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the sequence by whitespace. For example, if the question asks for the first
ten powers of two (starting at 1), then the following answer is acceptable:

     1 2 4 8 16 32 64 128 256 512

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contains the correct answer).

Question 1

(seed = 19289)
Consider the following edge-weighted graph with 10 vertices and 17 edges:

    v-w  weight
    -----------
    A-B    13
    F-A     8
    C-B    16
    B-G     9
    B-F     7
    H-C    12
    C-D     4
    C-G     2
    D-E    17
    D-J    14
    I-D     3
    H-D     1
    E-J    15
    F-G    10
    G-H     6
    I-H     5
    I-J    11


Here is a graphical representation of the same graph:

    (A)------13-----(B)------16-----(C)------4------(D)------17-----(E)
     |              /|              /|              /|\              | 
     |             / |             / |             / | \             | 
     |            /  |            /  |            /  |  \            | 
     |           /   |           /   |           /   |   \           | 
     |          /    |          /    |          /    |    \          | 
     |         /     |         /     |         /     |     \         | 
     |        /      |        /      |        /      |      \        | 
     8       7       9       2       12      1       3       14      15
     |      /        |      /        |      /        |        \      | 
     |     /         |     /         |     /         |         \     | 
     |    /          |    /          |    /          |          \    | 
     |   /           |   /           |   /           |           \   | 
     |  /            |  /            |  /            |            \  | 
     | /             | /             | /             |             \ | 
     |/              |/              |/              |              \| 
    (F)------10-----(G)------6------(H)------5------(I)------11-----(J)



Give the sequence of edges in the MST in the order that Kruskal's algorithm discovers them.
To specify an edge, use its weight.

Question 2

(seed = 219031)
Consider the following edge-weighted graph with 10 vertices and 17 edges.

    v-w  weight
    -----------
    A-B      16
    F-A      12
    G-A       1
    C-B      11
    G-B       2
    D-C      14
    C-G       9
    C-H       7
    C-I       3
    I-D      13
    J-D       5
    E-D       4
    E-J      10
    F-G      15
    G-H      17
    I-H       6
    I-J       8


Here is a graphical representation of the same graph:

    (A)------16-----(B)------11-----(C)------14-----(D)------4------(E)
     |\              |              /|\              |\              | 
     | \             |             / | \             | \             | 
     |  \            |            /  |  \            |  \            | 
     |   \           |           /   |   \           |   \           | 
     |    \          |          /    |    \          |    \          | 
     |     \         |         /     |     \         |     \         | 
     |      \        |        /      |      \        |      \        | 
     12      1       2       9       7       3       13      5       10
     |        \      |      /        |        \      |        \      | 
     |         \     |     /         |         \     |         \     | 
     |          \    |    /          |          \    |          \    | 
     |           \   |   /           |           \   |           \   | 
     |            \  |  /            |            \  |            \  | 
     |             \ | /             |             \ |             \ | 
     |              \|/              |              \|              \| 
    (F)------15-----(G)------17-----(H)------6------(I)------8------(J)



Give the sequence of edges in the MST in the order that Prim's algorithm adds them to the MST,
when starting Prim's algorithm from vertex C. To specify an edge, use its weight.

Question 3

(seed = 581770)
Which of the following statements about minimum spanning trees (MSTs) are guaranteed to be true in any edge-weighted graph G? Assume that G is connected and has no parallel edge or self-loops. Do not assume the edge weights are distinct unless this is specifically stated. Check all that apply.
    
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