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Discrete Structures 310213

Section 8.4. Connectivity. Paths. ???? (path) ??? k ???? ??????? G = (V,E) ????????????????? v0 , v1, v2 ,

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Discrete Structures 310213

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    1. Discrete Structures 310213

    2. Section 8.4

    3. Paths ???? (path) ??? k ???? ??????? G = (V,E) ????????????????? v0 , v1, v2 , ,vk ???? {vi-1 ,Vi } ?E , i = 1,, k . ??????????????? ( cycle ???? circuit ) ???????????????????????????????????????????????????????????????? ?????????????????????????????? 3 ????

    4. Paths

    5. Circuits, Simple Path or Circuit

    6. 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. Euler & Hamilton Paths

    10. 8.5: Euler & Hamilton Paths 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. A Hamilton circuit is a circuit that traverses each vertex in G exactly once. A Hamilton path is a path that traverses each vertex in G exactly once.

    11. Bridges of Knigsberg Problem Can we walk through town, crossing each bridge exactly once, and return to start?

    12. Euler Path Theorems Theorem: A connected multigraph has an Euler circuit iff each vertex has even degree. Proof: (?) The circuit contributes 2 to degree of each node. (?) By construction using algorithm on p. 580-581 Theorem: A connected multigraph has an Euler path (but not an Euler circuit) iff it has exactly 2 vertices of odd degree. One is the start, the other is the end.

    13. Euler Circuit Algorithm Begin with any arbitrary node. Construct a simple path from it till you get back to start. Repeat for each remaining subgraph, splicing results back into original cycle.

    14. 8.5: Euler & Hamilton Paths 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. A Hamilton circuit is a circuit that traverses each vertex in G exactly once. A Hamilton path is a path that traverses each vertex in G exactly once.

    15. Bridges of Knigsberg Problem Can we walk through town, crossing each bridge exactly once, and return to start?

    16. Euler Path Theorems Theorem: A connected multigraph has an Euler circuit iff each vertex has even degree. Proof: (?) The circuit contributes 2 to degree of each node. (?) By construction using algorithm on p. 580-581 Theorem: A connected multigraph has an Euler path (but not an Euler circuit) iff it has exactly 2 vertices of odd degree. One is the start, the other is the end.

    17. Euler Circuit Algorithm Begin with any arbitrary node. Construct a simple path from it till you get back to start. Repeat for each remaining subgraph, splicing results back into original cycle.

    18. Round-the-World Puzzle Can we traverse all the vertices of a dodecahedron, visiting each once?`

    19. Hamiltonian Path Theorems Diracs theorem: If (but not only if) G is connected, simple, has n?3 vertices, and ?v deg(v)?n/2, then G has a Hamilton circuit. Ores corollary: If G is connected, simple, has n=3 nodes, and deg(u)+deg(v)=n for every pair u,v of non-adjacent nodes, then G has a Hamilton circuit.

    20. HAM-CIRCUIT is NP-complete Let HAM-CIRCUIT be the problem: Given a simple graph G, does G contain a Hamiltonian circuit? This problem has been proven to be NP-complete! This means, if an algorithm for solving it in polynomial time were found, it could be used to solve all NP problems in polynomial time.

    21. 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.

    22. 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.

    23. 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.

    24. 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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