to an edge (a,b)
the vertex a is said to be adjacent to the vertex
b and the vertex b is said to be adjacent from
the vertex a. A vertex is said to be an isolated
vertex if there is no edge incident With it.
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Graphs - Basic Terminology

• An undirected graph G is defined abstractly as an
  ordered pair (V, E), where V is a set and E is a
  set of multi sets of two elements from V.

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• For example
• G (a, b, c, d, a, b, a, d, b, c, b, d,
  c, c) is an undirected graph. An undirected
  graph can be represented geometrically as a set
  of marked points V with a set of lines E between
  the points.
• From now on, we shall use the term graph to mean
  either a directed graph or an undirected graph or
  both.

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• Let V a, b, c, d be the four players in a
  round-robin tennis tournament.
• Let E (a, b),(a, d),(b, d),(c, a),(c, b),(d,
  c) be a binary relation on V so that (x, y) in
  E means that x beats y in the match between them.
  
• Let V 1,2,3,4 be the four chapters in a
  book. Let E(1,2), (2,3), (3,1), (3,4), (4,1),
  (4,2) be a binary relation on V such that (1,2)
  in E means that the material in chapter 1 refers
  to that in chapter 2 and so on.

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Definition

• Two Graphs are said to be isomorphic if there is
  a one-to-one correspondence between their
  vertices and between their edges such that
  incidences are preserved. In other words, there
  is an edge between two vertices in one graph if
  and only if there is a corresponding edge between
  the corresponding vertices in the other graph.

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• Let G(V, E) be a graph. A Graph G(V, E) is
  said to be a sub graph of G if E is a subset of
  E and V is a subset of V such that the edges in
  E are incident only with the vertices in V.
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• A sub graph of G is said to be a spanning sub
  graph if it contains all the vertices of G. The
  complement of a sub graph G (V, E) with
  respect to the graph G is another sub graph G
  (V, E) such that E is equal to E-E and V
  contains only the vertices with which the edges
  in E are incident.

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• The undirected complete graph of n vertices,
• denoted K n
• is a graph with n vertices in which there is an
  edge between each pair of distinct vertices
• The complement of a graph G of n vertices is
  defined to be its complement with respect to K n
  and is denoted
• _
• G.

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• Let G be a graph of n vertices. Let the n
  vertices in G represent n people and let the set
  of edges in G represent a compatible relationship
  such that an edge between two vertices means that
  the two corresponding persons can work
  cooperatively as a team. Clearly, the set of
  edges in
• _
• G will represent the incompatibility
  relationship among the n people. We also define a
  directed complete graph of n vertices to be a
  graph with n vertices in which there is exactly
  one arrow between each pair of distinct vertices.
•