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 Mathematics and Its Applications (5th Edition)

Kenneth H. Rosen

Chapter 8

Graphs

Section 8.4

Connectivity

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.

a,b

,d

,g

,f

Paths

e

g

a

b

d

f

c

e

g

a

b

d

f

c

Circuits, Simple Path or Circuit

b

c

d

e

a

f

Paths in Directed Graphs

Acquaintanceship Graphs

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

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

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

Connected Undirected

Simple path between every pair of distinct vertices

Connected Directed

Strongly Connected

Weakly Connected

Euler & Hamilton Paths

Bridges ofKonigsberg

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

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

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

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.