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MCIS 502
Mathematics in Computing
Exam 4
Due: Aug. 12, 2013, 11:55 PM <-- NOTE!!
Cutoff: Aug 12, 2013, 11:55 PM <-- NOTE!!!
The following questions are based on Chapters 10 and 11 of the
Rosen text.
Do your own work. Don't plagiarize!
Submit the completed exam to Blackboard, Assignments, Exam 4.
DO NOT INCLUDE PROOFS OR COMMENTS FOR YOUR WORK IN THIS SECTION.
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
PART A
Each question is worth 2 points. Choose True of False as the most appropriate answer.
1. The degree(v) of a pendant vertex may be either one or zero.
T or F
2. A tree is any connected, undirected graph with an odd number
of vertices.
T or F
3. A simple graph is an undirected graph with multiple edges but
no loops.
T or F
4. A multigraph is an directed graph with multiple edges and no
loops.
T or F
5. Consider the following directed relations on {1, 2, 3, 4} :
R = {(1,1), (2,2), (3,3), (4,4)}
T = {2,1), (3,1), (3,2)}
R is reflexive and T is transitive
T or F

6. Set A is divided into several disjoint partitions. The
UNION of these partitions is the original set.
T or F
7. A W16 has 17 vertices and 32 edges.
T or F
8. The root of any tree must be at either level 1 (one) or level
0 (zero).
T or F
9. A leaf is a vertex with no children.
T or F
10. A weighted graph has a value assigned to each edge.
T or F
11. The minimum spanning tree of a weighted graph is a graph
That is drawn with the length of each edge roughly
proportional to the value assigned to each edge.
T or F
12. Siblings must have the same parent but not necessarily the
same level.
T or F
13. Since Prim's and Kruskal's algorithms generate the minimum
spanning tree of a given weighted graph, each algorithm
would always provide identical MST solutions.
T or F
14. Bipartite graph, Kn,m, has (n+m) vertices and a maximum of
(n*m) edges.
T or F
PART B
Each question is worth 6 points unless noted otherwise. Provide your interim solutions to all problems.
Partial credit will be given for incorrect solutions if the process was shown and I can identify the error. Problems with incorrect answers and no intermediate work will be graded as zero.

Problems that require a particular approach - such as Prim and Kruskal - need a reasonable amount of intermediate work to identify and verify the solution process.
1. Form a binary search tree from the words of the following
sentence using alphabetical order and inserting words as they
appear in the sentence:
This test is easier than before because it is much shorter.
2. The expression below is in postfix expression form.
Determine its numerical value.
{ 3, -7, +, 6, 3, /, 2, *, / }
3. Determine if Graph Z is bipartite. Defend your answer.
4. Define a postorder and preorder traversal of the following:
(3 points each)
[(-3 – 2y) / 5 ] + [(-y + 3) * 6) ]
a. postorder:
b. preorder:
5. Determine the Minimal Spanning Tree in Graph Y using
Kruskal’s Algorithm. All edges must be labeled from lower to
higher named vertices, e.g., from "c" to "d" but not from "d"
to "c".
6. Given the coding scheme:
a:001, b:0001, e:1, r:0000, s: 0100, t:011, x:01010
Find the words represented by: (1 point each)
a. 0001110000
b. 001010101
c. 01110100011
d. 0010000011
e. What is the best compression ratio (versus ASCII 8-bit
encoding) of the words in a through d above? (2 points).
Defend your answer.

7. Determine the Minimum Spanning Tree in Graph X. Use Prim's
Algorithm in which all edges must be labeled from lower to
higher named vertices, e.g., from "c" to "d" but not from "d"
to "c"
8. Construct a postorder, inorder and preorder transversal of
Tree T.(2 points each)
a. postorder:
b. inorder:
c: preorder:
9. Are Graphs G and H isomorphic? Defend your answer.
10. Suppose that a full 37-ary tree has 4 internal vertices.
How many leaves does it have? Defend your answer.
11. What is the shortest path in Graph S between "a" and "z".
Use Dijkstra's algorithm.
a. the shortest path is:
b. the shortest distance between "a" and "z" is:
12. A tree has 52 edges. How many vertices does it have?
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
EXTRA CREDIT - OPTIONAL
DO ONE of the following:
A.
(3 points or no points)
Use a greedy algorithm to determine the shortest path in
Graph S. The algorithm starts at vertex “a” and ends at vertex “z” always selecting the shortest edge. The selection must be in ascending lexicographic order, i.e., m to n - not n to m. See discussion on pages 195, 232, and
798.

B.
(3 points or no points)
Develop the Basis Step of the algorithm to determine the
number of terms (cardinality) of the union of n mutually
intersecting sets. Show your work.
For example, the cardinality of the union of three
mutually intersecting sets is
C(3,1) + C(3,2) + C(3,3) = 3+3+1 = 7.
C.
(1 point or no points)
Is the solution using Prim ’s Algorithm in Question B.5 the
same topology and length as the required Kruskal solution?
Defend your answer.
GRAPH INFORMATION
Graph G
Initially draw a hexagon with vertices a-b-d-f-e-c-a.
Connect vertices a to f; b to c; d to e.
b d a f
c e
Graph H
Initially draw a hexagon with vertices u-v-w-x-y-z-u.
Connect vertices u to x; v to y; w to z.
There is no connection in the center.
u
z v
y w
x

Graph S
Initially draw a hexagon with vertices a-b-d-z-e-c-a.
Connect vertices b to c; b to e; c to d; d to e.
Edge values are:
a-b = 3; a-c = 4;
b-c = 1; b-d = 5; b-e = 5
c-d = 2; c-e = 4;
d-e = 2; d-z = 5; e-z = 3.
b d
a z
c e
Tree T
Construct a Tree with
vertex a at level 0;
vertices b, c and d at level 1;
vertices e, f, i, j, and k at level 2;
vertices g, h, l and m at level 3.
Connect vertex a to b, a to c, and a to d.
Connect vertex b to e and f.
Connect vertex c (no further connection).
Connect vertex d to i, j and k.
Connect vertex e to g.
Connect vertex f (no further connection).
Connect vertex i to h.
Connect vertex j to l and m.
Connect vertex k (no further connection).
Connect vertex g, h, l and m (no further connection).
a
b c d
e f i j k
g h l m

Graph X
Initially draw a rectangle with vertices a-c-e-z-d-b-a.
Connect vertices a to d; c to d; d to e.
Edge values are:
a-b = 1; a-c = 4; a-d =3;
b-d = 3; c-d = 2; c-e = 2;
d-e = 2; d-z = 2; e-z = 2.
a c e
b d z
Graph Y
Draw a hexagon with vertices a-b-d-z-e-c-a.
Connect vertices b to c; b to z; d to e.
Edge values are:
a-b = 3; a-c = 3;
b-c = 2; b-d = 5; b-z = 4;
c-e = 5;
d-e = 1; d-z = 7; e-z = 3.
b d
a z
c e
Graph Z
Graph Z is a five-pointed figure.
Connect a to b, a to c and a to e.
Connect b to d.
Connect c to d.
Connect d to e.
b c
a d
e

Good Luck!
END.
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