Task 6 : TREE


    • THE TASK:
  • 1. Give the definition and example for each of the following term:
    a) Euler Path
    b) Euler Circuit

    2. Determine what are the properties that differentiate between (a) and (b) in Question 1?

    3. What are the algorithm or step by step to determine (a) and (b) in Question 1?

    4. Give the definition and example for each of the following term:
    a) Hamilton Path
    b) Hamilton Circuit

    5. Determine what are the properties that differentiate between (a) and (b) in Question 4?

    6. What are the algorithm or step by step to determine (a) and (b) in Question 4?


    1. a)
    • EULAR PATH:
    • Eulerian path is a trail in a graph which visits every edge exactly once.





b)
    • EULAR CIRCUIT
      An Euler circuit is a circuit that uses every edge of a graph exactly once.
    • I An Euler circuit starts and ends at the same vertex.





2.
    • EULAR PATH:
    • Starting at initial vertex and ending at other vertex.
    • Some degree vertex are odd.
    • Odd degree vertex => 2.
    • EULAR CIRCUIT: 
    • Starting with initial vertex and ending at initial vertex.
    • All degree vertex are even.


3.
    • Euler Path :
      • Pick any vertex to start 
      • From that vertex pick an edge to traverse 
      • Darken that edge, as a reminder that you can't  traverse it again
      • Travel that edge, coming to the next vertex 
      • Repeat 2-4 until all edges have been traversed, and you are back at the starting vertex
    • Euler Circuit 
      • Step One: Randomly moves from node to node, until stuck.  Since all nodes had even degree, the circuit must have stopped at its starting point. (It is a circuit.)
      • Step Two: If any of the arcs have not been included in our circuit, find an arc that touches our partial circuit, and add in a new circuit.
      • Each time we add a new circuit, we have included more nodes.  
      • Since there are only a finite number of nodes, eventually the whole graph is included.
4.
    • Example:
    • Hamilton circuit:                                                        

                                           a-b-e-d-c-a





    • Hamilton Path 

    • is a  path visits each vertex of a graph once and only once.
      • Not all edges is passes through because the vertex is passes olny once.





5.
    • HAMILTON CIRCUIT:
      • When this initial vertex is connected to each vertex until it  meet the end at initial vertex
    • HAMILTON PATH:
      • The initial vertex is connected each vertex until it meet the last vertex before initial vertex
6.
    •                TO DETERMINE HAMILTON CIRCUIT
      • DIRAC’S THEOREM:  If G is a simple graph with n vertices with n ≥ 3 such that the
      • degree of every vertex in G is at least n/2, then G has a Hamilton circuit
      • ORE’S THEOREM : If G is a simple graph with n vertices with 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.
    • Randomized algorithm
      • A randomized algorithm for Hamiltonian path that is fast on most graphs is the following: Start from a random vertex, and continue if there is a neighbor not visited. If there are no more unvisited neighbors, and the path formed isn't Hamiltonian, pick a neighbor uniformly at random, and rotate using that neighbor as a pivot. (That is, add an edge to that neighbor, and remove one of the existing edges from that neighbor so as not to form a loop.) Then, continue the algorithm at the new end of the path.





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