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

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Discrete Structures – CNS2300

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Discrete structures cns2300

Discrete Structures – CNS2300

Text

Discrete Mathematics and Its Applications (5th Edition)

Kenneth H. Rosen

Chapter 8

Graphs


Section 8 4

Section 8.4

Connectivity


Paths

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.


Paths1

a,b

,d

,g

,f

Paths

e

g

a

b

d

f

c


Circuits simple path or circuit

e

g

a

b

d

f

c

Circuits, Simple Path or Circuit


Paths in directed graphs

b

c

d

e

a

f

Paths in Directed Graphs


Acquaintanceship graphs

Acquaintanceship Graphs

http://www.cs.virginia.edu/oracle/

http://www.brunching.com/bacondegrees.html


Counting paths between vertices

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


Connectedness

Connectedness

  • Connected Undirected

    • Simple path between every pair of distinct vertices

  • Connected Directed

    • Strongly Connected

    • Weakly Connected


Euler hamilton paths

Euler & Hamilton Paths

Bridges ofKonigsberg


Euler circuit

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.


Necessary sufficient conditions

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.


Hamilton paths and circuits

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.


Conditions

Conditions

  • If G is a simple graph with n vertices n>=3 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 n>=3 such that deg(u)+deg(v)>=n for every pair of nonadjacent vertices u and v in G, then G has a Hamilton circuit.


Finished

finished


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