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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
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

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.
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