Discrete Structures

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Discrete Structures CNS2300
Text Discrete Mathematics and Its Applications
(5th Edition) Kenneth H. Rosen Chapter 8 Graphs
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Section 8.4
Connectivity
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Paths

• A path is a sequence of edges that begins at a
  vertex of a graph and travels along edges of the
  graph, always connecting pairs of adjacent
  vertices.
• The path is a circuit if it begins and ends at
  the same vertex.
• The path or circuit is said to pass through the
  vertices or traverse the edges
• A path or circuit is simple if it does not
  contain the same edge more than once.

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Paths
e
g
a
b
d
f
c
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Circuits, Simple Path or Circuit
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Paths in Directed Graphs
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Acquaintanceship Graphs

• http//www.cs.virginia.edu/oracle/
• http//www.brunching.com/bacondegrees.html

Bacon No. No. People
012344678910 11479115204285929650214535534812811
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Counting Paths Between Vertices

• Let G be a graph with adjacency matrix A. The
  number of different paths of length r from vi to
  vj, where r is a positive integer, equals the (i,
  j)th entry of Ar

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Connectedness

• Connected Undirected
• Simple path between every pair of distinct
  vertices
• Connected Directed
• Strongly Connected
• Weakly Connected

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Euler Hamilton Paths
Bridges ofKonigsberg
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Euler Circuit

• An Euler circuit in a graph G is a simple circuit
  containing every edge of G.
• An Euler path in G is a simple path containing
  every edge of G.

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Necessary Sufficient Conditions

• A connected multigraph has an Euler circuit if
  and only if each of its vertices has even degree
• A connected multigraph has an Euler path but not
  an Euler circuit if and only if it has exactly
  two vertices of odd degree.

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Hamilton Paths and Circuits

• A Hamilton circuit in a graph G is a simple
  circuit passing through every vertex of G,
  exactly once.
• An Hamilton Path in G is a simple path passing
  through every vertex of G, exactly once.

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Conditions

• If G is a simple graph with n vertices ngt3 such
  that the degree of every vertex in G is at least
  n/2, then G has a Hamilton circuit.
• If G is a simple graph with n vertices ngt3 such
  that deg(u)deg(v)gtn for every pair of
  nonadjacent vertices u and v in G, then G has a
  Hamilton circuit.

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finished