Hamiltonian Path Theory

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Telecoms, Architectures and Business Models

Graph theory

Hamiltonian Path, Minimum Spanning Tree

Serafeim (Makis) Gravanis 4413512
 1) Is finding Hamiltonian path NP-complete? Why? Also explain in your own words what NP- complete means. 
 Before we explain why the Hamiltonian path is NP-complete, it will be worth mentioning to refer on the NP-completeness. NP stands for “nondeterministic polynomial time” and its roots are coming form the complexity theory. More specific, this means that although it is possible to verify the correctness of a solution to a problem in polynomial time (quick verification), the use of exponential time is needed in order to compute the exact solution. In other words, the required time to calculate the solution of the problem increases rapidly as the size of the of the
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Indeed in a chess board 3X3 the knight can has 8 possible moves. In Figure 1, can be seen a 3X3 chessboard, which has labeled every square with numbers. The possible moves that the knight can do are: (1,8), (1,6), (2,9), (2,7), (3,4), (3,8), (4,9), (4,3), (6,1), (6,7), (7,2), (7,6), (8,3), (8,1), (9,2), (9,4). We can notice that the square number 5 is not in the aforementioned list of moves. For the same reason the knight’s tour problem cannot be solved in a 4X4 chessboard. More specifically, in the 4X4 chessboard, not even an open tour is possible. In the Figure 2 a knight open tour can be seen. The 4X4 chessboard does not have a knight tour, which would force the four edges 1-2, 2-3, 3-4 and 4-1 to be in any Hamiltonian path of the whole graph. It is clear that after the step to vertex 4 whichever choice we made (left or right) we end up missing several vertices to the way to number 16. Therefore, an open knight tour in a 4X4 chessboard is

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