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 single edges and 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 = {1,3), (2,1), (2,3)}
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 W22 has 23 vertices and 44 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 vertex.
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 the same MST total length.
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 the last because it is much shorter.
2. The expression below is in postfix expression form.
Determine its numerical value.
{ -2, 5, +, 4, 1, /, 3, /, * }
3. Determine if Graph Z is bipartite. Defend your answer.
4. Define a postorder and preorder traversal of the following:
[(-3y + 2) / 4 ] - [(y - 3) * 5) ]
a. postorder:
b. preorder:
5. Determine the Minimal Spanning Tree in Graph X 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:
a. 001010101
b. 0001110000
c. 0010000011
d. 01110100011
e. What is the best compression ratio (versus ASCII 8-bit encoding) of the words in a through d above?
Defend your answer.
7. Determine the Minimum Spanning Tree in Graph Y. 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.
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 49 edges. How many vertices does it have?
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.
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 (no further connection).
Connect vertex f (no further connection).
Connect vertex i to h.
Connect vertex j (no further connection).
Connect vertex k to l and m.
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