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Discrete Structures – CNS2300

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

  2. Section 8.4 Connectivity

  3. 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.

  4. a,b ,d ,g ,f Paths e g a b d f c

  5. e g a b d f c Circuits, Simple Path or Circuit

  6. b c d e a f Paths in Directed Graphs

  7. Acquaintanceship Graphs http://www.cs.virginia.edu/oracle/ http://www.brunching.com/bacondegrees.html

  8. 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

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

  10. Euler & Hamilton Paths Bridges ofKonigsberg

  11. 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.

  12. 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.

  13. 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.

  14. 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.

  15. finished

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