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Euler circuit

Euler circuit. Theorem 1 If a graph G has an Eulerian path, then it must have exactly two odd vertices. Theorem 2 If a graph G has an Eulerian circuit, then all of its vertices must be even vertices. Euler circuit. Hierholzer's algorithm : Create a circuit C 1 2 3 4 5 6 1

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Euler circuit

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  1. Euler circuit Theorem 1 If a graph G has an Eulerian path, then it must have exactly two odd vertices. Theorem 2 If a graph G has an Eulerian circuit, then all of its vertices must be even vertices.

  2. Euler circuit Hierholzer's algorithm: Create a circuit C 1 2 3 4 5 6 1 Delete edges from the graph Find a vertex v  C, v  C1 and create new circuit 1 2 3 4 5 6 1 2 5 3 6 2

  3. Hamiltonian Circuit If there is a vertex of degree one in a graph then it is impossible for it to have a Hamiltonian circuit. Dirac’s Theorem- “If G is a simple graph with n vertices where n>==3 such that the degree of every vertex in G is at least n/2, then G has a Hamiltonian circuit.” Ore’s Theorem- “If G is a simple graph with n vertices with n>=3 such that the sum of degrees for every pair of non-adjacent vertices is greater than n, then G has a Hamiltonian circuit.”

  4. Hamiltonian Circuit The Brute force algorithm: 1.     List all possible Hamiltonian circuits 2.     Find the length of each circuit by adding the edge weights 3.     Select the circuit with minimal total weight.

  5. The Brute force algorithm Example: 13

  6. Nearest Neighbor Algorithm (NNA) 1.     Select a starting point. 2.     Move to the nearest unvisited vertex (the edge with smallest weight). 3.     Repeat until the circuit is complete. 13 ADCBA with a total weight of 1+8+13+4=26

  7. Sorted Edges algorithm 13

  8. Sorted Edges algorithm • AD

  9. Sorted Edges algorithm • AD • AC

  10. Sorted Edges algorithm • AD • AC • BD

  11. Sorted Edges algorithm • AD • AC • BD • BC

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