Graphs

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Graphs

• Irena Pevac
• CS501

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Graphs Set of nodes Set of edges
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Directed Graphs

• Definition A directed graph is a pair G (V,
  E),
• where V v1, v2, ..., vn is a set of vertices
  or nodes, and a set of directed edges (also
  called arcs) E ? VxV .
• Set of edges is a subset of Cartesian product VxV
  of a set of vertices i.e.
• E ? (x,y) x in V, y in V

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Example of Directed Graph

• Example G (V, E), V 1, 2, 3, 4 and E
  (1,2), (2,3), (3,4), (1,3)

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Undirected Graphs

• Definition An undirected graph is a pair G(V,
  E), where V v1, v2, ..., vn is a set of
  vertices, and E is set of undirected edges or
  lines connecting the nodes. .
• Set of edges is a subset of a set of all two
  element sets over the set of vertices i.e.
• E ? x,y x,y in V

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Example of Undirected Graph

• Example G (V, E), V 1, 2, 3, 4 and E
  1,2, 2,3, 3,4, 1,3

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Complete Graph

• Definition Graph G (V, E) with all possible
  edges present (i.e. E VxV) is called a complete
  graph.
• Example
• E1,2,3,4
• V1,2, 1,3,
• 1,4,2,3,
• 2,4,3,4

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Labeled Graphs

• Directed or undirected graphs can have labels or
  weights associated with edges.
• Example1
• Vertices may be towns, edges are road connections
  between the towns and labels associated with
  edges may represent shortest road traveling
  distance among towns.
• Example 2
• Vertices may be towns, edges represnt direct
  flights between towns and labels may represent
  cost of the flight.

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Definitions

• A graph G is subgraph of a graph G(V, E) if G
  (V, E), and V? V and E? E.
• A graph with a small number of edges present (
  ?E? lt ?V? log ?V?) is
• called a sparse graph. ?E? denotes the
  cardinality of E.
• A graph that is not sparse is called a dense
  graph.

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Labels for Vertices

• Vertices also may have labels.
• The names of vertices must be distinct but two or
  more vertices may have the same label.
• When each vertex has different label we usually
  use labels as names.

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Path

• A path in a directed graph is a list of nodes
  (n1,n2,n3,nk) where ni is in V each consecutive
  two ni,ni1 are connected with with an arc.
• A length of a path is number of arcs along the
  path.
• Any node v by itself is a path of length zero
  from v to v. This path has no arcs.

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Cycle

• A cycle in a directed graph is a path of length
  one or more that begins and ends at the same
  node.
• The length of the cycle is the length of the
  path.
• Trivial path (v) of length 0 is not a cycle.
• A path (v,v) consisting of a single arc v-gtv is a
  cycle of length 1.

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Adjacent Vertices

• Let G (V, E), and let x, y be two vertices
  from V. If x,y belongs to E we say that x and
  y are adjacent. Also we say that x and y are
  called incident to the edge x, y.

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Adjacency Matrix

• Graph G(V,E) can be represented with adjacency
  matrix by creating a two-dimensional array x, in
  which the value xuv is true iff (u,v) is an
  edge in E.
• For undirected graphs adjacency matrix is
  symmetric.

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Adjacency List

• Graph G(V,E) can be represented with adjacency
  list by creating an array x with ?V? items. Each
  value xu has linked list of items
  correcponding to nodes adjacent to u.

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Example

• Let G (V,E) V a, b, c, d and
• E a,b, c,d, b,c, a,c.

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Adjacency Matrix

• In adjacency matrix we use 0 for false and 1 for
  true.

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Adjacency List
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Representing Labeled Graphs

• Labeled graph has labels or weights on its arcs
  (if directed) or on its edges (if undirected.
• In adjacency matrix we can replace the 1 that
  represents the presence of an arc by its label.
  It is necessary to have a special value (that is
  different than any label) to be matrix entry
  indicating the absence of an arc.

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Example Labeled Graph
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Adjacency Matrix

• In adjacency matrix we use -1 for absence of the
  arc and label on the arc when arc is present.

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Adjacency List for Labeled Graph